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TD-R/Ma feuille d'exercice - Bibm@th.net.html
Félix MARQUET 0eb717651e feat: add exercise sheets for Bernoulli, integrals, binomial distribution, and cardinality 
- Added exercise sheets for Bernoulli experiments including probability of success, failure, and simulation of multiple experiments.
- Added exercise sheets for calculating integrals using different methods such as rectangles, Simpson's rule, and the integrate function.
- Added exercise sheets for binomial distribution including probability calculations and cumulative probability.
- Added exercise sheets for calculating the cardinality of sets including unique elements and empty sets.
2024-12-20 08:26:12 +01:00

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Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Triangles<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1199"> [Signaler une erreur]</a><a id="exo1" href="javascript:ajouteexo(1199,1,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1199&#39;,&#39;imageenonce1199&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1199"></a><div class="inner" id="innerenonce1199">On trace dans un plan <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4" class="mjx-mrow"><span id="MJXc-Node-5" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-6" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≥</span></span><span id="MJXc-Node-7" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-2">n\geq 3</script> droites en position générale (c'est-à-dire que deux droites ne sont jamais parallèles, et 3 droites ne sont jamais concourantes). Combien de triangles a-t-on ainsi tracé?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1199&#39;,&#39;imageindication1199&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1199"></a><div class="inner" id="innerindication1199" style="display:none;">Comment définir un triangle?</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1199&#39;,&#39;imagecorrige1199&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas(1).png" id="imagecorrige1199"></a><div class="inner" id="innercorrige1199" style="">Un triangle est déterminé par 3 droites (ses côtés). Il y a autant de triangles que de possibilités de choisir 3 droites parmi <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-3-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-9" class="mjx-mrow"><span id="MJXc-Node-10" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-3">n</script>, c'est-à-dire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-4-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-11" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-12" class="mjx-mrow"><span id="MJXc-Node-13" class="mjx-mrow"><span id="MJXc-Node-14" class="mjx-TeXmathchoice"><span id="MJXc-Node-15" class="mjx-texatom"><span id="MJXc-Node-16" class="mjx-mrow"><span id="MJXc-Node-17" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-18" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.424em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.6em; top: -1.095em;"><span id="MJXc-Node-19" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.6em; bottom: -0.535em;"><span id="MJXc-Node-20" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.152em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-21" class="mjx-TeXmathchoice"><span id="MJXc-Node-22" class="mjx-texatom"><span id="MJXc-Node-23" class="mjx-mrow"><span id="MJXc-Node-24" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mi>n</mi><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-4">\binom n3</script>.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>2</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Podium!<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1201"> [Signaler une erreur]</a><a id="exo2" href="javascript:ajouteexo(1201,2,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1201&#39;,&#39;imageenonce1201&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1201"></a><div class="inner" id="innerenonce1201">Une course oppose 20 concurrents, dont Émile.
<ol class="enumeratechiffre">
<li> Combien y-a-t-il de podiums possibles?
</li><li> Combien y-a-t-il de podiums possibles où Émile est premier?
</li><li> Combien y-a-t-il de podiums possibles dont Émile fait partie?
</li><li> On souhaite récompenser les 3 premiers en leur offrant un prix identique à chacun. Combien y-a-t-il de distributions de récompenses possibles?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1201&#39;,&#39;imageindication1201&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1201"></a><div class="inner" id="innerindication1201" style="display:none;"></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1201&#39;,&#39;imagecorrige1201&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1201"></a><div class="inner" id="innercorrige1201" style="display:none;"><ol class="enumeratechiffre">
<li> Pour le premier, on a 20 choix possibles, pour le second 19, pour le troisième 18. Le nombre de podiums possibles est donc
égal à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-5-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;19&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;18&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;6840&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-25" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-26" class="mjx-mrow"><span id="MJXc-Node-27" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">20</span></span><span id="MJXc-Node-28" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-29" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">19</span></span><span id="MJXc-Node-30" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-31" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">18</span></span><span id="MJXc-Node-32" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-33" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">6840</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo>×</mo><mn>19</mn><mo>×</mo><mn>18</mn><mo>=</mo><mn>6840</mn></math></span></span><script type="math/tex" id="MathJax-Element-5">20\times 19\times 18=6840</script>.
</li><li> Le premier concurrent est Emile. Pour les autres places, il y a 19 puis 18 choix possibles; Le nombre de podiums ainsi constitués est de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;19&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;18&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-34" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-35" class="mjx-mrow"><span id="MJXc-Node-36" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">19</span></span><span id="MJXc-Node-37" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-38" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">18</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>19</mn><mo>×</mo><mn>18</mn></math></span></span><script type="math/tex" id="MathJax-Element-6">19\times 18</script>.
</li><li> Il y a trois choix possibles pour la place d'Emile. Une fois ce choix fixé, il y a 19 choix possibles pour la première des deux autres places, puis 18 choix possibles pour la seconde des deux autres places. Le nombre de podiums vérifiant ces conditions est donc de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-7-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;19&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;18&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-39" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-40" class="mjx-mrow"><span id="MJXc-Node-41" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-42" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-43" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">19</span></span><span id="MJXc-Node-44" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-45" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">18</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>19</mn><mo>×</mo><mn>18</mn></math></span></span><script type="math/tex" id="MathJax-Element-7">3\times 19\times 18</script>.
</li><li> L'ordre n'est plus important, et on cherche
le nombre de choix de 3 concurrents parmi <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-8-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-46" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-47" class="mjx-mrow"><span id="MJXc-Node-48" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">20</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math></span></span><script type="math/tex" id="MathJax-Element-8">20</script>, c'est-à-dire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-9-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1140&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-49" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-50" class="mjx-mrow"><span id="MJXc-Node-51" class="mjx-mrow"><span id="MJXc-Node-52" class="mjx-TeXmathchoice"><span id="MJXc-Node-53" class="mjx-texatom"><span id="MJXc-Node-54" class="mjx-mrow"><span id="MJXc-Node-55" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-56" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-57" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">20</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-58" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-59" class="mjx-TeXmathchoice"><span id="MJXc-Node-60" class="mjx-texatom"><span id="MJXc-Node-61" class="mjx-mrow"><span id="MJXc-Node-62" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-63" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-64" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">1140</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>20</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>1140</mn></math></span></span><script type="math/tex" id="MathJax-Element-9">\binom {20}3=1140</script>.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>3</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Les boulangeries<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2362"> [Signaler une erreur]</a><a id="exo3" href="javascript:ajouteexo(2362,3,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2362&#39;,&#39;imageenonce2362&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2362"></a><div class="inner" id="innerenonce2362">Dans une ville, il y a quatre boulangeries qui ferment un jour par semaine.
<ol class="enumeratechiffre">
<li> Déterminer le nombre de façons d'attribuer un jour de fermeture hebdomadaire à chaque boulangerie.
</li><li> Déterminer le nombre de façons d'attribuer un jour de fermeture hebdomadaire à chaque boulangerie si plusieurs boulangeries ne peuvent fermer le même jour.
</li><li> Déterminer le nombre de façons d'attribuer un jour de fermeture hebdomadaire à chaque boulangerie si chaque jour, il doit y avoir au moins une boulangerie ouverte.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2362&#39;,&#39;imageindication2362&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2362"></a><div class="inner" id="innerindication2362" style="display:none;"><ol class="enumeratechiffre">
<li> On dénombre une liste.
</li><li> On dénombre un arrangement.
</li><li> Procéder par différence.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2362&#39;,&#39;imagecorrige2362&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2362"></a><div class="inner" id="innercorrige2362" style="display:none;"><ol class="enumeratechiffre">
<li> Pour chaque boulangerie, il y a 7 choix possibles. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-10-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-65" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-66" class="mjx-mrow"><span id="MJXc-Node-67" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-68" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-69" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>7</mn><mn>4</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-10">7^4</script> façons d'attribuer un jour de fermeture hebdomadaire à chaque boulangerie.
</li><li> On peut procéder comme suit pour dénombrer le nombre de possibilités. La première boulangerie peut fermer n'importe quel jour de la semaine, ce qui lui laisse 7 choix. La seconde boulangerie peut fermer n'importe quel autre jour : 6 choix. La troisième ne peut pas fermer l'un des jours déjà choisi, ce qui lui laisse 5 choix, et pour la dernière, il ne reste que 4 choix. Le nombre de possibilités est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-11-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-70" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-71" class="mjx-mrow"><span id="MJXc-Node-72" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-73" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-74" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-75" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-76" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-77" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-78" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-11">7\times 6\times 5\times 4</script>.
</li><li> On va raisonner par différence, et compter plutôt le nombre de possibilités pour que toutes les boulangeries ferment le même jour : il y a 7 choix (on choisit juste le jour de fermeture commun). Le nombre de possibilités pour qu'il y ait au moins une boulangerie ouverte chaque jour est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-12-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-79" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-80" class="mjx-mrow"><span id="MJXc-Node-81" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-82" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-83" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span id="MJXc-Node-84" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-85" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>7</mn><mn>4</mn></msup><mo>−</mo><mn>7</mn></math></span></span><script type="math/tex" id="MathJax-Element-12">7^4-7</script>.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>4</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Autour d'une table<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2376"> [Signaler une erreur]</a><a id="exo4" href="javascript:ajouteexo(2376,4,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2376&#39;,&#39;imageenonce2376&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2376"></a><div class="inner" id="innerenonce2376">Dans une pièce, il y a deux tables. La première dispose de 3 chaises, numérotées de 1 à 3, la seconde dispose de 4 chaises,
numérotées de 1 à 4.
Sept personnes entrent. Combien y-a-t-il de possibilités de les distribuer autour de ces deux tables?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2376&#39;,&#39;imageindication2376&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2376"></a><div class="inner" id="innerindication2376" style="display:none;">Choisir les trois personnes qui vont sur la petite table, puis disposer les personnes sur chaque table.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2376&#39;,&#39;imagecorrige2376&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2376"></a><div class="inner" id="innercorrige2376" style="display:none;">On commence par choisir les personnes qui vont s'installer autour de la première table. Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-13-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-86" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-87" class="mjx-mrow"><span id="MJXc-Node-88" class="mjx-mrow"><span id="MJXc-Node-89" class="mjx-TeXmathchoice"><span id="MJXc-Node-90" class="mjx-texatom"><span id="MJXc-Node-91" class="mjx-mrow"><span id="MJXc-Node-92" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-93" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.329em;"><span id="MJXc-Node-94" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-95" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.317em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-96" class="mjx-TeXmathchoice"><span id="MJXc-Node-97" class="mjx-texatom"><span id="MJXc-Node-98" class="mjx-mrow"><span id="MJXc-Node-99" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>7</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-13">\binom 73</script> possibilités.
Ensuite, les 3 personnes qui sont autour de la première table peuvent choisir librement leur place. Il y a 3! choix
(autant que de permutations des 3 chaises). De même, il y 4! choix pour les personnes qui s'installent autour de la deuxième table.
Le nombre total de possibilités est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-14-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-100" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-101" class="mjx-mrow"><span id="MJXc-Node-102" class="mjx-mrow"><span id="MJXc-Node-103" class="mjx-TeXmathchoice"><span id="MJXc-Node-104" class="mjx-texatom"><span id="MJXc-Node-105" class="mjx-mrow"><span id="MJXc-Node-106" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-107" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.329em;"><span id="MJXc-Node-108" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-109" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.317em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-110" class="mjx-TeXmathchoice"><span id="MJXc-Node-111" class="mjx-texatom"><span id="MJXc-Node-112" class="mjx-mrow"><span id="MJXc-Node-113" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-114" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-115" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-116" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-117" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-118" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-119" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-120" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-121" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-122" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>7</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mn>3</mn><mo>!</mo><mo>×</mo><mn>4</mn><mo>!</mo><mo>=</mo><mn>7</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-14">\binom 73\times 3!\times 4!=7!</script>.
<br>
Le fait de trouver <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-15-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-123" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-124" class="mjx-mrow"><span id="MJXc-Node-125" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-126" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-15">7!</script> montre que le dénombrement que nous avons fait, qui suit les données de l'énoncé, peut être simplifié. En effet, le fait d'imposer deux tables ne change en réalité rien au problème : on doit placer 7 personnes sur 7 chaises, et il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-16-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-127" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-128" class="mjx-mrow"><span id="MJXc-Node-129" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-130" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-16">7!</script> façons différentes de le faire. </div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>5</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Addition, multiplication et combinaison<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=3196"> [Signaler une erreur]</a><a id="exo5" href="javascript:ajouteexo(3196,5,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce3196&#39;,&#39;imageenonce3196&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce3196"></a><div class="inner" id="innerenonce3196">Une entreprise comporte 18 employés, dont 8 femmes. Pour un sondage, on choisit 3 personnes au hasard. Quel est le nombre d'échantillons comportant au moins 2 hommes?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication3196&#39;,&#39;imageindication3196&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication3196"></a><div class="inner" id="innerindication3196" style="display:none;">Compter le nombre d'échantillons comportant exactement 3 hommes et le nombre d'échantillons comportant exactement 2 hommes.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige3196&#39;,&#39;imagecorrige3196&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige3196"></a><div class="inner" id="innercorrige3196" style="display:none;">On commence par compter le nombre d'échantillons comportant exactement 3 hommes. On choisit 3 hommes parmi 10. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-17-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;120&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-131" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-132" class="mjx-mrow"><span id="MJXc-Node-133" class="mjx-mrow"><span id="MJXc-Node-134" class="mjx-TeXmathchoice"><span id="MJXc-Node-135" class="mjx-texatom"><span id="MJXc-Node-136" class="mjx-mrow"><span id="MJXc-Node-137" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-138" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-139" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-140" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-141" class="mjx-TeXmathchoice"><span id="MJXc-Node-142" class="mjx-texatom"><span id="MJXc-Node-143" class="mjx-mrow"><span id="MJXc-Node-144" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-145" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-146" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">120</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>10</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>120</mn></math></span></span><script type="math/tex" id="MathJax-Element-17">\binom{10}3=120</script> tels échantillons. On compte ensuite le nombre d'échantillons comportant exactement 2 hommes. On choisit 2 hommes parmi 10, puis une femme parmi 8. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-18-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;360&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-147" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-148" class="mjx-mrow"><span id="MJXc-Node-149" class="mjx-mrow"><span id="MJXc-Node-150" class="mjx-TeXmathchoice"><span id="MJXc-Node-151" class="mjx-texatom"><span id="MJXc-Node-152" class="mjx-mrow"><span id="MJXc-Node-153" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-154" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-155" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.513em;"><span id="MJXc-Node-156" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-157" class="mjx-TeXmathchoice"><span id="MJXc-Node-158" class="mjx-texatom"><span id="MJXc-Node-159" class="mjx-mrow"><span id="MJXc-Node-160" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-161" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-162" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-163" class="mjx-TeXmathchoice"><span id="MJXc-Node-164" class="mjx-texatom"><span id="MJXc-Node-165" class="mjx-mrow"><span id="MJXc-Node-166" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-167" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-168" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-169" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-170" class="mjx-TeXmathchoice"><span id="MJXc-Node-171" class="mjx-texatom"><span id="MJXc-Node-172" class="mjx-mrow"><span id="MJXc-Node-173" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-174" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-175" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">360</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>10</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>1</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>360</mn></math></span></span><script type="math/tex" id="MathJax-Element-18">\binom{10}2\times \binom 81=360</script> tels échantillons. Finalement, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-19-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;120&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;360&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;480&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-176" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-177" class="mjx-mrow"><span id="MJXc-Node-178" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">120</span></span><span id="MJXc-Node-179" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-180" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">360</span></span><span id="MJXc-Node-181" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-182" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">480</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>120</mn><mo>+</mo><mn>360</mn><mo>=</mo><mn>480</mn></math></span></span><script type="math/tex" id="MathJax-Element-19">120+360=480</script>
échantillons comportant au moins deux hommes.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>6</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Le cadenas<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2327"> [Signaler une erreur]</a><a id="exo6" href="javascript:ajouteexo(2327,6,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2327&#39;,&#39;imageenonce2327&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2327"></a><div class="inner" id="innerenonce2327">Un cadenas possède un code à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-20-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-183" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-184" class="mjx-mrow"><span id="MJXc-Node-185" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-20">3</script> chiffres, chacun des chiffres pouvant être un chiffre de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-21-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-186" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-187" class="mjx-mrow"><span id="MJXc-Node-188" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-21">1</script> à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-22-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-189" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-190" class="mjx-mrow"><span id="MJXc-Node-191" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">9</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></span></span><script type="math/tex" id="MathJax-Element-22">9</script>.
<ol class="enumeratechiffre">
<li>
<ol class="enumeratechiffre">
<li> Combien y-a-t-il de codes possibles?
</li><li> Combien y-a-t-il de codes se terminant par un chiffre pair?
</li><li> Combien y-a-t-il de codes contenant au moins un chiffre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-23-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-192" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-193" class="mjx-mrow"><span id="MJXc-Node-194" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-23">4</script>?
</li><li> Combien y-a-t-il de codes contenant exactement un chiffre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-24-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-195" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-196" class="mjx-mrow"><span id="MJXc-Node-197" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-24">4</script>?
</li></ol>
</li><li> Dans les questions suivantes, on souhaite que le code comporte obligatoirement trois chiffres distincts.
<ol class="enumeratechiffre">
<li> Combien y-a-t-il de codes possibles?
</li><li> Combien y-a-t-il de codes se terminant par un chiffre impair?
</li><li> Combien y-a-t-il de codes comprenant le chiffre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-25-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-198" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-199" class="mjx-mrow"><span id="MJXc-Node-200" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-25">6</script>?
</li></ol>
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2327&#39;,&#39;imageindication2327&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2327"></a><div class="inner" id="innerindication2327" style="display:none;"></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2327&#39;,&#39;imagecorrige2327&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2327"></a><div class="inner" id="innercorrige2327" style="display:none;"><ol class="enumeratechiffre">
<li> <ol class="enumeratechiffre">
<li> Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-26-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;729&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-201" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-202" class="mjx-mrow"><span id="MJXc-Node-203" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-204" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-205" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-206" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-207" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-208" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-209" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-210" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-211" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-212" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-213" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">729</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>9</mn><mn>3</mn></msup><mo>=</mo><mn>9</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>9</mn><mo>=</mo><mn>729</mn></math></span></span><script type="math/tex" id="MathJax-Element-26">9^3=9\times 9\times 9=729</script> codes possibles.
</li><li> Pour chacun des deux premiers chiffres, il y a 9 choix possibles. Pour le dernier, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-27-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-214" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-215" class="mjx-mrow"><span id="MJXc-Node-216" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-27">4</script> choix possibles (on peut choisir 2,4,6,8). Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-28-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-217" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-218" class="mjx-mrow"><span id="MJXc-Node-219" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-220" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-221" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-222" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-223" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-28">9\times 9\times 4</script> tels codes.
</li><li> On va compter par différence. Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-29-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-224" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-225" class="mjx-mrow"><span id="MJXc-Node-226" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-227" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-228" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-229" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-230" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>8</mn></math></span></span><script type="math/tex" id="MathJax-Element-29">8\times 8\times 8</script> codes ne contenant pas du tout le chiffre 4. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-30-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;217&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-231" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-232" class="mjx-mrow"><span id="MJXc-Node-233" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-234" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-235" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-236" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-237" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-238" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-239" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-240" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-241" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-242" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-243" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-244" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-245" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">217</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>9</mn><mo>−</mo><mn>8</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>8</mn><mo>=</mo><mn>217</mn></math></span></span><script type="math/tex" id="MathJax-Element-30">9\times 9\times 9-8\times 8\times 8=217</script> codes comprenant au moins une fois le chiffre 4.
</li><li> Il y a 3 choix pour la place dans le nombre où se situe le chiffre 4. Pour chacun des deux autres chiffres, il y a 8 choix possibles. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-31-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-246" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-247" class="mjx-mrow"><span id="MJXc-Node-248" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-249" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-250" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-251" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-252" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>8</mn></math></span></span><script type="math/tex" id="MathJax-Element-31">3\times 8\times 8</script> tels codes.
</li></ol>
</li><li> <ol class="enumeratechiffre">
<li> On cherche cette fois un arrangement de 3 chiffres parmi 9. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-32-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-253" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-254" class="mjx-mrow"><span id="MJXc-Node-255" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-256" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-257" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-258" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-259" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>7</mn></math></span></span><script type="math/tex" id="MathJax-Element-32">9\times 8\times 7</script> choix possibles.
</li><li> Il y a cinq choix pour le dernier chiffre. Celui-ci choisi, il reste huit choix pour le premier chiffre, puis sept pour le deuxième. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-33-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-260" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-261" class="mjx-mrow"><span id="MJXc-Node-262" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-263" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-264" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-265" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-266" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><mn>7</mn><mo>×</mo><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-33">8\times 7\times 5</script> tels codes.
</li><li> Il y a 3 choix pour la place dans le nombre où on place le chiffre 6. Pour les autres chiffres, il y a d'abord 8 choix, puis 7 choix possibles (rappelons que tous les chiffres sont distincts). Le nombre de tels codes est donc de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-34-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-267" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-268" class="mjx-mrow"><span id="MJXc-Node-269" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-270" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-271" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-272" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-273" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><mn>7</mn><mo>×</mo><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-34">8\times 7\times 3</script>.
</li></ol>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>7</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Les hélicoptères<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2900"> [Signaler une erreur]</a><a id="exo7" href="javascript:ajouteexo(2900,7,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2900&#39;,&#39;imageenonce2900&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2900"></a><div class="inner" id="innerenonce2900">On dispose de 4 hélicoptères de tourisme, de 4 pilotes et de 8 hôtesses de l'air. Combien de façons différentes y a-t-il d'attribuer les pilotes et hôtesses de l'air aux hélicoptères de manière que chaque hélicoptère ait un pilote et deux hôtesses de l'air ?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2900&#39;,&#39;imageindication2900&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2900"></a><div class="inner" id="innerindication2900" style="display:none;">Choisir d'abord les hôtesses et le pilote du premier hélicoptère, puis les hôtesses et le pilote du deuxième hélicoptère, etc...</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2900&#39;,&#39;imagecorrige2900&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2900"></a><div class="inner" id="innercorrige2900" style="display:none;">On fait les choix successifs suivants :
<ul class="rien">
<li> on choisit 2 hôtesses parmi 8 et un pilote parmi 4 pour le premier hélicoptère. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-35-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;112&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-274" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-275" class="mjx-mrow"><span id="MJXc-Node-276" class="mjx-mrow"><span id="MJXc-Node-277" class="mjx-TeXmathchoice"><span id="MJXc-Node-278" class="mjx-texatom"><span id="MJXc-Node-279" class="mjx-mrow"><span id="MJXc-Node-280" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-281" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-282" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-283" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-284" class="mjx-TeXmathchoice"><span id="MJXc-Node-285" class="mjx-texatom"><span id="MJXc-Node-286" class="mjx-mrow"><span id="MJXc-Node-287" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-288" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-289" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-290" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-291" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span><span id="MJXc-Node-292" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-293" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-294" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-295" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">112</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mn>4</mn><mo>=</mo><mn>28</mn><mo>×</mo><mn>4</mn><mo>=</mo><mn>112</mn></math></span></span><script type="math/tex" id="MathJax-Element-35">\binom 82\times 4=28\times 4=112</script> tels choix.
</li><li> pour le deuxième hélicoptère, on choisit 2 hôtesses parmi les 6 restantes, puis un pilote parmi les 3 restants. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-36-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;45&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-296" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-297" class="mjx-mrow"><span id="MJXc-Node-298" class="mjx-mrow"><span id="MJXc-Node-299" class="mjx-TeXmathchoice"><span id="MJXc-Node-300" class="mjx-texatom"><span id="MJXc-Node-301" class="mjx-mrow"><span id="MJXc-Node-302" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-303" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-304" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-305" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-306" class="mjx-TeXmathchoice"><span id="MJXc-Node-307" class="mjx-texatom"><span id="MJXc-Node-308" class="mjx-mrow"><span id="MJXc-Node-309" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-310" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-311" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-312" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-313" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">15</span></span><span id="MJXc-Node-314" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-315" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-316" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-317" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">45</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mn>3</mn><mo>=</mo><mn>15</mn><mo>×</mo><mn>3</mn><mo>=</mo><mn>45</mn></math></span></span><script type="math/tex" id="MathJax-Element-36">\binom 62\times 3=15\times 3=45</script> tels choix.
</li><li> pour le troisième hélicoptère, on choisit 2 hôtesses parmi les 4 restantes, puis un pilote parmi les 2 restants. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-37-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-318" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-319" class="mjx-mrow"><span id="MJXc-Node-320" class="mjx-mrow"><span id="MJXc-Node-321" class="mjx-TeXmathchoice"><span id="MJXc-Node-322" class="mjx-texatom"><span id="MJXc-Node-323" class="mjx-mrow"><span id="MJXc-Node-324" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-325" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-326" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-327" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-328" class="mjx-TeXmathchoice"><span id="MJXc-Node-329" class="mjx-texatom"><span id="MJXc-Node-330" class="mjx-mrow"><span id="MJXc-Node-331" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-332" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-333" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-334" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-335" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-336" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-337" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-338" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-339" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mn>2</mn><mo>=</mo><mn>6</mn><mo>×</mo><mn>2</mn><mo>=</mo><mn>12</mn></math></span></span><script type="math/tex" id="MathJax-Element-37">\binom 42\times 2=6\times 2=12</script> tels choix.
</li><li> pour le dernier hélicoptère, on n'a plus de choix à faire : on lui affecte les deux dernières hôtesses et le dernier pilote.
</li></ul>
Le nombre total de choix est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-38-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;112&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;45&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;60480&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-340" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-341" class="mjx-mrow"><span id="MJXc-Node-342" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">112</span></span><span id="MJXc-Node-343" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-344" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">45</span></span><span id="MJXc-Node-345" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-346" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-347" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-348" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">60480</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>112</mn><mo>×</mo><mn>45</mn><mo>×</mo><mn>12</mn><mo>=</mo><mn>60480</mn></math></span></span><script type="math/tex" id="MathJax-Element-38">112\times 45\times 12=60480</script>.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>8</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Comité de joueurs<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2328"> [Signaler une erreur]</a><a id="exo8" href="javascript:ajouteexo(2328,8,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2328&#39;,&#39;imageenonce2328&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2328"></a><div class="inner" id="innerenonce2328">Fred et Émile font partie d'une équipe de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-39-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-349" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-350" class="mjx-mrow"><span id="MJXc-Node-351" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">8</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span></span><script type="math/tex" id="MathJax-Element-39">8</script> joueurs (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-40-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-352" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-353" class="mjx-mrow"><span id="MJXc-Node-354" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-40">6</script> garçons et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-41-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-355" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-356" class="mjx-mrow"><span id="MJXc-Node-357" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-41">2</script> filles). On décide de fabriquer un comité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-42-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-358" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-359" class="mjx-mrow"><span id="MJXc-Node-360" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-42">3</script> joueurs.
<ol class="enumeratechiffre">
<li> Combien y-a-t-il de comités possibles?
</li><li> Combien y-a-t-il de comités contenant exactement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-43-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-361" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-362" class="mjx-mrow"><span id="MJXc-Node-363" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-43">2</script> garçons et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-44-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-364" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-365" class="mjx-mrow"><span id="MJXc-Node-366" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-44">1</script> fille?
</li><li> Combien y-a-t-il de comités contenant au moins deux garçons?
</li><li> On veut que Fred et Émile soient ensemble dans le comité. Combien y-a-t-il de comités possibles?
</li><li> On ne veut pas que Fred et Émile soient ensemble dans le comité. Combien y-a-t-il de comités possibles?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2328&#39;,&#39;imageindication2328&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2328"></a><div class="inner" id="innerindication2328" style="display:none;"></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2328&#39;,&#39;imagecorrige2328&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2328"></a><div class="inner" id="innercorrige2328" style="display:none;"><ol class="enumeratechiffre">
<li> Il s'agit de choisir trois joueurs parmi 8. Le nombre de comités possibles est donc de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-45-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;56&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-367" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-368" class="mjx-mrow"><span id="MJXc-Node-369" class="mjx-mrow"><span id="MJXc-Node-370" class="mjx-TeXmathchoice"><span id="MJXc-Node-371" class="mjx-texatom"><span id="MJXc-Node-372" class="mjx-mrow"><span id="MJXc-Node-373" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-374" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-375" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-376" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-377" class="mjx-TeXmathchoice"><span id="MJXc-Node-378" class="mjx-texatom"><span id="MJXc-Node-379" class="mjx-mrow"><span id="MJXc-Node-380" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-381" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-382" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">56</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>56</mn></math></span></span><script type="math/tex" id="MathJax-Element-45">\binom 83=56</script>.
</li><li> Il s'agit de choisir deux garçons parmi 6, puis une fille parmi 2. Le nombre de choix possibles est donc de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-46-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-383" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-384" class="mjx-mrow"><span id="MJXc-Node-385" class="mjx-mrow"><span id="MJXc-Node-386" class="mjx-TeXmathchoice"><span id="MJXc-Node-387" class="mjx-texatom"><span id="MJXc-Node-388" class="mjx-mrow"><span id="MJXc-Node-389" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-390" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-391" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-392" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-393" class="mjx-TeXmathchoice"><span id="MJXc-Node-394" class="mjx-texatom"><span id="MJXc-Node-395" class="mjx-mrow"><span id="MJXc-Node-396" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-397" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-398" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-399" class="mjx-TeXmathchoice"><span id="MJXc-Node-400" class="mjx-texatom"><span id="MJXc-Node-401" class="mjx-mrow"><span id="MJXc-Node-402" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-403" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-404" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-405" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-406" class="mjx-TeXmathchoice"><span id="MJXc-Node-407" class="mjx-texatom"><span id="MJXc-Node-408" class="mjx-mrow"><span id="MJXc-Node-409" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>2</mn><mn>1</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-46">\binom 62\times\binom 21</script>.
</li><li> On compte le nombre de comités comprenant 3 garçons : il vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-47-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-410" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-411" class="mjx-mrow"><span id="MJXc-Node-412" class="mjx-mrow"><span id="MJXc-Node-413" class="mjx-TeXmathchoice"><span id="MJXc-Node-414" class="mjx-texatom"><span id="MJXc-Node-415" class="mjx-mrow"><span id="MJXc-Node-416" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-417" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-418" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-419" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-420" class="mjx-TeXmathchoice"><span id="MJXc-Node-421" class="mjx-texatom"><span id="MJXc-Node-422" class="mjx-mrow"><span id="MJXc-Node-423" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-47">\binom 63</script> (il faut choisir trois garçons parmi 6). On a déjà compté le nombre de comités comprenant exactement deux garçons. Donc le nombre de comités comprenant au moins deux garçons vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-48-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-424" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-425" class="mjx-mrow"><span id="MJXc-Node-426" class="mjx-mrow"><span id="MJXc-Node-427" class="mjx-TeXmathchoice"><span id="MJXc-Node-428" class="mjx-texatom"><span id="MJXc-Node-429" class="mjx-mrow"><span id="MJXc-Node-430" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-431" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-432" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-433" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-434" class="mjx-TeXmathchoice"><span id="MJXc-Node-435" class="mjx-texatom"><span id="MJXc-Node-436" class="mjx-mrow"><span id="MJXc-Node-437" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-438" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-439" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-440" class="mjx-TeXmathchoice"><span id="MJXc-Node-441" class="mjx-texatom"><span id="MJXc-Node-442" class="mjx-mrow"><span id="MJXc-Node-443" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-444" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-445" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-446" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-447" class="mjx-TeXmathchoice"><span id="MJXc-Node-448" class="mjx-texatom"><span id="MJXc-Node-449" class="mjx-mrow"><span id="MJXc-Node-450" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-451" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-452" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-453" class="mjx-TeXmathchoice"><span id="MJXc-Node-454" class="mjx-texatom"><span id="MJXc-Node-455" class="mjx-mrow"><span id="MJXc-Node-456" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-457" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-458" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-459" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-460" class="mjx-TeXmathchoice"><span id="MJXc-Node-461" class="mjx-texatom"><span id="MJXc-Node-462" class="mjx-mrow"><span id="MJXc-Node-463" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>2</mn><mn>1</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>+</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-48">\binom 62\times\binom 21+\binom 63</script>.
</li><li> Il ne reste qu'à choisir le dernier membre du comité : il y a 6 comités comprenant à la fois Fred et Émile.
</li><li> On compte les comités comprenant Fred, mais pas Émile, et les comités comprenant Émile, mais pas Fred. Dans le premier cas, on trouve <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-49-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-464" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-465" class="mjx-mrow"><span id="MJXc-Node-466" class="mjx-mrow"><span id="MJXc-Node-467" class="mjx-TeXmathchoice"><span id="MJXc-Node-468" class="mjx-texatom"><span id="MJXc-Node-469" class="mjx-mrow"><span id="MJXc-Node-470" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-471" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-472" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-473" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-474" class="mjx-TeXmathchoice"><span id="MJXc-Node-475" class="mjx-texatom"><span id="MJXc-Node-476" class="mjx-mrow"><span id="MJXc-Node-477" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-49">\binom 62</script> comités (il reste à choisir deux joueurs parmi 6, puisqu'on ne peut plus prendre ni Fred, ni Émile). Dans le second cas, on a aussi <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-50-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-478" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-479" class="mjx-mrow"><span id="MJXc-Node-480" class="mjx-mrow"><span id="MJXc-Node-481" class="mjx-TeXmathchoice"><span id="MJXc-Node-482" class="mjx-texatom"><span id="MJXc-Node-483" class="mjx-mrow"><span id="MJXc-Node-484" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-485" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-486" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-487" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-488" class="mjx-TeXmathchoice"><span id="MJXc-Node-489" class="mjx-texatom"><span id="MJXc-Node-490" class="mjx-mrow"><span id="MJXc-Node-491" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-50">\binom 62</script> comités. On compte enfin les comités ne comprenant ni Fred, ni Émile. Il y en a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-51-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-492" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-493" class="mjx-mrow"><span id="MJXc-Node-494" class="mjx-mrow"><span id="MJXc-Node-495" class="mjx-TeXmathchoice"><span id="MJXc-Node-496" class="mjx-texatom"><span id="MJXc-Node-497" class="mjx-mrow"><span id="MJXc-Node-498" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-499" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-500" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-501" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-502" class="mjx-TeXmathchoice"><span id="MJXc-Node-503" class="mjx-texatom"><span id="MJXc-Node-504" class="mjx-mrow"><span id="MJXc-Node-505" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-51">\binom 63</script>. Finalement, le nombre total de comités ne comprenant pas simultanément Émile et Fred est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-52-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-506" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-507" class="mjx-mrow"><span id="MJXc-Node-508" class="mjx-mrow"><span id="MJXc-Node-509" class="mjx-TeXmathchoice"><span id="MJXc-Node-510" class="mjx-texatom"><span id="MJXc-Node-511" class="mjx-mrow"><span id="MJXc-Node-512" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-513" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-514" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-515" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-516" class="mjx-TeXmathchoice"><span id="MJXc-Node-517" class="mjx-texatom"><span id="MJXc-Node-518" class="mjx-mrow"><span id="MJXc-Node-519" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-520" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-521" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-522" class="mjx-TeXmathchoice"><span id="MJXc-Node-523" class="mjx-texatom"><span id="MJXc-Node-524" class="mjx-mrow"><span id="MJXc-Node-525" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-526" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-527" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-528" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-529" class="mjx-TeXmathchoice"><span id="MJXc-Node-530" class="mjx-texatom"><span id="MJXc-Node-531" class="mjx-mrow"><span id="MJXc-Node-532" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-533" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-534" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-535" class="mjx-TeXmathchoice"><span id="MJXc-Node-536" class="mjx-texatom"><span id="MJXc-Node-537" class="mjx-mrow"><span id="MJXc-Node-538" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-539" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-540" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-541" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-542" class="mjx-TeXmathchoice"><span id="MJXc-Node-543" class="mjx-texatom"><span id="MJXc-Node-544" class="mjx-mrow"><span id="MJXc-Node-545" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-546" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-547" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">50</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>+</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>+</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>50</mn></math></span></span><script type="math/tex" id="MathJax-Element-52">\binom 62+\binom 62+\binom 63=50</script>. Plus simplement, on pouvait aussi soustraire du nombre total de comités (56, cf question 1) le nombre de comités comprenant à la fois Fred et Émile (6, cf question 4), et on retrouve bien 50 comités ne comprenant pas simultanément Fred et Émile.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>9</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Les trois mousquetaires<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2329"> [Signaler une erreur]</a><a id="exo9" href="javascript:ajouteexo(2329,9,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2329&#39;,&#39;imageenonce2329&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2329"></a><div class="inner" id="innerenonce2329">Les trois mousquetaires (donc quatre personnes avec d'Artagnan), ont mélangé leurs bottes dans le couloir de l'auberge. D'Artagnan se lève en premier et prend deux bottes au hasard.
<ol class="enumeratechiffre">
<li> Combien de possibilités s'offrent à lui?
</li><li> Combien de choix a-t-il tels que les deux bottes forment une paire (une droite et une gauche quelconques)?
</li><li> Combien de choix a-t-il tels que les deux bottes appartiennent à deux personnes différentes?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2329&#39;,&#39;imageindication2329&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2329"></a><div class="inner" id="innerindication2329" style="display:none;"><ol class="enumeratechiffre">
<li>
</li><li>
</li><li> Compter par différence en cherchant le nombre de choix pour que les deux bottes appartiennent à la même personne.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2329&#39;,&#39;imagecorrige2329&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2329"></a><div class="inner" id="innercorrige2329" style="display:none;"><ol class="enumeratechiffre">
<li> Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-53-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-548" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-549" class="mjx-mrow"><span id="MJXc-Node-550" class="mjx-mrow"><span id="MJXc-Node-551" class="mjx-TeXmathchoice"><span id="MJXc-Node-552" class="mjx-texatom"><span id="MJXc-Node-553" class="mjx-mrow"><span id="MJXc-Node-554" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-555" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-556" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-557" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-558" class="mjx-TeXmathchoice"><span id="MJXc-Node-559" class="mjx-texatom"><span id="MJXc-Node-560" class="mjx-mrow"><span id="MJXc-Node-561" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-562" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-563" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>28</mn></math></span></span><script type="math/tex" id="MathJax-Element-53">\binom 82=28</script> choix possibles (il s'agit de choisir deux éléments parmi 8).
</li><li> Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-54-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-564" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-565" class="mjx-mrow"><span id="MJXc-Node-566" class="mjx-mrow"><span id="MJXc-Node-567" class="mjx-TeXmathchoice"><span id="MJXc-Node-568" class="mjx-texatom"><span id="MJXc-Node-569" class="mjx-mrow"><span id="MJXc-Node-570" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-571" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-572" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-573" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-574" class="mjx-TeXmathchoice"><span id="MJXc-Node-575" class="mjx-texatom"><span id="MJXc-Node-576" class="mjx-mrow"><span id="MJXc-Node-577" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>1</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-54">\binom 41</script> choix possibles pour une botte gauche et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-55-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-578" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-579" class="mjx-mrow"><span id="MJXc-Node-580" class="mjx-mrow"><span id="MJXc-Node-581" class="mjx-TeXmathchoice"><span id="MJXc-Node-582" class="mjx-texatom"><span id="MJXc-Node-583" class="mjx-mrow"><span id="MJXc-Node-584" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-585" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-586" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-587" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-588" class="mjx-TeXmathchoice"><span id="MJXc-Node-589" class="mjx-texatom"><span id="MJXc-Node-590" class="mjx-mrow"><span id="MJXc-Node-591" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>1</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-55">\binom 41</script> choix possibles pour une botte droite. Le nombre total de choix formant une paire est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-56-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-592" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-593" class="mjx-mrow"><span id="MJXc-Node-594" class="mjx-mrow"><span id="MJXc-Node-595" class="mjx-TeXmathchoice"><span id="MJXc-Node-596" class="mjx-texatom"><span id="MJXc-Node-597" class="mjx-mrow"><span id="MJXc-Node-598" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-599" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-600" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-601" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-602" class="mjx-TeXmathchoice"><span id="MJXc-Node-603" class="mjx-texatom"><span id="MJXc-Node-604" class="mjx-mrow"><span id="MJXc-Node-605" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-606" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-607" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-608" class="mjx-TeXmathchoice"><span id="MJXc-Node-609" class="mjx-texatom"><span id="MJXc-Node-610" class="mjx-mrow"><span id="MJXc-Node-611" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-612" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-613" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-614" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-615" class="mjx-TeXmathchoice"><span id="MJXc-Node-616" class="mjx-texatom"><span id="MJXc-Node-617" class="mjx-mrow"><span id="MJXc-Node-618" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-619" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-620" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">16</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>1</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>1</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>16</mn></math></span></span><script type="math/tex" id="MathJax-Element-56">\binom 41\times\binom 41=16</script>.
</li><li> Il y a de nombreuses façons de répondre à cette question mais la plus simple est de raisonner par différence. Il y a en effet <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-57-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-621" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-622" class="mjx-mrow"><span id="MJXc-Node-623" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-57">4</script> choix pour que le deux bottes choisies appartiennent à la même personne. Comme il y a d'après la première question <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-58-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-624" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-625" class="mjx-mrow"><span id="MJXc-Node-626" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn></math></span></span><script type="math/tex" id="MathJax-Element-58">28</script> choix possible de deux bottes, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-59-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;24&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-627" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-628" class="mjx-mrow"><span id="MJXc-Node-629" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span><span id="MJXc-Node-630" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-631" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-632" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-633" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">24</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn><mo>−</mo><mn>4</mn><mo>=</mo><mn>24</mn></math></span></span><script type="math/tex" id="MathJax-Element-59">28-4=24</script> choix possibles de sorte que les deux bottes appartiennent à deux personnes différentes.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>10</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Chaussures dans mon armoire<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=3195"> [Signaler une erreur]</a><a id="exo10" href="javascript:ajouteexo(3195,10,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce3195&#39;,&#39;imageenonce3195&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce3195"></a><div class="inner" id="innerenonce3195">Dans mon armoire, il y a 5 paires de chaussures noires, 3 paires de chaussures marrons et 2 paires de chaussures blanches. Je peux distinguer toutes ces chaussures les unes des autres. Un matin, mal réveillé, je choisis deux chaussures au hasard.
<ol class="enumeratechiffre">
<li> Combien y-a-t-il de choix possibles ?
</li><li> Combien y-a-t-il de choix où j'obtiens deux chaussures de même couleur ?
</li><li> Combien de choix amènent un pied gauche et un pied droit ?
</li><li> Combien de choix amènent une chaussure droite et une chaussure gauche de même couleur ?
</li><li> Combien de choix amènent à deux chaussures qui ne sont pas de la même paire?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication3195&#39;,&#39;imageindication3195&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication3195"></a><div class="inner" id="innerindication3195" style="display:none;"><ol class="enumeratechiffre">
<li> On choisit deux chaussures parmi 20.
</li><li> Commencer par compter le nombre de choix qui amènent deux chaussures noires.
</li><li> Combien de choix pour choisir une chaussure pied gauche? pied droit? Faire un arbre...
</li><li> Commencer par compter le nombre de choix qui amènent une chaussure droite et une chaussure gauche qui sont toutes les deux noires.
</li><li> Dénombrer le complémentaire.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige3195&#39;,&#39;imagecorrige3195&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige3195"></a><div class="inner" id="innercorrige3195" style="display:none;"><ol class="enumeratechiffre">
<li> Les chaussures étant discernables, j'ai choisi 2 chaussures parmi les 20 de mon armoire. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-60-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;190&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-634" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-635" class="mjx-mrow"><span id="MJXc-Node-636" class="mjx-mrow"><span id="MJXc-Node-637" class="mjx-TeXmathchoice"><span id="MJXc-Node-638" class="mjx-texatom"><span id="MJXc-Node-639" class="mjx-mrow"><span id="MJXc-Node-640" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-641" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-642" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">20</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.513em;"><span id="MJXc-Node-643" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-644" class="mjx-TeXmathchoice"><span id="MJXc-Node-645" class="mjx-texatom"><span id="MJXc-Node-646" class="mjx-mrow"><span id="MJXc-Node-647" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-648" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-649" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">190</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>20</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>190</mn></math></span></span><script type="math/tex" id="MathJax-Element-60">\binom{20}2=190</script> choix possibles.
</li><li> Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-61-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-650" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-651" class="mjx-mrow"><span id="MJXc-Node-652" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math></span></span><script type="math/tex" id="MathJax-Element-61">E</script> les tirages qui m'amènent à des chaussures de même couleur, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-62-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-653" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-654" class="mjx-mrow"><span id="MJXc-Node-655" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-656" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-657" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>N</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-62">E_N</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-63-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-658" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-659" class="mjx-mrow"><span id="MJXc-Node-660" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-661" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-662" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>M</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-63">E_M</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-64-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-663" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-664" class="mjx-mrow"><span id="MJXc-Node-665" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-666" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-667" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>B</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-64">E_B</script> les tirages qui m'amènent à deux chaussures noires (resp. marron, resp. blanches). Alors <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-65-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-668" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-669" class="mjx-mrow"><span id="MJXc-Node-670" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-671" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-672" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-673" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-674" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-675" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-676" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-677" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-678" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span id="MJXc-Node-679" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-680" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-681" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-682" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><msub><mi>E</mi><mi>N</mi></msub><mo>∪</mo><msub><mi>E</mi><mi>M</mi></msub><mo>∪</mo><msub><mi>E</mi><mi>B</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-65">E=E_N\cup E_M\cup E_B</script> et cette réunion est disjointe. On a donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-66-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;card&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;card&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;card&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;card&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-683" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-684" class="mjx-mrow"><span id="MJXc-Node-685" class="mjx-texatom"><span id="MJXc-Node-686" class="mjx-mrow"><span id="MJXc-Node-687" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span></span></span><span id="MJXc-Node-688" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-689" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-690" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-691" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-692" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-693" class="mjx-mrow"><span id="MJXc-Node-694" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span></span></span><span id="MJXc-Node-695" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-696" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-697" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-698" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-699" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-700" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-701" class="mjx-texatom MJXc-space2"><span id="MJXc-Node-702" class="mjx-mrow"><span id="MJXc-Node-703" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span></span></span><span id="MJXc-Node-704" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-705" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-706" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-707" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span id="MJXc-Node-708" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-709" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-710" class="mjx-texatom MJXc-space2"><span id="MJXc-Node-711" class="mjx-mrow"><span id="MJXc-Node-712" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span></span></span><span id="MJXc-Node-713" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-714" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-715" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-716" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span id="MJXc-Node-717" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-718" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mtext>card</mtext></mrow><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mtext>card</mtext></mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mi>N</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mtext>card</mtext></mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mi>M</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mtext>card</mtext></mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-66">\textrm{card}(E)=\textrm{card}(E_N)+\textrm{card}(E_M)+\textrm{card}(E_B).</script>
Or, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-67-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;card&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-719" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-720" class="mjx-mrow"><span id="MJXc-Node-721" class="mjx-texatom"><span id="MJXc-Node-722" class="mjx-mrow"><span id="MJXc-Node-723" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span></span></span><span id="MJXc-Node-724" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-725" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.026em;"><span id="MJXc-Node-726" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-727" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-728" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-729" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-730" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-731" class="mjx-TeXmathchoice"><span id="MJXc-Node-732" class="mjx-texatom"><span id="MJXc-Node-733" class="mjx-mrow"><span id="MJXc-Node-734" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-735" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-736" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.513em;"><span id="MJXc-Node-737" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-738" class="mjx-TeXmathchoice"><span id="MJXc-Node-739" class="mjx-texatom"><span id="MJXc-Node-740" class="mjx-mrow"><span id="MJXc-Node-741" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext>card</mtext></mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mi>N</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>10</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-67">\textrm{card}(E_N)=\binom{10}2</script>, puisque je peux choisir 2 chaussures parmi 10. Finalement,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-68-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;card&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;66.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-742" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-743" class="mjx-mrow"><span id="MJXc-Node-744" class="mjx-texatom"><span id="MJXc-Node-745" class="mjx-mrow"><span id="MJXc-Node-746" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span></span></span><span id="MJXc-Node-747" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-748" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-749" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-750" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-751" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-752" class="mjx-TeXmathchoice"><span id="MJXc-Node-753" class="mjx-texatom"><span id="MJXc-Node-754" class="mjx-mrow"><span id="MJXc-Node-755" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-756" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1em;"><span class="mjx-numerator" style="width: 1em; top: -1.368em;"><span id="MJXc-Node-757" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-denominator" style="width: 1em; bottom: -0.711em;"><span id="MJXc-Node-758" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-759" class="mjx-TeXmathchoice"><span id="MJXc-Node-760" class="mjx-texatom"><span id="MJXc-Node-761" class="mjx-mrow"><span id="MJXc-Node-762" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-763" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-764" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-765" class="mjx-TeXmathchoice"><span id="MJXc-Node-766" class="mjx-texatom"><span id="MJXc-Node-767" class="mjx-mrow"><span id="MJXc-Node-768" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-769" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.5em;"><span class="mjx-numerator" style="width: 0.5em; top: -1.368em;"><span id="MJXc-Node-770" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="width: 0.5em; bottom: -0.711em;"><span id="MJXc-Node-771" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-772" class="mjx-TeXmathchoice"><span id="MJXc-Node-773" class="mjx-texatom"><span id="MJXc-Node-774" class="mjx-mrow"><span id="MJXc-Node-775" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-776" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-777" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-778" class="mjx-TeXmathchoice"><span id="MJXc-Node-779" class="mjx-texatom"><span id="MJXc-Node-780" class="mjx-mrow"><span id="MJXc-Node-781" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-782" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.5em;"><span class="mjx-numerator" style="width: 0.5em; top: -1.379em;"><span id="MJXc-Node-783" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 0.5em; bottom: -0.711em;"><span id="MJXc-Node-784" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-785" class="mjx-TeXmathchoice"><span id="MJXc-Node-786" class="mjx-texatom"><span id="MJXc-Node-787" class="mjx-mrow"><span id="MJXc-Node-788" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-789" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-790" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">66.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mtext>card</mtext></mrow><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>10</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>+</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>+</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>=</mo><mn>66.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-68">\textrm{card}(E)=\binom{10}2+\binom{6}2+\binom 42=66.</script>
</li><li> Il y a 10 choix possibles pour la chaussure droite. Une fois ce choix réalisé, il y a 10 choix possibles pour la chaussure gauche. Au total, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-69-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-791" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-792" class="mjx-mrow"><span id="MJXc-Node-793" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-794" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-795" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-796" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-797" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">100</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>×</mo><mn>10</mn><mo>=</mo><mn>100</mn></math></span></span><script type="math/tex" id="MathJax-Element-69">10\times 10=100</script> choix possibles.
</li><li> Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-70-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-798" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-799" class="mjx-mrow"><span id="MJXc-Node-800" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-70">A</script> les tirages qui amènent une chaussure droite et une chaussure gauche de la même couleur. Comme précédemment, on va décomposer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-71-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-801" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-802" class="mjx-mrow"><span id="MJXc-Node-803" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-71">A</script> en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-72-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-804" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-805" class="mjx-mrow"><span id="MJXc-Node-806" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-807" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-808" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-809" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-810" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-811" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-812" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span id="MJXc-Node-813" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-814" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-815" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-816" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>N</mi></msub><mo>∪</mo><msub><mi>A</mi><mi>M</mi></msub><mo>∪</mo><msub><mi>A</mi><mi>B</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-72">A_N\cup A_M\cup A_B</script>, où <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-73-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-817" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-818" class="mjx-mrow"><span id="MJXc-Node-819" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-820" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-821" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>N</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-73">A_N</script> est l'ensemble des tirages amenant une chaussure droite noire et une chaussure gauche noire. On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-74-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;card&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-822" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-823" class="mjx-mrow"><span id="MJXc-Node-824" class="mjx-texatom"><span id="MJXc-Node-825" class="mjx-mrow"><span id="MJXc-Node-826" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span></span></span><span id="MJXc-Node-827" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-828" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-829" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-830" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-831" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-832" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-833" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-834" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-835" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext>card</mtext></mrow><mo stretchy="false">(</mo><msub><mi>A</mi><mi>N</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mn>5</mn><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-74">\textrm{card}(A_N)=5^2</script> (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-75-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-836" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-837" class="mjx-mrow"><span id="MJXc-Node-838" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-75">5</script> choix pour la chaussure droite, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-76-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-839" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-840" class="mjx-mrow"><span id="MJXc-Node-841" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-76">5</script> choix pour la chaussure gauche) et finalement
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-77-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;card&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;38.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-842" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-843" class="mjx-mrow"><span id="MJXc-Node-844" class="mjx-texatom"><span id="MJXc-Node-845" class="mjx-mrow"><span id="MJXc-Node-846" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span></span></span><span id="MJXc-Node-847" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-848" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-849" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-850" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-851" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-852" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-853" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-854" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-855" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-856" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-857" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-858" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-859" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-860" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-861" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-862" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-863" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">38.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mtext>card</mtext></mrow><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mn>5</mn><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>=</mo><mn>38.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-77">\textrm{card}(A)=5^2+3^2+2^2=38.</script>
</li><li> On compte d'abord le nombre de choix de deux chaussures qui amènent la même paire : il y en a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-78-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-864" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-865" class="mjx-mrow"><span id="MJXc-Node-866" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math></span></span><script type="math/tex" id="MathJax-Element-78">10</script> (autant que de paires de chaussures). Par différence, le nombre de choix qui amènent à deux chaussures qui ne sont pas de la même paire est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-79-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;20&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;180&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-867" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-868" class="mjx-mrow"><span id="MJXc-Node-869" class="mjx-mrow"><span id="MJXc-Node-870" class="mjx-TeXmathchoice"><span id="MJXc-Node-871" class="mjx-texatom"><span id="MJXc-Node-872" class="mjx-mrow"><span id="MJXc-Node-873" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-874" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-875" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">20</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.513em;"><span id="MJXc-Node-876" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-877" class="mjx-TeXmathchoice"><span id="MJXc-Node-878" class="mjx-texatom"><span id="MJXc-Node-879" class="mjx-mrow"><span id="MJXc-Node-880" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-881" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-882" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-883" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-884" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">180</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>20</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>−</mo><mn>10</mn><mo>=</mo><mn>180</mn></math></span></span><script type="math/tex" id="MathJax-Element-79">\binom{20}2-10=180</script>.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>11</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Le tournoi de tennis<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=3197"> [Signaler une erreur]</a><a id="exo11" href="javascript:ajouteexo(3197,11,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce3197&#39;,&#39;imageenonce3197&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce3197"></a><div class="inner" id="innerenonce3197">Un tournoi de tennis comporte <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-80-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-885" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-886" class="mjx-mrow"><span id="MJXc-Node-887" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span id="MJXc-Node-888" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-80">2n</script> joueurs. De combien de façons peut-on organiser le premier tour dans le cas où :
<ol class="enumeratechiffre">
<li> on s’intéresse à la fois aux joueurs qui sont opposés et à l’ordre des matches ;
</li><li> on ne s’intéresse qu’à la connaissance des joueurs opposés.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication3197&#39;,&#39;imageindication3197&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication3197"></a><div class="inner" id="innerindication3197" style="display:none;"><ol class="enumeratechiffre">
<li> Choisir les deux premiers joueurs, puis les deux suivants, etc...
</li><li>
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige3197&#39;,&#39;imagecorrige3197&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige3197"></a><div class="inner" id="innercorrige3197" style="display:none;"><ol class="enumeratechiffre">
<li> On choisit d'abord <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-81-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-889" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-890" class="mjx-mrow"><span id="MJXc-Node-891" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-81">2</script> joueurs parmi les <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-82-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-892" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-893" class="mjx-mrow"><span id="MJXc-Node-894" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-895" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-82">2n</script> joueurs pour le match 1, puis <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-83-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-896" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-897" class="mjx-mrow"><span id="MJXc-Node-898" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-83">2</script> joueurs parmi les <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-84-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-899" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-900" class="mjx-mrow"><span id="MJXc-Node-901" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-902" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-903" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-904" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-84">2n-2</script> joueurs restants pour le match 2, et ainsi de suite. Le nombre de façons d'organiser le premier tour, en tenant compte de l'ordre des matchs, est donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-85-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-905" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-906" class="mjx-mrow"><span id="MJXc-Node-907" class="mjx-mrow"><span id="MJXc-Node-908" class="mjx-TeXmathchoice"><span id="MJXc-Node-909" class="mjx-texatom"><span id="MJXc-Node-910" class="mjx-mrow"><span id="MJXc-Node-911" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-912" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.1em;"><span class="mjx-numerator" style="width: 1.1em; top: -1.368em;"><span id="MJXc-Node-913" class="mjx-mrow"><span id="MJXc-Node-914" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-915" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="mjx-denominator" style="width: 1.1em; bottom: -0.711em;"><span id="MJXc-Node-916" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-917" class="mjx-TeXmathchoice"><span id="MJXc-Node-918" class="mjx-texatom"><span id="MJXc-Node-919" class="mjx-mrow"><span id="MJXc-Node-920" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-921" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-922" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-923" class="mjx-TeXmathchoice"><span id="MJXc-Node-924" class="mjx-texatom"><span id="MJXc-Node-925" class="mjx-mrow"><span id="MJXc-Node-926" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-927" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.822em;"><span class="mjx-numerator" style="width: 2.822em; top: -1.368em;"><span id="MJXc-Node-928" class="mjx-mrow"><span id="MJXc-Node-929" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-930" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-931" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-932" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-denominator" style="width: 2.822em; bottom: -0.711em;"><span id="MJXc-Node-933" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-934" class="mjx-TeXmathchoice"><span id="MJXc-Node-935" class="mjx-texatom"><span id="MJXc-Node-936" class="mjx-mrow"><span id="MJXc-Node-937" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-938" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-939" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-940" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-941" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-942" class="mjx-TeXmathchoice"><span id="MJXc-Node-943" class="mjx-texatom"><span id="MJXc-Node-944" class="mjx-mrow"><span id="MJXc-Node-945" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-946" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.5em;"><span class="mjx-numerator" style="width: 0.5em; top: -1.368em;"><span id="MJXc-Node-947" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 0.5em; bottom: -0.711em;"><span id="MJXc-Node-948" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-949" class="mjx-TeXmathchoice"><span id="MJXc-Node-950" class="mjx-texatom"><span id="MJXc-Node-951" class="mjx-mrow"><span id="MJXc-Node-952" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-953" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-954" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.356em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.356em; top: -1.542em;"><span id="MJXc-Node-955" class="mjx-mrow"><span id="MJXc-Node-956" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-957" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-958" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-959" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-960" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-denominator" style="width: 2.356em; bottom: -0.766em;"><span id="MJXc-Node-961" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-962" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-963" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.356em;"></span></span><span class="mjx-vsize" style="height: 2.308em; vertical-align: -0.766em;"></span></span><span id="MJXc-Node-964" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mrow><mn>2</mn><mi>n</mi></mrow><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>×</mo><mo>⋯</mo><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>2</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo><mo>!</mo></mrow><msup><mn>2</mn><mi>n</mi></msup></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-85">\binom{2n}2\times\binom{2n-2}2\times\cdots\times\binom{2}2=\frac{(2n)!}{2^n}.</script>
</li><li> Si l'ordre des matchs n'est plus pris en compte, alors il faut diviser en plus par le nombre de permutations possibles des matchs, c'est-à-dire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-86-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-965" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-966" class="mjx-mrow"><span id="MJXc-Node-967" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-968" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-86">n!</script>. Le nombre de façons d'organiser le premier tour, sans tenir compte de l'ordre des matchs, est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-87-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x22C5;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-969" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-970" class="mjx-mrow"><span id="MJXc-Node-971" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.666em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.356em; top: -1.652em;"><span id="MJXc-Node-972" class="mjx-mrow" style=""><span id="MJXc-Node-973" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-974" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-975" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-976" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-977" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.356em; bottom: -0.679em;"><span id="MJXc-Node-978" class="mjx-mrow" style=""><span id="MJXc-Node-979" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-980" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-981" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-982" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋅</span></span><span id="MJXc-Node-983" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-984" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.666em;"></span></span><span class="mjx-vsize" style="height: 1.649em; vertical-align: -0.48em;"></span></span><span id="MJXc-Node-985" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo><mo>!</mo></mrow><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>⋅</mo><mi>n</mi><mo>!</mo></mrow></mfrac><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-87">\frac{(2n)!}{2^n\cdot n!}.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>12</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Dénombrement de tirages successifs et simultanés, avec ou sans remise<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=3198"> [Signaler une erreur]</a><a id="exo12" href="javascript:ajouteexo(3198,12,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce3198&#39;,&#39;imageenonce3198&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce3198"></a><div class="inner" id="innerenonce3198">Un sac contient 5 jetons blancs et 8 jetons noirs. On suppose que les jetons sont discernables (numérotés par exemple) et on effectue un tirage de 6 jetons de ce sac.
<ol class="enumeratechiffre">
<li> On suppose que les jetons sont tirés successivement en remettant à chaque fois le jeton tiré.
<ol class="enumeratechiffre">
<li> Donner le nombre de résultats possibles.
</li><li> Combien de ces résultats amènent
<ol class="enumeratechiffre">
<li> exactement 1 jeton noir ?
</li><li> au moins 1 jeton noir ?
</li><li> au plus un jeton noir ?
</li><li> 2 fois plus de jetons noirs que de jetons blancs ?
</li></ol>
</li></ol>
</li><li> Mêmes questions en supposant que les jetons sont tirés successivement sans remise.
</li><li> Mêmes questions en supposant que les jetons sont tirés simultanément.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication3198&#39;,&#39;imageindication3198&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication3198"></a><div class="inner" id="innerindication3198" style="display:none;"></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige3198&#39;,&#39;imagecorrige3198&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige3198"></a><div class="inner" id="innercorrige3198" style="display:none;"><ol class="enumeratechiffre">
<li> <ol class="enumeratechiffre">
<li> On est dans un cas où l’ordre et la répétition interviennent puisque les jetons sont tirés successivement et avec remise. A chaque tirage, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-88-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-986" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-987" class="mjx-mrow"><span id="MJXc-Node-988" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math></span></span><script type="math/tex" id="MathJax-Element-88">13</script> choix possibles, et donc on obtient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-89-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-989" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-990" class="mjx-mrow"><span id="MJXc-Node-991" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-992" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-993" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>13</mn><mn>6</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-89">13^6</script> résultats possibles (on a dénombré des listes).
</li><li>
<ol class="enumeratechiffre">
<li> Pour obtenir exactement un jeton noir, on doit choisir à quel tirage on va tirer le jeton noir (6 choix possibles). Ensuite pour chaque choix de numéro de tirage, on a 8 choix possibles de jetons noirs et pour les 5 autres tirages, on a 5 possibilités à chaque fois puisqu'il y a 5 jetons blancs. Ainsi, on obtient : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-90-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-994" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-995" class="mjx-mrow"><span id="MJXc-Node-996" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-997" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-998" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-999" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1000" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1001" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1002" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>8</mn><mo>×</mo><msup><mn>5</mn><mn>5</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-90">6 \times 8 \times 5^5</script> résultats possibles.
</li><li> On va dénombrer l’ensemble complémentaire. En effet, il y a exactement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-91-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1003" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1004" class="mjx-mrow"><span id="MJXc-Node-1005" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1006" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1007" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>5</mn><mn>6</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-91">5^6</script> tirages sans jeton noir (à chaque fois, on a tiré un jeton blanc).
Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-92-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1008" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1009" class="mjx-mrow"><span id="MJXc-Node-1010" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1011" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1012" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span id="MJXc-Node-1013" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1014" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1015" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1016" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>13</mn><mn>6</mn></msup><mo>−</mo><msup><mn>5</mn><mn>6</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-92">13^6-5^6</script> tirages avec au moins un jeton noir.
</li><li> On a déjà dénombré les tirages avec exactement un jeton noir et les tirages sans jeton noir. L'ensemble recherché étant l'union disjointe des ensembles précédents, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-93-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1017" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1018" class="mjx-mrow"><span id="MJXc-Node-1019" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1020" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1021" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-1022" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1023" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1024" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1025" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span id="MJXc-Node-1026" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1027" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1028" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1029" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>8</mn><mo>×</mo><msup><mn>5</mn><mn>5</mn></msup><mo>+</mo><msup><mn>5</mn><mn>6</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-93">6 \times 8 \times 5^5+5^6</script> tirages amenant au plus un jeton noir.
</li><li> Si on a deux fois plus de jetons noirs que de jetons blancs, c'est qu'on a tiré 4 jetons noirs et 2 jetons blancs. On commence par fixer les 2 tirages parmi 6 pour lesquels on a trouvé un jeton blanc : il y en a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-94-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1030" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1031" class="mjx-mrow"><span id="MJXc-Node-1032" class="mjx-mrow"><span id="MJXc-Node-1033" class="mjx-TeXmathchoice"><span id="MJXc-Node-1034" class="mjx-texatom"><span id="MJXc-Node-1035" class="mjx-mrow"><span id="MJXc-Node-1036" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1037" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1038" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1039" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1040" class="mjx-TeXmathchoice"><span id="MJXc-Node-1041" class="mjx-texatom"><span id="MJXc-Node-1042" class="mjx-mrow"><span id="MJXc-Node-1043" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-94">\binom 62</script>.
Ce choix fixé, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-95-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1044" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1045" class="mjx-mrow"><span id="MJXc-Node-1046" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1047" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1048" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>5</mn><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-95">5^2</script> choix pour les jetons blancs et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-96-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1049" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1050" class="mjx-mrow"><span id="MJXc-Node-1051" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1052" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1053" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>8</mn><mn>4</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-96">8^4</script> choix possibles pour les jetons noirs. On obtient au final <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-97-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1054" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1055" class="mjx-mrow"><span id="MJXc-Node-1056" class="mjx-mrow"><span id="MJXc-Node-1057" class="mjx-TeXmathchoice"><span id="MJXc-Node-1058" class="mjx-texatom"><span id="MJXc-Node-1059" class="mjx-mrow"><span id="MJXc-Node-1060" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1061" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1062" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1063" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1064" class="mjx-TeXmathchoice"><span id="MJXc-Node-1065" class="mjx-texatom"><span id="MJXc-Node-1066" class="mjx-mrow"><span id="MJXc-Node-1067" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1068" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1069" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1070" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1071" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-1072" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1073" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1074" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1075" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><msup><mn>5</mn><mn>2</mn></msup><mo>×</mo><msup><mn>8</mn><mn>4</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-97">\binom 62\times 5^2\times 8^4</script> tirages amenant 2 fois plus de jetons noirs que de jetons blancs.
</li></ol>
</li></ol>
</li><li> <ol class="enumeratechiffre">
<li>
Si on tire les jetons successivement et sans remise, l'ordre intervient mais il n'y a plus de répétition. On dénombre des arrangements et il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-98-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;11&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1076" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1077" class="mjx-mrow"><span id="MJXc-Node-1078" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1079" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1080" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-1081" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1082" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">11</span></span><span id="MJXc-Node-1083" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1084" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-1085" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1086" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-1087" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1088" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-1089" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1090" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1091" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1092" class="mjx-texatom"><span id="MJXc-Node-1093" class="mjx-mrow"><span id="MJXc-Node-1094" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-1095" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-1096" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mo>×</mo><mn>12</mn><mo>×</mo><mn>11</mn><mo>×</mo><mn>10</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn><mo>=</mo><mn>13</mn><mo>!</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>7</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-98">13\times 12\times 11\times 10\times 9\times 8=13!/7!</script> tirages possibles.
</li><li>
<ol class="enumeratechiffre">
<li> Pour obtenir exactement un jeton noir, on doit choisir à quel tirage on va tirer le jeton noir (6 choix possibles). Ensuite pour chaque choix de numéro de tirage, on a 8 choix possibles de jetons noirs et pour les 5 autres tirages, on doit choisir 5 jetons parmi 5 sans remise mais avec ordre : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-99-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1097" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1098" class="mjx-mrow"><span id="MJXc-Node-1099" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1100" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1101" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1102" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1103" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-1104" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1105" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1106" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1107" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-1108" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1109" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1110" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn><mo>=</mo><mn>5</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-99">5 \times 4 \times 3 \times 2 \times 1 = 5!</script>. Ainsi, on obtient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-100-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1111" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1112" class="mjx-mrow"><span id="MJXc-Node-1113" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1114" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1115" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-1116" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1117" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1118" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>5</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-100">6 \times 8 \times 5!</script>
résultats possibles.
</li><li> On remarque qu'il n'y a aucun tirage sans jeton noir, puisque l'on fait 6 tirages, sans remise, et qu'il n'y a que 5 jetons blancs. Le nombre de tirages avec au moins un jeton noir est égal au nombre total de tirages, soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-101-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1119" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1120" class="mjx-mrow"><span id="MJXc-Node-1121" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1122" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1123" class="mjx-texatom"><span id="MJXc-Node-1124" class="mjx-mrow"><span id="MJXc-Node-1125" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-1126" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-1127" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mo>!</mo><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>7</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-101">13!/7!</script>.
</li><li> Comme il n'y a aucun tirage sans jeton noir, le nombre de tirages avec au plus un jeton noir est égal au nombre de tirages avec exactement un jeton noir, soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-102-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1128" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1129" class="mjx-mrow"><span id="MJXc-Node-1130" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1131" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1132" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-1133" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1134" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1135" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>5</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-102">6 \times 8 \times 5!</script> résultats possibles.
</li><li> Comme précédemment, on tire 2 jetons blancs et 4 jetons noirs. On commence par fixer la place des 2 jetons blancs parmi les 6 tirages, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-103-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1136" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1137" class="mjx-mrow"><span id="MJXc-Node-1138" class="mjx-mrow"><span id="MJXc-Node-1139" class="mjx-TeXmathchoice"><span id="MJXc-Node-1140" class="mjx-texatom"><span id="MJXc-Node-1141" class="mjx-mrow"><span id="MJXc-Node-1142" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1143" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1144" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1145" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1146" class="mjx-TeXmathchoice"><span id="MJXc-Node-1147" class="mjx-texatom"><span id="MJXc-Node-1148" class="mjx-mrow"><span id="MJXc-Node-1149" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-103">\binom 6 2</script> choix. Puis il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-104-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1150" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1151" class="mjx-mrow"><span id="MJXc-Node-1152" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1153" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1154" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>×</mo><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-104">5\times 4</script> choix pour les jetons blancs et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-105-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1155" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1156" class="mjx-mrow"><span id="MJXc-Node-1157" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-1158" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1159" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-1160" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1161" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1162" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1163" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><mn>7</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-105">8\times 7\times 6\times 5</script> choix possibles pour les jetons noirs. Finalement, on obtient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-106-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1164" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1165" class="mjx-mrow"><span id="MJXc-Node-1166" class="mjx-mrow"><span id="MJXc-Node-1167" class="mjx-TeXmathchoice"><span id="MJXc-Node-1168" class="mjx-texatom"><span id="MJXc-Node-1169" class="mjx-mrow"><span id="MJXc-Node-1170" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1171" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1172" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1173" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1174" class="mjx-TeXmathchoice"><span id="MJXc-Node-1175" class="mjx-texatom"><span id="MJXc-Node-1176" class="mjx-mrow"><span id="MJXc-Node-1177" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1178" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1179" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1180" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1181" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1182" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1183" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-1184" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1185" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-1186" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1187" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1188" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1189" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>6</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>7</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-106">\binom 62\times 5\times 4\times 8\times 7\times 6\times 5</script> choix possibles.
</li></ol>
</li></ol>
</li><li> <ol class="enumeratechiffre">
On est dans le cas où ni l'ordre ni la répétition n'interviennent, on dénombre donc des combinaisons. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-107-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1190" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1191" class="mjx-mrow"><span id="MJXc-Node-1192" class="mjx-mrow"><span id="MJXc-Node-1193" class="mjx-TeXmathchoice"><span id="MJXc-Node-1194" class="mjx-texatom"><span id="MJXc-Node-1195" class="mjx-mrow"><span id="MJXc-Node-1196" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1197" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1198" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1199" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1200" class="mjx-TeXmathchoice"><span id="MJXc-Node-1201" class="mjx-texatom"><span id="MJXc-Node-1202" class="mjx-mrow"><span id="MJXc-Node-1203" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>13</mn><mn>6</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-107">\binom{13}6</script> résultats possibles.
<li> <ol class="enumeratechiffre">
<li> On tire forcément tous les jetons blancs et un jeton noir parmi les 8 : il y a donc 8 tirages possibles.
</li><li> On remarque qu'il n'y a aucun tirage sans jeton noir, puisque l'on fait 6 tirages, sans remise, et qu'il n'y a que 5 jetons blancs. Le nombre de tirages avec au moins un jeton noir est égal au nombre total de tirages, soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-108-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1204" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1205" class="mjx-mrow"><span id="MJXc-Node-1206" class="mjx-mrow"><span id="MJXc-Node-1207" class="mjx-TeXmathchoice"><span id="MJXc-Node-1208" class="mjx-texatom"><span id="MJXc-Node-1209" class="mjx-mrow"><span id="MJXc-Node-1210" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1211" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1212" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1213" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1214" class="mjx-TeXmathchoice"><span id="MJXc-Node-1215" class="mjx-texatom"><span id="MJXc-Node-1216" class="mjx-mrow"><span id="MJXc-Node-1217" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>13</mn><mn>6</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-108">\binom{13}6</script>.
</li><li> Comme il n'y a aucun tirage sans jeton noir, le nombre de tirages avec au plus un jeton noir est égal au nombre de tirages avec exactement un jeton noir, soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-109-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1218" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1219" class="mjx-mrow"><span id="MJXc-Node-1220" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span></span><script type="math/tex" id="MathJax-Element-109">8</script> résultats possibles.
</li><li> On doit tirer 2 jetons blancs parmi 5 et 4 jetons noirs parmi 8. Le nombre de tirages recherché est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-110-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1221" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1222" class="mjx-mrow"><span id="MJXc-Node-1223" class="mjx-mrow"><span id="MJXc-Node-1224" class="mjx-TeXmathchoice"><span id="MJXc-Node-1225" class="mjx-texatom"><span id="MJXc-Node-1226" class="mjx-mrow"><span id="MJXc-Node-1227" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1228" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1229" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1230" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1231" class="mjx-TeXmathchoice"><span id="MJXc-Node-1232" class="mjx-texatom"><span id="MJXc-Node-1233" class="mjx-mrow"><span id="MJXc-Node-1234" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1235" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1236" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1237" class="mjx-TeXmathchoice"><span id="MJXc-Node-1238" class="mjx-texatom"><span id="MJXc-Node-1239" class="mjx-mrow"><span id="MJXc-Node-1240" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1241" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1242" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1243" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1244" class="mjx-TeXmathchoice"><span id="MJXc-Node-1245" class="mjx-texatom"><span id="MJXc-Node-1246" class="mjx-mrow"><span id="MJXc-Node-1247" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>5</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>4</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-110">\binom 52\times \binom 84</script>.
</li></ol>
</li></ol>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>13</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Bridge<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2587"> [Signaler une erreur]</a><a id="exo13" href="javascript:ajouteexo(2587,13,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2587&#39;,&#39;imageenonce2587&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2587"></a><div class="inner" id="innerenonce2587">Dans cet exercice, on attend des réponses qui ne sont pas des valeurs numériques, mais des expressions en termes de factorielles, sous la forme
la plus simple possible.
<ol class="enumeratechiffre">
<li> Combien y-a-t-il de façons de répartir les 52 cartes d'un jeu entre 4 joueurs N, S, E, O, chacun possédant 13 cartes.
</li><li> Parmi ces façons, combien y-en-a-t-il qui sont telles que chaque joueur n'a qu'une seule couleur (par exemple, N a les 13 piques, S a les 13 coeurs,...)?
</li><li> Combien y-a-t-il de façons que deux joueurs quelconques aient chacun une seule couleur?
</li><li> Combien y-a-t-il de façons que deux partenaires, c'est-à-dire (N,S) ou (E,O), aient chacun une seule couleur, les deux autres partenaires ayant des cartes quelconques.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2587&#39;,&#39;imageindication2587&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2587"></a><div class="inner" id="innerindication2587" style="display:none;"><ol class="enumeratechiffre">
<li> Compter le nombre de choix possibles pour le premier joueur, puis pour le deuxième...
</li><li> Choisir une couleur parmi 4 pour N, puis une couleur parmi...
</li><li> Choisir d'abord les deux joueurs qui ont une seule couleur, puis choisir pour chaque joueur la couleur.
</li><li> Raisonner de la même façon.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2587&#39;,&#39;imagecorrige2587&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2587"></a><div class="inner" id="innercorrige2587" style="display:none;"><ol class="enumeratechiffre">
<li> Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-111-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;52&lt;/mn&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1248" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1249" class="mjx-mrow"><span id="MJXc-Node-1250" class="mjx-mrow"><span id="MJXc-Node-1251" class="mjx-TeXmathchoice"><span id="MJXc-Node-1252" class="mjx-texatom"><span id="MJXc-Node-1253" class="mjx-mrow"><span id="MJXc-Node-1254" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1255" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1256" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">52</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1257" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1258" class="mjx-TeXmathchoice"><span id="MJXc-Node-1259" class="mjx-texatom"><span id="MJXc-Node-1260" class="mjx-mrow"><span id="MJXc-Node-1261" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>52</mn><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-111">\binom{52}{13}</script> choix possibles pour le premier joueur, puis <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-112-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mrow&gt;&lt;mn&gt;52&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;39&lt;/mn&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1262" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1263" class="mjx-mrow"><span id="MJXc-Node-1264" class="mjx-mrow"><span id="MJXc-Node-1265" class="mjx-TeXmathchoice"><span id="MJXc-Node-1266" class="mjx-texatom"><span id="MJXc-Node-1267" class="mjx-mrow"><span id="MJXc-Node-1268" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1269" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.964em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.778em; top: -1.319em;"><span id="MJXc-Node-1270" class="mjx-mrow" style=""><span id="MJXc-Node-1271" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">52</span></span><span id="MJXc-Node-1272" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1273" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.778em; bottom: -0.535em;"><span id="MJXc-Node-1274" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1275" class="mjx-TeXmathchoice"><span id="MJXc-Node-1276" class="mjx-texatom"><span id="MJXc-Node-1277" class="mjx-mrow"><span id="MJXc-Node-1278" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1279" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1280" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-1281" class="mjx-TeXmathchoice"><span id="MJXc-Node-1282" class="mjx-texatom"><span id="MJXc-Node-1283" class="mjx-mrow"><span id="MJXc-Node-1284" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1285" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1286" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">39</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1287" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1288" class="mjx-TeXmathchoice"><span id="MJXc-Node-1289" class="mjx-texatom"><span id="MJXc-Node-1290" class="mjx-mrow"><span id="MJXc-Node-1291" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mrow><mn>52</mn><mo>−</mo><mn>13</mn></mrow><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>39</mn><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-112">\binom{52-13}{13}=\binom{39}{13}</script> choix possibles pour le deuxième joueur,
et encore <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-113-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mrow&gt;&lt;mn&gt;39&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1292" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1293" class="mjx-mrow"><span id="MJXc-Node-1294" class="mjx-mrow"><span id="MJXc-Node-1295" class="mjx-TeXmathchoice"><span id="MJXc-Node-1296" class="mjx-texatom"><span id="MJXc-Node-1297" class="mjx-mrow"><span id="MJXc-Node-1298" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1299" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.964em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.778em; top: -1.319em;"><span id="MJXc-Node-1300" class="mjx-mrow" style=""><span id="MJXc-Node-1301" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">39</span></span><span id="MJXc-Node-1302" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1303" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.778em; bottom: -0.535em;"><span id="MJXc-Node-1304" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1305" class="mjx-TeXmathchoice"><span id="MJXc-Node-1306" class="mjx-texatom"><span id="MJXc-Node-1307" class="mjx-mrow"><span id="MJXc-Node-1308" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1309" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1310" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-1311" class="mjx-TeXmathchoice"><span id="MJXc-Node-1312" class="mjx-texatom"><span id="MJXc-Node-1313" class="mjx-mrow"><span id="MJXc-Node-1314" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1315" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1316" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1317" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1318" class="mjx-TeXmathchoice"><span id="MJXc-Node-1319" class="mjx-texatom"><span id="MJXc-Node-1320" class="mjx-mrow"><span id="MJXc-Node-1321" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mrow><mn>39</mn><mo>−</mo><mn>13</mn></mrow><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>26</mn><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-113">\binom{39-13}{13}=\binom{26}{13}</script> pour le troisième. Le dernier joueur prend les 13 cartes restantes. Au total on trouve
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-114-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;52&lt;/mn&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;39&lt;/mn&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;52&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;39&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;39&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;52&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-1322" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1323" class="mjx-mrow"><span id="MJXc-Node-1324" class="mjx-mrow"><span id="MJXc-Node-1325" class="mjx-TeXmathchoice"><span id="MJXc-Node-1326" class="mjx-texatom"><span id="MJXc-Node-1327" class="mjx-mrow"><span id="MJXc-Node-1328" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-1329" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1em;"><span class="mjx-numerator" style="width: 1em; top: -1.368em;"><span id="MJXc-Node-1330" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">52</span></span></span><span class="mjx-denominator" style="width: 1em; bottom: -0.733em;"><span id="MJXc-Node-1331" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1332" class="mjx-TeXmathchoice"><span id="MJXc-Node-1333" class="mjx-texatom"><span id="MJXc-Node-1334" class="mjx-mrow"><span id="MJXc-Node-1335" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-1336" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1337" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1338" class="mjx-TeXmathchoice"><span id="MJXc-Node-1339" class="mjx-texatom"><span id="MJXc-Node-1340" class="mjx-mrow"><span id="MJXc-Node-1341" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-1342" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1em;"><span class="mjx-numerator" style="width: 1em; top: -1.368em;"><span id="MJXc-Node-1343" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">39</span></span></span><span class="mjx-denominator" style="width: 1em; bottom: -0.733em;"><span id="MJXc-Node-1344" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1345" class="mjx-TeXmathchoice"><span id="MJXc-Node-1346" class="mjx-texatom"><span id="MJXc-Node-1347" class="mjx-mrow"><span id="MJXc-Node-1348" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-1349" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1350" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1351" class="mjx-TeXmathchoice"><span id="MJXc-Node-1352" class="mjx-texatom"><span id="MJXc-Node-1353" class="mjx-mrow"><span id="MJXc-Node-1354" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-1355" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1em;"><span class="mjx-numerator" style="width: 1em; top: -1.368em;"><span id="MJXc-Node-1356" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span></span><span class="mjx-denominator" style="width: 1em; bottom: -0.733em;"><span id="MJXc-Node-1357" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1358" class="mjx-TeXmathchoice"><span id="MJXc-Node-1359" class="mjx-texatom"><span id="MJXc-Node-1360" class="mjx-mrow"><span id="MJXc-Node-1361" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-1362" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1363" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.756em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.756em; top: -1.418em;"><span id="MJXc-Node-1364" class="mjx-mrow"><span id="MJXc-Node-1365" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">52</span></span><span id="MJXc-Node-1366" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-denominator" style="width: 2.756em; bottom: -0.78em;"><span id="MJXc-Node-1367" class="mjx-mrow"><span id="MJXc-Node-1368" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">39</span></span><span id="MJXc-Node-1369" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1370" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1371" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.756em;"></span></span><span class="mjx-vsize" style="height: 2.198em; vertical-align: -0.78em;"></span></span><span id="MJXc-Node-1372" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1373" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.756em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.756em; top: -1.418em;"><span id="MJXc-Node-1374" class="mjx-mrow"><span id="MJXc-Node-1375" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">39</span></span><span id="MJXc-Node-1376" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-denominator" style="width: 2.756em; bottom: -0.78em;"><span id="MJXc-Node-1377" class="mjx-mrow"><span id="MJXc-Node-1378" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span><span id="MJXc-Node-1379" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1380" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1381" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.756em;"></span></span><span class="mjx-vsize" style="height: 2.198em; vertical-align: -0.78em;"></span></span><span id="MJXc-Node-1382" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1383" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 3.978em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.978em; top: -1.418em;"><span id="MJXc-Node-1384" class="mjx-mrow"><span id="MJXc-Node-1385" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span><span id="MJXc-Node-1386" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-denominator" style="width: 3.978em; bottom: -0.78em;"><span id="MJXc-Node-1387" class="mjx-mrow"><span id="MJXc-Node-1388" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1389" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1390" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1391" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1392" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.978em;"></span></span><span class="mjx-vsize" style="height: 2.198em; vertical-align: -0.78em;"></span></span><span id="MJXc-Node-1393" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1394" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.645em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.645em; top: -1.418em;"><span id="MJXc-Node-1395" class="mjx-mrow"><span id="MJXc-Node-1396" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">52</span></span><span id="MJXc-Node-1397" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-denominator" style="width: 2.645em; bottom: -1.053em;"><span id="MJXc-Node-1398" class="mjx-mrow"><span id="MJXc-Node-1399" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1400" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1401" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1402" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1403" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1404" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.645em;"></span></span><span class="mjx-vsize" style="height: 2.47em; vertical-align: -1.053em;"></span></span><span id="MJXc-Node-1405" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>52</mn><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>39</mn><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>26</mn><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mn>52</mn><mo>!</mo></mrow><mrow><mn>39</mn><mo>!</mo><mn>13</mn><mo>!</mo></mrow></mfrac><mo>×</mo><mfrac><mrow><mn>39</mn><mo>!</mo></mrow><mrow><mn>26</mn><mo>!</mo><mn>13</mn><mo>!</mo></mrow></mfrac><mo>×</mo><mfrac><mrow><mn>26</mn><mo>!</mo></mrow><mrow><mn>13</mn><mo>!</mo><mo>×</mo><mn>13</mn><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>52</mn><mo>!</mo></mrow><mrow><mo stretchy="false">(</mo><mn>13</mn><mo>!</mo><msup><mo stretchy="false">)</mo><mn>4</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-114">\binom{52}{13}\times\binom{39}{13}\times\binom{26}{13}=\frac{52!}{39!13!}\times\frac{39!}{26!13!}\times \frac{26!}{13!\times 13!}=\frac{52!}{(13!)^4}.</script>
</li><li> On choisit une couleur parmi 4 pour le premier joueur, puis une couleur parmi 3 pour le deuxième et enfin une couleur parmi 2 pour le troisième. Cela fait <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-115-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1406" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1407" class="mjx-mrow"><span id="MJXc-Node-1408" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1409" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-115">4!</script> possibilités.
On peut aussi argumenter à l'aide du nombre de permutation de 4 couleurs.
</li><li> On commence par choisir les deux joueurs qui ont la même couleur. Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-116-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1410" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1411" class="mjx-mrow"><span id="MJXc-Node-1412" class="mjx-mrow"><span id="MJXc-Node-1413" class="mjx-TeXmathchoice"><span id="MJXc-Node-1414" class="mjx-texatom"><span id="MJXc-Node-1415" class="mjx-mrow"><span id="MJXc-Node-1416" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1417" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-1418" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1419" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1420" class="mjx-TeXmathchoice"><span id="MJXc-Node-1421" class="mjx-texatom"><span id="MJXc-Node-1422" class="mjx-mrow"><span id="MJXc-Node-1423" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-116">\binom 42</script> tels choix. On choisit ensuite une couleur parmi 4 pour le premier joueur, et une couleur parmi 3 pour le deuxième
joueur. Il reste pour le 3ème joueur à choisir 13 cartes parmi les 26 restantes tandis que le quatrième prend le reste. On trouve donc :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-117-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;72&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-1424" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1425" class="mjx-mrow"><span id="MJXc-Node-1426" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.422em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.422em; top: -1.379em;"><span id="MJXc-Node-1427" class="mjx-mrow"><span id="MJXc-Node-1428" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1429" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1430" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-denominator" style="width: 2.422em; bottom: -0.711em;"><span id="MJXc-Node-1431" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.422em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-1432" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1433" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1434" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1435" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-1436" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1437" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1438" class="mjx-TeXmathchoice"><span id="MJXc-Node-1439" class="mjx-texatom"><span id="MJXc-Node-1440" class="mjx-mrow"><span id="MJXc-Node-1441" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-1442" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1em;"><span class="mjx-numerator" style="width: 1em; top: -1.368em;"><span id="MJXc-Node-1443" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span></span><span class="mjx-denominator" style="width: 1em; bottom: -0.733em;"><span id="MJXc-Node-1444" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1445" class="mjx-TeXmathchoice"><span id="MJXc-Node-1446" class="mjx-texatom"><span id="MJXc-Node-1447" class="mjx-mrow"><span id="MJXc-Node-1448" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-1449" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1450" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.478em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.478em; top: -1.542em;"><span id="MJXc-Node-1451" class="mjx-mrow"><span id="MJXc-Node-1452" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">72</span></span><span id="MJXc-Node-1453" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1454" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1455" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span><span id="MJXc-Node-1456" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1457" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-denominator" style="width: 4.478em; bottom: -1.045em;"><span id="MJXc-Node-1458" class="mjx-mrow"><span id="MJXc-Node-1459" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1460" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1461" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1462" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1463" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1464" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.478em;"></span></span><span class="mjx-vsize" style="height: 2.587em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-1465" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mn>4</mn><mo>×</mo><mn>3</mn></mrow><mn>2</mn></mfrac><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>26</mn><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>=</mo><mfrac><mrow><mn>72</mn><mo>×</mo><mo stretchy="false">(</mo><mn>26</mn><mo stretchy="false">)</mo><mo>!</mo></mrow><mrow><mo stretchy="false">(</mo><mn>13</mn><mo>!</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-117"> \frac{4\times 3}{2}\times 4\times 3\times\binom{26}{13}=\frac{72\times (26)!}{(13!)^2}.</script>
Mais ce faisant, on a compté trop les tirages où les 4 joueurs ont une seule couleur : en effet, un tel tirage est compté <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-118-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1466" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1467" class="mjx-mrow"><span id="MJXc-Node-1468" class="mjx-mrow"><span id="MJXc-Node-1469" class="mjx-TeXmathchoice"><span id="MJXc-Node-1470" class="mjx-texatom"><span id="MJXc-Node-1471" class="mjx-mrow"><span id="MJXc-Node-1472" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1473" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-1474" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1475" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1476" class="mjx-TeXmathchoice"><span id="MJXc-Node-1477" class="mjx-texatom"><span id="MJXc-Node-1478" class="mjx-mrow"><span id="MJXc-Node-1479" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-118">\binom 4 2</script> fois, puisqu'on a choisit deux joueurs qui avaient la même couleur et que cela n'a pas lieu d'être dans ce cas là puisque tous les joueurs ont la même couleur.
Il faut donc retirer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-119-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1480" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1481" class="mjx-mrow"><span id="MJXc-Node-1482" class="mjx-mrow"><span id="MJXc-Node-1483" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">(</span></span><span id="MJXc-Node-1484" class="mjx-mrow"><span id="MJXc-Node-1485" class="mjx-mrow"><span id="MJXc-Node-1486" class="mjx-TeXmathchoice"><span id="MJXc-Node-1487" class="mjx-texatom"><span id="MJXc-Node-1488" class="mjx-mrow"><span id="MJXc-Node-1489" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1490" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-1491" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1492" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1493" class="mjx-TeXmathchoice"><span id="MJXc-Node-1494" class="mjx-texatom"><span id="MJXc-Node-1495" class="mjx-mrow"><span id="MJXc-Node-1496" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1497" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1498" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span id="MJXc-Node-1499" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">)</span></span></span><span id="MJXc-Node-1500" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1501" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>4</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-119">\left(\binom 4 2-1\right)4!</script> et le résultat final est :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-120-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;72&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-1502" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1503" class="mjx-mrow"><span id="MJXc-Node-1504" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.422em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.422em; top: -1.379em;"><span id="MJXc-Node-1505" class="mjx-mrow"><span id="MJXc-Node-1506" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1507" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1508" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-denominator" style="width: 2.422em; bottom: -0.711em;"><span id="MJXc-Node-1509" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.422em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-1510" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1511" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1512" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1513" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-1514" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1515" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1516" class="mjx-TeXmathchoice"><span id="MJXc-Node-1517" class="mjx-texatom"><span id="MJXc-Node-1518" class="mjx-mrow"><span id="MJXc-Node-1519" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-1520" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1em;"><span class="mjx-numerator" style="width: 1em; top: -1.368em;"><span id="MJXc-Node-1521" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span></span><span class="mjx-denominator" style="width: 1em; bottom: -0.733em;"><span id="MJXc-Node-1522" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-1523" class="mjx-TeXmathchoice"><span id="MJXc-Node-1524" class="mjx-texatom"><span id="MJXc-Node-1525" class="mjx-mrow"><span id="MJXc-Node-1526" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-1527" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1528" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1529" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-1530" class="mjx-mrow"><span id="MJXc-Node-1531" class="mjx-mrow"><span id="MJXc-Node-1532" class="mjx-TeXmathchoice"><span id="MJXc-Node-1533" class="mjx-texatom"><span id="MJXc-Node-1534" class="mjx-mrow"><span id="MJXc-Node-1535" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-1536" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.5em;"><span class="mjx-numerator" style="width: 0.5em; top: -1.379em;"><span id="MJXc-Node-1537" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="width: 0.5em; bottom: -0.711em;"><span id="MJXc-Node-1538" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-1539" class="mjx-TeXmathchoice"><span id="MJXc-Node-1540" class="mjx-texatom"><span id="MJXc-Node-1541" class="mjx-mrow"><span id="MJXc-Node-1542" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-1543" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1544" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span id="MJXc-Node-1545" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span><span id="MJXc-Node-1546" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1547" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1548" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1549" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.478em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.478em; top: -1.542em;"><span id="MJXc-Node-1550" class="mjx-mrow"><span id="MJXc-Node-1551" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">72</span></span><span id="MJXc-Node-1552" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1553" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1554" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span><span id="MJXc-Node-1555" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1556" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-denominator" style="width: 4.478em; bottom: -1.045em;"><span id="MJXc-Node-1557" class="mjx-mrow"><span id="MJXc-Node-1558" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1559" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1560" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1561" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1562" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1563" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.478em;"></span></span><span class="mjx-vsize" style="height: 2.587em; vertical-align: -1.045em;"></span></span><span id="MJXc-Node-1564" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1565" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-1566" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1567" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1568" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1569" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mn>4</mn><mo>×</mo><mn>3</mn></mrow><mn>2</mn></mfrac><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>26</mn><mn>13</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>4</mn><mo>!</mo><mo>=</mo><mfrac><mrow><mn>72</mn><mo>×</mo><mo stretchy="false">(</mo><mn>26</mn><mo stretchy="false">)</mo><mo>!</mo></mrow><mrow><mo stretchy="false">(</mo><mn>13</mn><mo>!</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo>!</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-120"> \frac{4\times 3}{2}\times 4\times 3\times\binom{26}{13}-\left(\binom 4 2-1\right)4!=\frac{72\times (26)!}{(13!)^2}-5\times 4!.</script>
</li><li> Le dénombrement à réaliser est très similaire, mais il y a moins de choix pour choisir les deux joueurs qui ont la même couleur puisqu'on ne choisit plus deux joueurs quelconques,
mais deux partenaires. Il n'y a donc que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-121-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1570" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1571" class="mjx-mrow"><span id="MJXc-Node-1572" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-121">2</script> choix au lieu de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-122-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1573" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1574" class="mjx-mrow"><span id="MJXc-Node-1575" class="mjx-mrow"><span id="MJXc-Node-1576" class="mjx-TeXmathchoice"><span id="MJXc-Node-1577" class="mjx-texatom"><span id="MJXc-Node-1578" class="mjx-mrow"><span id="MJXc-Node-1579" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1580" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-1581" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1582" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1583" class="mjx-TeXmathchoice"><span id="MJXc-Node-1584" class="mjx-texatom"><span id="MJXc-Node-1585" class="mjx-mrow"><span id="MJXc-Node-1586" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-122">\binom 42</script>. Finalement, on trouve <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-123-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;24&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1587" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1588" class="mjx-mrow"><span id="MJXc-Node-1589" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.852em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 4.034em; top: -1.652em;"><span id="MJXc-Node-1590" class="mjx-mrow" style=""><span id="MJXc-Node-1591" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">24</span></span><span id="MJXc-Node-1592" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1593" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1594" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">26</span></span><span id="MJXc-Node-1595" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-1596" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 4.034em; bottom: -0.948em;"><span id="MJXc-Node-1597" class="mjx-mrow" style=""><span id="MJXc-Node-1598" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-1599" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1600" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1601" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1602" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1603" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.852em;"></span></span><span class="mjx-vsize" style="height: 1.838em; vertical-align: -0.67em;"></span></span><span id="MJXc-Node-1604" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1605" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1606" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-1607" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>24</mn><mo>×</mo><mo stretchy="false">(</mo><mn>26</mn><mo stretchy="false">)</mo><mo>!</mo></mrow><mrow><mo stretchy="false">(</mo><mn>13</mn><mo>!</mo><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mn>4</mn><mo>!</mo><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-123">\frac{24\times (26)!}{(13!)^2}-4!.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>14</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Le poker<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2361"> [Signaler une erreur]</a><a id="exo14" href="javascript:ajouteexo(2361,14,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2361&#39;,&#39;imageenonce2361&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2361"></a><div class="inner" id="innerenonce2361">Une main au poker est formée de 5 cartes extraites d'un jeu de 52 cartes. Traditionnellement,trèfle, carreau, coeur, pique sont appelées couleurs et les valeurs des cartes sont rangées dans l'ordre : as, roi, dame, valet, 10, 9, 8, 7, 6, 5, 4, 3, 2, de la plus forte à la plus faible. Dénombrer les mains suivantes :
<ol class="enumeratechiffre">
<li> quinte flush : main formée de 5 cartes consécutives de la même couleur (Attention! la suite as, 2, 3, 4 et 5 est une quinte flush).
</li><li> carré : main contenant 4 cartes de la même valeur (4 as par exemple).
</li><li> full : main formée de 3 cartes de la même valeur et de deux autres cartes de même valeur (par exemple, 3 as et 2 rois).
</li><li> quinte : main formée de 5 cartes consécutives et qui ne sont pas toutes de la même couleur.
</li><li> brelan : main comprenant 3 cartes de même valeur et qui n'est ni un carré, ni un full (par exemple, 3 as, 1 valet, 1 dix).
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2361&#39;,&#39;imageindication2361&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2361"></a><div class="inner" id="innerindication2361" style="display:none;"><ol class="enumeratechiffre">
<li> On choisit la couleur puis la hauteur de la plus haute carte.
</li><li> On choisit la hauteur, puis la dernière carte.
</li><li> Choisir les deux hauteurs, puis 3 cartes parmi 4, et 2 cartes parmi 4.
</li><li> Raisonner par différence en utilisant le résultat pour les quintes flush.
</li><li> Raisonner par différence.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2361&#39;,&#39;imagecorrige2361&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2361"></a><div class="inner" id="innercorrige2361" style="display:none;"><ol class="enumeratechiffre">
<li> Il y a 4 choix pour la couleur. La couleur choisie, il suffit de choisir la plus haute carte de la main : il s'agit de n'importe quelle carte entre le 5 et l'as. Il y a donc 10 choix choix possibles pour la hauteur de la plus haute carte, et donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-124-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1608" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1609" class="mjx-mrow"><span id="MJXc-Node-1610" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1611" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1612" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-1613" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1614" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">40</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><mn>10</mn><mo>=</mo><mn>40</mn></math></span></span><script type="math/tex" id="MathJax-Element-124">4\times 10=40</script> mains conduisant à une quinte flush.
</li><li> Il y a 13 choix possibles pour la hauteur de la carte constituant le carré (carré d'as, carré de rois,...). Cette hauteur fixée, il faut encore choisir une carte parmi les 48 autres pour compléter la main. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-125-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;48&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;624&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1615" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1616" class="mjx-mrow"><span id="MJXc-Node-1617" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1618" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1619" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">48</span></span><span id="MJXc-Node-1620" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1621" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">624</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mo>×</mo><mn>48</mn><mo>=</mo><mn>624</mn></math></span></span><script type="math/tex" id="MathJax-Element-125">13\times 48=624</script> mains contenant un carré.
</li><li> On choisit d'abord la hauteur des 3 cartes de même valeur. Il y a 13 choix. On choisit alors ces 3 cartes de même valeur. Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-126-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1622" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1623" class="mjx-mrow"><span id="MJXc-Node-1624" class="mjx-mrow"><span id="MJXc-Node-1625" class="mjx-TeXmathchoice"><span id="MJXc-Node-1626" class="mjx-texatom"><span id="MJXc-Node-1627" class="mjx-mrow"><span id="MJXc-Node-1628" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1629" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-1630" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-1631" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.318em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1632" class="mjx-TeXmathchoice"><span id="MJXc-Node-1633" class="mjx-texatom"><span id="MJXc-Node-1634" class="mjx-mrow"><span id="MJXc-Node-1635" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1636" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1637" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-126">\binom 43=4</script> choix possibles. On choisit ensuite la hauteur des deux autres cartes. Il reste 12 choix. Puis on choisit ces 2 cartes de même valeur. Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-127-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1638" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1639" class="mjx-mrow"><span id="MJXc-Node-1640" class="mjx-mrow"><span id="MJXc-Node-1641" class="mjx-TeXmathchoice"><span id="MJXc-Node-1642" class="mjx-texatom"><span id="MJXc-Node-1643" class="mjx-mrow"><span id="MJXc-Node-1644" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1645" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-1646" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1647" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1648" class="mjx-TeXmathchoice"><span id="MJXc-Node-1649" class="mjx-texatom"><span id="MJXc-Node-1650" class="mjx-mrow"><span id="MJXc-Node-1651" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1652" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1653" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-127">\binom 42=6</script> choix. Finalement, le nombre de mains constituant un full est
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-128-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3744.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-1654" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1655" class="mjx-mrow"><span id="MJXc-Node-1656" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1657" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1658" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1659" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1660" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-1661" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1662" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-1663" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1664" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">3744.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>13</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>12</mn><mo>×</mo><mn>6</mn><mo>=</mo><mn>3744.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-128">13\times 4\times 12\times 6=3744.</script>
</li><li> On va dénombrer d'abord le nombre de mains constituée de 5 cartes consécutives, qu'elles soient ou non de la même couleur. On retirera ensuite le nombres de quintes flush pour obtenir le nombres de mains constituées de 5 cartes qui ne sont pas toutes de la même couleur. Si on compte le nombre de mains constituées de 5 cartes consécutives, qu'elles soient ou non de la même couleur, alors on doit choisir la hauteur de la carte la plus haute. Comme précédemment, il y a 10 choix possibles. Ensuite, pour chacune des cinq hauteurs consécutives, on choisit l'une des 4 couleurs, ce qui fait <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-129-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1665" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1666" class="mjx-mrow"><span id="MJXc-Node-1667" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1668" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1669" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>4</mn><mn>5</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-129">4^5</script> choix. Finalement, le nombre de mains correspondant est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-130-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;10240&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1670" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1671" class="mjx-mrow"><span id="MJXc-Node-1672" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-1673" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1674" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1675" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1676" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span id="MJXc-Node-1677" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1678" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">10240</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>×</mo><msup><mn>4</mn><mn>5</mn></msup><mo>=</mo><mn>10240</mn></math></span></span><script type="math/tex" id="MathJax-Element-130">10\times 4^5=10240</script> ce qui fait que le nombre de quintes est exactement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-131-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10240&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;40&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;10200&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1679" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1680" class="mjx-mrow"><span id="MJXc-Node-1681" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">10240</span></span><span id="MJXc-Node-1682" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1683" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">40</span></span><span id="MJXc-Node-1684" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1685" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10200</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10240</mn><mo>−</mo><mn>40</mn><mo>=</mo><mn>10200</mn></math></span></span><script type="math/tex" id="MathJax-Element-131">10240-40=10200</script>.
</li><li> On compte le nombre de mains comprenant 3 cartes de même niveau et qui ne contient pas un carré : il y a 13 choix pour le niveau, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-132-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1686" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1687" class="mjx-mrow"><span id="MJXc-Node-1688" class="mjx-mrow"><span id="MJXc-Node-1689" class="mjx-TeXmathchoice"><span id="MJXc-Node-1690" class="mjx-texatom"><span id="MJXc-Node-1691" class="mjx-mrow"><span id="MJXc-Node-1692" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1693" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-1694" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-1695" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.318em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1696" class="mjx-TeXmathchoice"><span id="MJXc-Node-1697" class="mjx-texatom"><span id="MJXc-Node-1698" class="mjx-mrow"><span id="MJXc-Node-1699" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1700" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1701" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-132">\binom 43=4</script> choix de 3 couleurs parmi 4 pour ce niveau. Puis, pour les deux autres cartes, il faut choisir deux cartes parmi 48 (on doit exclure la dernière carte de même hauteur, sinon on aurait un carré), soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-133-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;48&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;48&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;47&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1128&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1702" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1703" class="mjx-mrow"><span id="MJXc-Node-1704" class="mjx-mrow"><span id="MJXc-Node-1705" class="mjx-TeXmathchoice"><span id="MJXc-Node-1706" class="mjx-texatom"><span id="MJXc-Node-1707" class="mjx-mrow"><span id="MJXc-Node-1708" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1709" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.33em;"><span id="MJXc-Node-1710" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">48</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.513em;"><span id="MJXc-Node-1711" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1712" class="mjx-TeXmathchoice"><span id="MJXc-Node-1713" class="mjx-texatom"><span id="MJXc-Node-1714" class="mjx-mrow"><span id="MJXc-Node-1715" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1716" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1717" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">48</span></span><span id="MJXc-Node-1718" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1719" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">47</span></span><span id="MJXc-Node-1720" class="mjx-texatom"><span id="MJXc-Node-1721" class="mjx-mrow"><span id="MJXc-Node-1722" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-1723" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1724" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1725" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1128</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>48</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>48</mn><mo>×</mo><mn>47</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo>=</mo><mn>1128</mn></math></span></span><script type="math/tex" id="MathJax-Element-133">\binom {48} 2=48\times 47/2=1128</script> choix. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-134-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1128&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;58656&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1726" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1727" class="mjx-mrow"><span id="MJXc-Node-1728" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1729" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1730" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1731" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1732" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1128</span></span><span id="MJXc-Node-1733" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1734" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">58656</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>1128</mn><mo>=</mo><mn>58656</mn></math></span></span><script type="math/tex" id="MathJax-Element-134">13\times 4\times 1128=58656</script> mains contenant 3 cartes de même niveau, mais pas de carrés. Pour dénombrer le nombre de brelans, il faut retirer les fulls. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-135-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;58656&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;3744&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;54912&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1735" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1736" class="mjx-mrow"><span id="MJXc-Node-1737" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">58656</span></span><span id="MJXc-Node-1738" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1739" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">3744</span></span><span id="MJXc-Node-1740" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1741" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">54912</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>58656</mn><mo>−</mo><mn>3744</mn><mo>=</mo><mn>54912</mn></math></span></span><script type="math/tex" id="MathJax-Element-135">58656-3744=54912</script> brelans.<br>
Attention au raisonnement faux suivant : on pourrait compter le nombre de mains comprenant 3 cartes de même niveau : il y a 13 choix pour le niveau, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-136-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1742" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1743" class="mjx-mrow"><span id="MJXc-Node-1744" class="mjx-mrow"><span id="MJXc-Node-1745" class="mjx-TeXmathchoice"><span id="MJXc-Node-1746" class="mjx-texatom"><span id="MJXc-Node-1747" class="mjx-mrow"><span id="MJXc-Node-1748" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1749" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.33em;"><span id="MJXc-Node-1750" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-1751" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.318em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1752" class="mjx-TeXmathchoice"><span id="MJXc-Node-1753" class="mjx-texatom"><span id="MJXc-Node-1754" class="mjx-mrow"><span id="MJXc-Node-1755" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1756" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1757" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>4</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-136">\binom 43=4</script> choix de 3 couleurs parmi 4 pour ce niveau. Puis, pour les deux autres cartes, il faut choisir deux cartes parmi 52-3=49, soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-137-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;49&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;49&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;48&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1176&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1758" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1759" class="mjx-mrow"><span id="MJXc-Node-1760" class="mjx-mrow"><span id="MJXc-Node-1761" class="mjx-TeXmathchoice"><span id="MJXc-Node-1762" class="mjx-texatom"><span id="MJXc-Node-1763" class="mjx-mrow"><span id="MJXc-Node-1764" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1765" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.33em;"><span id="MJXc-Node-1766" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">49</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.513em;"><span id="MJXc-Node-1767" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.303em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1768" class="mjx-TeXmathchoice"><span id="MJXc-Node-1769" class="mjx-texatom"><span id="MJXc-Node-1770" class="mjx-mrow"><span id="MJXc-Node-1771" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1772" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1773" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">49</span></span><span id="MJXc-Node-1774" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1775" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">48</span></span><span id="MJXc-Node-1776" class="mjx-texatom"><span id="MJXc-Node-1777" class="mjx-mrow"><span id="MJXc-Node-1778" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-1779" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1780" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1781" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">1176</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>49</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>49</mn><mo>×</mo><mn>48</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo>=</mo><mn>1176</mn></math></span></span><script type="math/tex" id="MathJax-Element-137">\binom {49} 2=49\times 48/2=1176</script> choix. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-138-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;1176&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;61152&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1782" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1783" class="mjx-mrow"><span id="MJXc-Node-1784" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-1785" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1786" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1787" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1788" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">1176</span></span><span id="MJXc-Node-1789" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1790" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">61152</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>1176</mn><mo>=</mo><mn>61152</mn></math></span></span><script type="math/tex" id="MathJax-Element-138">13\times 4\times 1176=61152</script> mains contenant 3 cartes de même niveau. Pour dénombrer le nombre de brelans, il faut retirer les carrés et les fulls. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-139-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;61152&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;624&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;3744&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;56784&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1791" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1792" class="mjx-mrow"><span id="MJXc-Node-1793" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">61152</span></span><span id="MJXc-Node-1794" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1795" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">624</span></span><span id="MJXc-Node-1796" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1797" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">3744</span></span><span id="MJXc-Node-1798" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1799" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">56784</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>61152</mn><mo>−</mo><mn>624</mn><mo>−</mo><mn>3744</mn><mo>=</mo><mn>56784</mn></math></span></span><script type="math/tex" id="MathJax-Element-139">61152-624-3744=56784</script> brelans. Ce raisonnement est faux car quand on a compté le nombre de mains contenant 3 cartes de même niveau, on a compté 4 fois chaque carré. En effet, un carré d'as peut apparaitre en choisissant d'abord as de coeur, de carreau, de trèfle, puis en complétant avec as de pique, ou en choisissant as de coeur, de carreau, de pique, puis en complétant avec as de trèfle, etc... Si on veut réaliser un dénombrement correct avec cette méthode, il faut retirer quatre fois le nombre de carrés, et on a bien <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-140-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;61152&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;624&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;3744&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;54912&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1800" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1801" class="mjx-mrow"><span id="MJXc-Node-1802" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">61152</span></span><span id="MJXc-Node-1803" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1804" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1805" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1806" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">624</span></span><span id="MJXc-Node-1807" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1808" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">3744</span></span><span id="MJXc-Node-1809" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1810" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">54912</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>61152</mn><mo>−</mo><mn>4</mn><mo>×</mo><mn>624</mn><mo>−</mo><mn>3744</mn><mo>=</mo><mn>54912</mn></math></span></span><script type="math/tex" id="MathJax-Element-140">61152-4\times 624-3744=54912</script>.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>15</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Tirages dans un jeu de cartes<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1202"> [Signaler une erreur]</a><a id="exo15" href="javascript:ajouteexo(1202,15,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1202&#39;,&#39;imageenonce1202&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1202"></a><div class="inner" id="innerenonce1202">On tire simultanément 5 cartes d'un jeu de 32 cartes. Combien de tirages différents peut-on obtenir :
<ol class="enumeratechiffre">
<li> sans imposer de contraintes sur les cartes.
</li><li> contenant 5 carreaux ou 5 piques.
</li><li> contenant 2 carreaux et 3 piques.
</li><li> contenant au moins un roi.
</li><li> contenant au plus un roi.
</li><li> contenant exactement 2 rois et exactement 3 piques.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1202&#39;,&#39;imageindication1202&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1202"></a><div class="inner" id="innerindication1202" style="display:none;"><ol class="enumeratechiffre">
<li> Il faut choisir une partie à 5 élements dans un ensemble à 32 éléments.
</li><li> Compter séparément le nombre de tirages comprenant 5 carreaux et 5 piques.
</li><li> Choisir les carreaux, puis les piques.
</li><li> Compter les tirages sans roi, et retirer du nombre total.
</li><li> Compter les tirages sans roi, puis ceux avec exactement un roi.
</li><li> Séparer les tirages contenant le roi de pique, et ceux ne contenant pas le roi de pique.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1202&#39;,&#39;imagecorrige1202&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1202"></a><div class="inner" id="innercorrige1202" style="display:none;"><ol class="enumeratechiffre">
<li> Il n'y a pas d'ordre et pas de répétition sur les cartes : un tirage est donc une combinaison de 5 cartes parmi 32. Il y a : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-141-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;32&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;201376&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1811" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1812" class="mjx-mrow"><span id="MJXc-Node-1813" class="mjx-mrow"><span id="MJXc-Node-1814" class="mjx-TeXmathchoice"><span id="MJXc-Node-1815" class="mjx-texatom"><span id="MJXc-Node-1816" class="mjx-mrow"><span id="MJXc-Node-1817" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1818" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1819" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">32</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1820" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1821" class="mjx-TeXmathchoice"><span id="MJXc-Node-1822" class="mjx-texatom"><span id="MJXc-Node-1823" class="mjx-mrow"><span id="MJXc-Node-1824" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1825" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1826" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">201376</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>32</mn><mn>5</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>201376</mn></math></span></span><script type="math/tex" id="MathJax-Element-141">\binom{32}5=201376</script> tirages différents.
</li><li> Pour obtenir 5 carreaux, il faut choisir 5 cartes parmi 8 : il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-142-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1827" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1828" class="mjx-mrow"><span id="MJXc-Node-1829" class="mjx-mrow"><span id="MJXc-Node-1830" class="mjx-TeXmathchoice"><span id="MJXc-Node-1831" class="mjx-texatom"><span id="MJXc-Node-1832" class="mjx-mrow"><span id="MJXc-Node-1833" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1834" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1835" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-1836" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1837" class="mjx-TeXmathchoice"><span id="MJXc-Node-1838" class="mjx-texatom"><span id="MJXc-Node-1839" class="mjx-mrow"><span id="MJXc-Node-1840" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>5</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-142">\binom85</script> tels tirages. De même pour obtenir 5 piques. Comme les deux cas sont disjoints, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-143-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;112&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1841" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1842" class="mjx-mrow"><span id="MJXc-Node-1843" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-1844" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1845" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1846" class="mjx-TeXmathchoice"><span id="MJXc-Node-1847" class="mjx-texatom"><span id="MJXc-Node-1848" class="mjx-mrow"><span id="MJXc-Node-1849" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1850" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1851" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-1852" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1853" class="mjx-TeXmathchoice"><span id="MJXc-Node-1854" class="mjx-texatom"><span id="MJXc-Node-1855" class="mjx-mrow"><span id="MJXc-Node-1856" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1857" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1858" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">112</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>5</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>112</mn></math></span></span><script type="math/tex" id="MathJax-Element-143">2\times \binom85=112</script> tels tirages différents.
</li><li> Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-144-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1859" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1860" class="mjx-mrow"><span id="MJXc-Node-1861" class="mjx-mrow"><span id="MJXc-Node-1862" class="mjx-TeXmathchoice"><span id="MJXc-Node-1863" class="mjx-texatom"><span id="MJXc-Node-1864" class="mjx-mrow"><span id="MJXc-Node-1865" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1866" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1867" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1868" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1869" class="mjx-TeXmathchoice"><span id="MJXc-Node-1870" class="mjx-texatom"><span id="MJXc-Node-1871" class="mjx-mrow"><span id="MJXc-Node-1872" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-144">\binom82</script> façons de choisir 2 carreaux parmi 8 puis, pour chacune de ces façons, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-145-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1873" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1874" class="mjx-mrow"><span id="MJXc-Node-1875" class="mjx-mrow"><span id="MJXc-Node-1876" class="mjx-TeXmathchoice"><span id="MJXc-Node-1877" class="mjx-texatom"><span id="MJXc-Node-1878" class="mjx-mrow"><span id="MJXc-Node-1879" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1880" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1881" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-1882" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1883" class="mjx-TeXmathchoice"><span id="MJXc-Node-1884" class="mjx-texatom"><span id="MJXc-Node-1885" class="mjx-mrow"><span id="MJXc-Node-1886" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-145">\binom83</script> façons de choisir 3 piques. Le nombre de tirages recherché est donc : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-146-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1568&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1887" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1888" class="mjx-mrow"><span id="MJXc-Node-1889" class="mjx-mrow"><span id="MJXc-Node-1890" class="mjx-TeXmathchoice"><span id="MJXc-Node-1891" class="mjx-texatom"><span id="MJXc-Node-1892" class="mjx-mrow"><span id="MJXc-Node-1893" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1894" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1895" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-1896" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1897" class="mjx-TeXmathchoice"><span id="MJXc-Node-1898" class="mjx-texatom"><span id="MJXc-Node-1899" class="mjx-mrow"><span id="MJXc-Node-1900" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1901" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-1902" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1903" class="mjx-TeXmathchoice"><span id="MJXc-Node-1904" class="mjx-texatom"><span id="MJXc-Node-1905" class="mjx-mrow"><span id="MJXc-Node-1906" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1907" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.319em;"><span id="MJXc-Node-1908" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-1909" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1910" class="mjx-TeXmathchoice"><span id="MJXc-Node-1911" class="mjx-texatom"><span id="MJXc-Node-1912" class="mjx-mrow"><span id="MJXc-Node-1913" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1914" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1915" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1568</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>8</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>1568</mn></math></span></span><script type="math/tex" id="MathJax-Element-146">\binom82\times \binom83=1568</script>.
</li><li> On compte le complémentaire, c'est-à-dire les tirages sans rois : il faut alors choisir 5 cartes parmi 28, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-147-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1916" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1917" class="mjx-mrow"><span id="MJXc-Node-1918" class="mjx-mrow"><span id="MJXc-Node-1919" class="mjx-TeXmathchoice"><span id="MJXc-Node-1920" class="mjx-texatom"><span id="MJXc-Node-1921" class="mjx-mrow"><span id="MJXc-Node-1922" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1923" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1924" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1925" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1926" class="mjx-TeXmathchoice"><span id="MJXc-Node-1927" class="mjx-texatom"><span id="MJXc-Node-1928" class="mjx-mrow"><span id="MJXc-Node-1929" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>28</mn><mn>5</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-147">\binom{28}5</script> tels tirages. Le nombre de tirages recherché est donc : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-148-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;32&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;103096&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1930" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1931" class="mjx-mrow"><span id="MJXc-Node-1932" class="mjx-mrow"><span id="MJXc-Node-1933" class="mjx-TeXmathchoice"><span id="MJXc-Node-1934" class="mjx-texatom"><span id="MJXc-Node-1935" class="mjx-mrow"><span id="MJXc-Node-1936" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1937" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1938" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">32</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1939" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1940" class="mjx-TeXmathchoice"><span id="MJXc-Node-1941" class="mjx-texatom"><span id="MJXc-Node-1942" class="mjx-mrow"><span id="MJXc-Node-1943" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1944" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-1945" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1946" class="mjx-TeXmathchoice"><span id="MJXc-Node-1947" class="mjx-texatom"><span id="MJXc-Node-1948" class="mjx-mrow"><span id="MJXc-Node-1949" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1950" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1951" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1952" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1953" class="mjx-TeXmathchoice"><span id="MJXc-Node-1954" class="mjx-texatom"><span id="MJXc-Node-1955" class="mjx-mrow"><span id="MJXc-Node-1956" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1957" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-1958" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">103096</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>32</mn><mn>5</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>−</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>28</mn><mn>5</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>103096</mn></math></span></span><script type="math/tex" id="MathJax-Element-148">\binom{32}5-\binom{28}5=103096</script>.
</li><li> On a déjà compté les tirages sans roi. Pour les tirages comprenant exactement un roi, il y a 4 façons de choisir le roi, puis, pour chacune de ces façons, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-149-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1959" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1960" class="mjx-mrow"><span id="MJXc-Node-1961" class="mjx-mrow"><span id="MJXc-Node-1962" class="mjx-TeXmathchoice"><span id="MJXc-Node-1963" class="mjx-texatom"><span id="MJXc-Node-1964" class="mjx-mrow"><span id="MJXc-Node-1965" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1966" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1967" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.513em;"><span id="MJXc-Node-1968" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1969" class="mjx-TeXmathchoice"><span id="MJXc-Node-1970" class="mjx-texatom"><span id="MJXc-Node-1971" class="mjx-mrow"><span id="MJXc-Node-1972" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>28</mn><mn>4</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-149">\binom{28}4</script> façons de choisir les autres cartes. On en déduit qu'il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-150-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;28&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;180180&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-1973" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1974" class="mjx-mrow"><span id="MJXc-Node-1975" class="mjx-mrow"><span id="MJXc-Node-1976" class="mjx-TeXmathchoice"><span id="MJXc-Node-1977" class="mjx-texatom"><span id="MJXc-Node-1978" class="mjx-mrow"><span id="MJXc-Node-1979" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1980" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1981" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.535em;"><span id="MJXc-Node-1982" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="mjx-vsize" style="height: 1.31em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-1983" class="mjx-TeXmathchoice"><span id="MJXc-Node-1984" class="mjx-texatom"><span id="MJXc-Node-1985" class="mjx-mrow"><span id="MJXc-Node-1986" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-1987" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-1988" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-1989" class="mjx-mrow"><span id="MJXc-Node-1990" class="mjx-TeXmathchoice"><span id="MJXc-Node-1991" class="mjx-texatom"><span id="MJXc-Node-1992" class="mjx-mrow"><span id="MJXc-Node-1993" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-1994" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.707em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1em; top: -1.319em;"><span id="MJXc-Node-1995" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">28</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1em; bottom: -0.513em;"><span id="MJXc-Node-1996" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="mjx-vsize" style="height: 1.295em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-1997" class="mjx-TeXmathchoice"><span id="MJXc-Node-1998" class="mjx-texatom"><span id="MJXc-Node-1999" class="mjx-mrow"><span id="MJXc-Node-2000" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-2001" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2002" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">180180</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>28</mn><mn>5</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>+</mo><mn>4</mn><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>28</mn><mn>4</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>180180</mn></math></span></span><script type="math/tex" id="MathJax-Element-150">\binom{28}5+4\binom{28}{4}=180180</script> tels tirages.
</li><li> On sépare les tirages contenant le roi de pique et ceux ne contenant pas le roi de pique. <ul class="rien">
<li> si le tirage ne contient pas le roi de pique, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-151-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2003" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2004" class="mjx-mrow"><span id="MJXc-Node-2005" class="mjx-mrow"><span id="MJXc-Node-2006" class="mjx-TeXmathchoice"><span id="MJXc-Node-2007" class="mjx-texatom"><span id="MJXc-Node-2008" class="mjx-mrow"><span id="MJXc-Node-2009" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-2010" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.318em;"><span id="MJXc-Node-2011" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-2012" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.294em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-2013" class="mjx-TeXmathchoice"><span id="MJXc-Node-2014" class="mjx-texatom"><span id="MJXc-Node-2015" class="mjx-mrow"><span id="MJXc-Node-2016" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>3</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-151">\binom32</script> choix différents de 2 rois parmi 3, puis <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-152-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2017" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2018" class="mjx-mrow"><span id="MJXc-Node-2019" class="mjx-mrow"><span id="MJXc-Node-2020" class="mjx-TeXmathchoice"><span id="MJXc-Node-2021" class="mjx-texatom"><span id="MJXc-Node-2022" class="mjx-mrow"><span id="MJXc-Node-2023" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-2024" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.329em;"><span id="MJXc-Node-2025" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.535em;"><span id="MJXc-Node-2026" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 1.317em; vertical-align: -0.378em;"></span></span><span id="MJXc-Node-2027" class="mjx-TeXmathchoice"><span id="MJXc-Node-2028" class="mjx-texatom"><span id="MJXc-Node-2029" class="mjx-mrow"><span id="MJXc-Node-2030" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>7</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-152">\binom73</script> choix de 3 piques parmi 7 (tous sauf le roi de pique).
</li><li> si le tirage contient le roi de pique, il reste 3 choix pour le roi différent du roi de pique, puis <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-153-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2031" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2032" class="mjx-mrow"><span id="MJXc-Node-2033" class="mjx-mrow"><span id="MJXc-Node-2034" class="mjx-TeXmathchoice"><span id="MJXc-Node-2035" class="mjx-texatom"><span id="MJXc-Node-2036" class="mjx-mrow"><span id="MJXc-Node-2037" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-2038" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.329em;"><span id="MJXc-Node-2039" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-2040" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.302em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-2041" class="mjx-TeXmathchoice"><span id="MJXc-Node-2042" class="mjx-texatom"><span id="MJXc-Node-2043" class="mjx-mrow"><span id="MJXc-Node-2044" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>7</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow></math></span></span><script type="math/tex" id="MathJax-Element-153">\binom{7}2</script> choix pour les deux autres piques. Cela faisant, on n'a tiré que 4 cartes. Il reste une carte à choisir qui n'est ni un roi, ni un pique, et donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-154-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;32&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2045" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2046" class="mjx-mrow"><span id="MJXc-Node-2047" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">32</span></span><span id="MJXc-Node-2048" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2049" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2050" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2051" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2052" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-2053" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2054" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2055" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn><mo>−</mo><mo stretchy="false">(</mo><mn>4</mn><mo>+</mo><mn>7</mn><mo stretchy="false">)</mo><mo>=</mo><mn>21</mn></math></span></span><script type="math/tex" id="MathJax-Element-154">32-(4+7)=21</script> choix (attention à ne pas compter à nouveau deux fois le roi de pique!).
</li></ul>
Finalement, le nombre de tirages possibles est :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-155-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;2.047em&quot; minsize=&quot;2.047em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;21&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1428.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2056" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2057" class="mjx-mrow"><span id="MJXc-Node-2058" class="mjx-mrow"><span id="MJXc-Node-2059" class="mjx-TeXmathchoice"><span id="MJXc-Node-2060" class="mjx-texatom"><span id="MJXc-Node-2061" class="mjx-mrow"><span id="MJXc-Node-2062" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-2063" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.5em;"><span class="mjx-numerator" style="width: 0.5em; top: -1.367em;"><span id="MJXc-Node-2064" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="width: 0.5em; bottom: -0.711em;"><span id="MJXc-Node-2065" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 2.077em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-2066" class="mjx-TeXmathchoice"><span id="MJXc-Node-2067" class="mjx-texatom"><span id="MJXc-Node-2068" class="mjx-mrow"><span id="MJXc-Node-2069" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-2070" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-2071" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-2072" class="mjx-TeXmathchoice"><span id="MJXc-Node-2073" class="mjx-texatom"><span id="MJXc-Node-2074" class="mjx-mrow"><span id="MJXc-Node-2075" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-2076" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.5em;"><span class="mjx-numerator" style="width: 0.5em; top: -1.378em;"><span id="MJXc-Node-2077" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-denominator" style="width: 0.5em; bottom: -0.733em;"><span id="MJXc-Node-2078" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-2079" class="mjx-TeXmathchoice"><span id="MJXc-Node-2080" class="mjx-texatom"><span id="MJXc-Node-2081" class="mjx-mrow"><span id="MJXc-Node-2082" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-2083" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2084" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2085" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-2086" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-2087" class="mjx-TeXmathchoice"><span id="MJXc-Node-2088" class="mjx-texatom"><span id="MJXc-Node-2089" class="mjx-mrow"><span id="MJXc-Node-2090" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span></span></span></span><span id="MJXc-Node-2091" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.5em;"><span class="mjx-numerator" style="width: 0.5em; top: -1.378em;"><span id="MJXc-Node-2092" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-denominator" style="width: 0.5em; bottom: -0.711em;"><span id="MJXc-Node-2093" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 2.088em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-2094" class="mjx-TeXmathchoice"><span id="MJXc-Node-2095" class="mjx-texatom"><span id="MJXc-Node-2096" class="mjx-mrow"><span id="MJXc-Node-2097" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span></span></span><span id="MJXc-Node-2098" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-2099" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">21</span></span><span id="MJXc-Node-2100" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2101" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">1428.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>3</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>7</mn><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>+</mo><mn>3</mn><mo>×</mo><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mn>7</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow><mo>×</mo><mn>21</mn><mo>=</mo><mn>1428.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-155">\binom32\times \binom73+3\times \binom{7}2\times 21=1428.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>16</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Ranger des livres<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1203"> [Signaler une erreur]</a><a id="exo16" href="javascript:ajouteexo(1203,16,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1203&#39;,&#39;imageenonce1203&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1203"></a><div class="inner" id="innerenonce1203">On souhaite ranger sur une étagère 4 livres de mathématiques (distincts), 6 livres de physique, et 3 de chimie. De combien de façons peut-on effectuer ce rangement :
<ol class="enumeratechiffre">
<li> si les livres doivent être groupés par matières.
</li><li> si on exige seulement que les livres mathématiques soient groupés (les livres des autres disciplines pouvant l'être ou non).
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1203&#39;,&#39;imageindication1203&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1203"></a><div class="inner" id="innerindication1203" style="display:none;"><ol class="enumeratechiffre">
<li> Choisir l'ordre des groupes, puis l'ordre des livres à l'intérieur de chacun des groupes.
</li><li> Choisir la position du groupe des livres de mathématiques, puis ranger les livres de mathématiques d'un côté, le reste des livres de l'autre.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1203&#39;,&#39;imagecorrige1203&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1203"></a><div class="inner" id="innercorrige1203" style="display:none;"><ol class="enumeratechiffre">
<li> Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-156-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2102" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2103" class="mjx-mrow"><span id="MJXc-Node-2104" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2105" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-156">3!</script> façons de choisir l'ordre des matières. Une telle façon choisie, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-157-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2106" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2107" class="mjx-mrow"><span id="MJXc-Node-2108" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2109" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-157">4!</script> façons de ranger les livres de mathématiques, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-158-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2110" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2111" class="mjx-mrow"><span id="MJXc-Node-2112" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-2113" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-158">6!</script> façons de ranger les livres de physique, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-159-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2114" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2115" class="mjx-mrow"><span id="MJXc-Node-2116" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2117" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-159">3!</script> façons de ranger les livres de chimie. Le nombre de rangements possible est donc : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-160-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2118" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2119" class="mjx-mrow"><span id="MJXc-Node-2120" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2121" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-2122" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2123" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-2124" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-2125" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-2126" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2127" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>!</mo><mn>4</mn><mo>!</mo><mn>6</mn><mo>!</mo><mn>3</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-160">3!4!6!3!</script>.
</li><li> Il peut y avoir <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-161-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2128" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2129" class="mjx-mrow"><span id="MJXc-Node-2130" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-2131" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2132" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2133" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2134" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-2135" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2136" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>9</mn></math></span></span><script type="math/tex" id="MathJax-Element-161">0,1,\dots,9</script> livres placés avant les livres de mathématiques. Il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-162-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2137" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2138" class="mjx-mrow"><span id="MJXc-Node-2139" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math></span></span><script type="math/tex" id="MathJax-Element-162">10</script> choix du nombre de livres placés avant le livre de mathématiques. Ce choix fait, il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-163-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2140" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2141" class="mjx-mrow"><span id="MJXc-Node-2142" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2143" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-163">4!</script> façons d'ordonner les livres de mathématiques, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-164-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2144" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2145" class="mjx-mrow"><span id="MJXc-Node-2146" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-2147" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-164">9!</script> façons d'ordonner les autres : il y a donc en tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-165-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2148" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2149" class="mjx-mrow"><span id="MJXc-Node-2150" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-2151" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-2152" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2153" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span><span id="MJXc-Node-2154" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-2155" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">!</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>×</mo><mn>4</mn><mo>!</mo><mn>9</mn><mo>!</mo></math></span></span><script type="math/tex" id="MathJax-Element-165">10\times 4!9!</script> rangements différents.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>17</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Autour d'une table!<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2360"> [Signaler une erreur]</a><a id="exo17" href="javascript:ajouteexo(2360,17,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2360&#39;,&#39;imageenonce2360&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2360"></a><div class="inner" id="innerenonce2360">Une table ronde comporte cinq places, numérotées de 1 à 5. On veut répartir Adélie, Brigitte, Chafik, Denis et Emilie autour de la table. Mais attention! Denis et Émilie ne s'entendent pas du tout, et il ne faut pas les placer côte à côte!!! Combien y-a-t-il de dispositions possibles?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2360&#39;,&#39;imageindication2360&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2360"></a><div class="inner" id="innerindication2360" style="display:none;">Compter le nombre de dispositions totales (sans contraintes) et le nombre de dispositions où Denis et Émilie sont côte à côte.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2360&#39;,&#39;imagecorrige2360&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2360"></a><div class="inner" id="innercorrige2360" style="display:none;">On va raisonner par différence. Si l'on ne met pas de contraintes, il y a 5!=120 façons de placer les gens autour de la table. Comptons maintenant le nombre de façons où Denis et Émilie sont côte à côte. On commence par choisir la position de ce couple. Il y a cinq positions possibles : (1,2), (2,3), (3,4), (4,5) et (5,1). Cette position fixée, il y a 2!=2 choix pour placer Denis et Émilie, puis 3!=6 choix pour placer les autres. Il y a donc 5*2*6=60 dispositions où Denis et Émilie sont côte à côte. Finalement, il y a 120-60=60 dispositions où Émilie et Denis ne sont pas côte à côte!<br>
Remarquons que ce raisonnement dépend du fait que l'on a numéroté les places et donc que, implicitement, la position Adélie 1, Brigitte 2, Chafik 3, Denis 4 et Emilie 5 est différente de la position Adélie 2, Brigitte 3, Chafik 4, Denis 5 et Emilie 1, alors que dans ces deux configurations, tout le monde a le même voisin. Si on ne numérote pas les places, alors on doit diviser le nombre de possibilités pas 5 et il y a 12 configurations différentes.<br>
Une autre possibilité, pour faire le dénombrement, est de choisir d'abord la place de Denis (5 choix), puis la place d'Emile (il ne reste que 2 choix), puis la place des 3 autres (respectivement, 3,2 et 1 choix). Le nombre de dispositions possibles vaut bien <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-166-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2156" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2157" class="mjx-mrow"><span id="MJXc-Node-2158" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-2159" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-2160" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2161" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-2162" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2163" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-2164" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2165" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2166" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">60</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>=</mo><mn>60</mn></math></span></span><script type="math/tex" id="MathJax-Element-166">5\times 2\times 3\times 2=60</script>.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>18</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Groupe d'étudiants<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1197"> [Signaler une erreur]</a><a id="exo18" href="javascript:ajouteexo(1197,18,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1197&#39;,&#39;imageenonce1197&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1197"></a><div class="inner" id="innerenonce1197">A leur entrée en L1, les étudiants choisissent une langue (anglais ou allemand) et une option (informatique, chimie ou astronomie).
Dans un groupe d'étudiants, 12 étudiants sont inscrits en astronomie, 15 en chimie, 16 étudient l'allemand. Par ailleurs, 8 inscrits en astronomie et 3 inscrits en informatique étudient l'anglais, 6 inscrits en chimie étudient l'allemand.<br>
Indiquer la répartition des étudiants par discipline, ainsi que le nombre total d'étudiants dans le groupe.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1197&#39;,&#39;imageindication1197&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1197"></a><div class="inner" id="innerindication1197" style="display:none;">Réaliser un tableau à double entrée.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1197&#39;,&#39;imagecorrige1197&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1197"></a><div class="inner" id="innercorrige1197" style="display:none;">Ce genre d'exercices se traite très facilement en utilisant un tableau à double entrée dans lequel on inscrit les informations
à notre disposition :
<div class="equation"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/etudiants1.png"></div>
On complète alors le tableau de proche en proche de sorte d'obtenir les bons totaux sur chaque ligne et chaque colonne. Par exemple, sur la première ligne de la colonne "Chimie", on peut facilement inscrire le chiffre 9. On obtient donc
<div class="equation"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/etudiants2.png"></div>
</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>19</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Dans une entreprise...<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1198"> [Signaler une erreur]</a><a id="exo19" href="javascript:ajouteexo(1198,19,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1198&#39;,&#39;imageenonce1198&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1198"></a><div class="inner" id="innerenonce1198">Dans une entreprise, il y a 800 employés. 300 sont des hommes, 352 sont membres d'un syndicat, 424 sont mariés, 188 sont des hommes syndiqués, 166 sont des hommes mariés, 208 sont syndiqués et mariés, 144 sont des hommes mariés syndiqués. Combien y-a-t-il de femmes célibataires non syndiquées?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1198&#39;,&#39;imageindication1198&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1198"></a><div class="inner" id="innerindication1198" style="display:none;">Donner un nom à chacun des ensembles de base, et exprimer les autres en fonction de ceux-ci.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1198&#39;,&#39;imagecorrige1198&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1198"></a><div class="inner" id="innercorrige1198" style="display:none;">Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-167-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2167" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2168" class="mjx-mrow"><span id="MJXc-Node-2169" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math></span></span><script type="math/tex" id="MathJax-Element-167">E</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-168-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2170" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2171" class="mjx-mrow"><span id="MJXc-Node-2172" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math></span></span><script type="math/tex" id="MathJax-Element-168">H</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-169-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2173" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2174" class="mjx-mrow"><span id="MJXc-Node-2175" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-169">M</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-170-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2176" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2177" class="mjx-mrow"><span id="MJXc-Node-2178" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math></span></span><script type="math/tex" id="MathJax-Element-170">S</script> les ensembles constitués respectivement des employés, des employés hommes, des employés mariés, des employés syndiqués. L'énoncé donne :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-171-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;800&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;300&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;352&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;424&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2179" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2180" class="mjx-mrow"><span id="MJXc-Node-2181" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2182" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2183" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2184" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-2185" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2186" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2187" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">800</span></span><span id="MJXc-Node-2188" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2189" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-2190" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2191" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2192" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2193" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2194" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2195" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2196" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">300</span></span><span id="MJXc-Node-2197" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2198" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-2199" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2200" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2201" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2202" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2203" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2204" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2205" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">352</span></span><span id="MJXc-Node-2206" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2207" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-2208" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2209" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2210" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2211" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2212" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2213" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2214" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">424</span></span><span id="MJXc-Node-2215" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mn>800</mn><mo>,</mo><mtext>&nbsp;</mtext><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><mn>300</mn><mo>,</mo><mtext>&nbsp;</mtext><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mn>352</mn><mo>,</mo><mtext>&nbsp;</mtext><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><mn>424</mn><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-171">\card (E)=800,\ \card(H)=300,\ \card(S)=352,\ \card(M)=424,</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-172-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;188&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;166&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;208&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;144.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2216" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2217" class="mjx-mrow"><span id="MJXc-Node-2218" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2219" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2220" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2221" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2222" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2223" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2224" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2225" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2226" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">188</span></span><span id="MJXc-Node-2227" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2228" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-2229" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2230" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2231" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2232" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2233" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2234" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2235" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2236" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2237" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">166</span></span><span id="MJXc-Node-2238" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2239" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-2240" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2241" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2242" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2243" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2244" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2245" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2246" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2247" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2248" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">208</span></span><span id="MJXc-Node-2249" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2250" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2251" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2252" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2253" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2254" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2255" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2256" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2257" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2258" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2259" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2260" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">144.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo>∩</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mn>188</mn><mo>,</mo><mtext>&nbsp;</mtext><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo>∩</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><mn>166</mn><mo>,</mo><mtext>&nbsp;</mtext><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>S</mi><mo>∩</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><mn>208</mn><mo>,</mo><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo>∩</mo><mi>M</mi><mo>∩</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mn>144.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-172">\card(H\cap S)=188,\ \card(H\cap M)=166,\ \card(S\cap M)=208,\card(H\cap M\cap S)=144.</script>
On cherche <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-173-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2261" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2262" class="mjx-mrow"><span id="MJXc-Node-2263" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2264" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2265" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2266" class="mjx-texatom"><span id="MJXc-Node-2267" class="mjx-mrow"><span id="MJXc-Node-2268" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.312em;"><span id="MJXc-Node-2270" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2269" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span></span></span></span></span></span><span id="MJXc-Node-2271" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2272" class="mjx-texatom MJXc-space2"><span id="MJXc-Node-2273" class="mjx-mrow"><span id="MJXc-Node-2274" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.436em;"><span id="MJXc-Node-2276" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2275" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span></span></span></span><span id="MJXc-Node-2277" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2278" class="mjx-texatom MJXc-space2"><span id="MJXc-Node-2279" class="mjx-mrow"><span id="MJXc-Node-2280" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.202em;"><span id="MJXc-Node-2282" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2281" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span></span></span></span></span></span><span id="MJXc-Node-2283" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">¯</mo></mover></mrow><mo>∩</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">¯</mo></mover></mrow><mo>∩</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>S</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-173">\card(\bar H\cap\bar M\cap \bar S)</script>, où <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-174-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2284" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2285" class="mjx-mrow"><span id="MJXc-Node-2286" class="mjx-texatom"><span id="MJXc-Node-2287" class="mjx-mrow"><span id="MJXc-Node-2288" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-2290" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2289" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-174">\bar A</script> désigne le complémentaire de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-175-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2291" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2292" class="mjx-mrow"><span id="MJXc-Node-2293" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-175">A</script> dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-176-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2294" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2295" class="mjx-mrow"><span id="MJXc-Node-2296" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math></span></span><script type="math/tex" id="MathJax-Element-176">E</script>. D'après les lois de Morgan :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-177-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2297" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2298" class="mjx-mrow"><span id="MJXc-Node-2299" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2300" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2301" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2302" class="mjx-texatom"><span id="MJXc-Node-2303" class="mjx-mrow"><span id="MJXc-Node-2304" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.312em;"><span id="MJXc-Node-2306" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2305" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span></span></span></span></span></span><span id="MJXc-Node-2307" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2308" class="mjx-texatom MJXc-space2"><span id="MJXc-Node-2309" class="mjx-mrow"><span id="MJXc-Node-2310" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.436em;"><span id="MJXc-Node-2312" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2311" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span></span></span></span><span id="MJXc-Node-2313" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2314" class="mjx-texatom MJXc-space2"><span id="MJXc-Node-2315" class="mjx-mrow"><span id="MJXc-Node-2316" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.202em;"><span id="MJXc-Node-2318" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2317" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span></span></span></span></span></span><span id="MJXc-Node-2319" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2320" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2321" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2322" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2323" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2324" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em; padding-left: 0.065em;"><span id="MJXc-Node-2331" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.032em 0px -0.171em; letter-spacing: -0.243em;">¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-2325" class="mjx-mrow"><span id="MJXc-Node-2326" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2327" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-2328" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2329" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-2330" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span></span></span></span></span><span id="MJXc-Node-2332" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2333" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">¯</mo></mover></mrow><mo>∩</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">¯</mo></mover></mrow><mo>∩</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>S</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mover><mrow><mi>H</mi><mo>∪</mo><mi>M</mi><mo>∪</mo><mi>S</mi></mrow><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-177">\card(\bar H\cap\bar M\cap \bar S)=\card(\overline{H\cup M\cup S}).</script>
On applique la formule du crible de Poincaré :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-178-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2334" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2335" class="mjx-mrow"><span id="MJXc-Node-2336" class="mjx-mtable" style="vertical-align: -0.975em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-2337" class="mjx-mtr" style="height: 1.225em;"><span id="MJXc-Node-2338" class="mjx-mtd" style="padding: 0px; text-align: right; width: 7.477em;"><span id="MJXc-Node-2339" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-2340" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2341" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2342" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2343" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2344" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-2345" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2346" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-2347" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2348" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2349" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-2350" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-2351" class="mjx-mi"></span><span id="MJXc-Node-2352" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2353" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 33.219em;"><span id="MJXc-Node-2354" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-2355" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2356" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2357" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2358" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2359" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2360" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2361" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2362" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2363" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2364" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2365" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2366" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2367" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2368" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2369" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2370" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2371" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2372" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2373" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2374" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2375" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2376" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2377" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2378" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2379" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2380" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2381" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2382" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2384" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2385" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2386" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2387" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2388" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2389" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2390" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2391" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2392" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2393" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2394" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2395" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-2396" class="mjx-mtr" style="height: 1.225em;"><span id="MJXc-Node-2397" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-2398" class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2399" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-2400" class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2401" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-2402" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-2403" class="mjx-mi"></span><span id="MJXc-Node-2404" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-2405" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2406" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2407" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2408" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2409" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2410" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2411" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2412" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2413" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2414" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo>∪</mo><mi>M</mi><mo>∪</mo><mi>S</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>+</mo><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>+</mo><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>−</mo><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo>∩</mo><mi>M</mi><mo stretchy="false">)</mo><mo>−</mo><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo>∩</mo><mi>S</mi><mo stretchy="false">)</mo><mo>−</mo><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>M</mi><mo>∩</mo><mi>S</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd></mtd><mtd><mi></mi><mo>+</mo><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo>∩</mo><mi>M</mi><mo>∩</mo><mi>S</mi><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-178">\begin{eqnarray*}
\card(H\cup M\cup S)&=&\card(H)+\card(M)+\card(S)-\card(H\cap M)-\card(H\cap S)-\card(M\cap S)\\
&&+\card(H\cap M\cap S).
\end{eqnarray*}</script>
On en déduit :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-179-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;658&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2415" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2416" class="mjx-mrow"><span id="MJXc-Node-2417" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2418" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2419" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2420" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-2421" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-2422" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-2423" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-2424" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span><span id="MJXc-Node-2425" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2426" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2427" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">658</span></span><span id="MJXc-Node-2428" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>H</mi><mo>∪</mo><mi>M</mi><mo>∪</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mn>658</mn><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-179">\card(H\cup M\cup S)=658,</script>
et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-180-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;card&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;800&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;658&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;142.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2429" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2430" class="mjx-mrow"><span id="MJXc-Node-2431" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">card</span></span><span id="MJXc-Node-2432" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-2433" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2434" class="mjx-texatom"><span id="MJXc-Node-2435" class="mjx-mrow"><span id="MJXc-Node-2436" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.312em;"><span id="MJXc-Node-2438" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2437" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span></span></span></span></span></span><span id="MJXc-Node-2439" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2440" class="mjx-texatom MJXc-space2"><span id="MJXc-Node-2441" class="mjx-mrow"><span id="MJXc-Node-2442" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.436em;"><span id="MJXc-Node-2444" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2443" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span></span></span></span><span id="MJXc-Node-2445" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2446" class="mjx-texatom MJXc-space2"><span id="MJXc-Node-2447" class="mjx-mrow"><span id="MJXc-Node-2448" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.202em;"><span id="MJXc-Node-2450" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-2449" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.032em;">S</span></span></span></span></span></span></span><span id="MJXc-Node-2451" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2452" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2453" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">800</span></span><span id="MJXc-Node-2454" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2455" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">658</span></span><span id="MJXc-Node-2456" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2457" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">142.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>card</mi><mo>⁡</mo><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>H</mi><mo stretchy="false">¯</mo></mover></mrow><mo>∩</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>M</mi><mo stretchy="false">¯</mo></mover></mrow><mo>∩</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>S</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>800</mn><mo>−</mo><mn>658</mn><mo>=</mo><mn>142.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-180">\card(\bar H\cap\bar M\cap \bar S)=800-658=142.</script>
Il y a donc 142 femmes célibataires non syndiquées.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>20</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Indépendance et contexte<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1234"> [Signaler une erreur]</a><a id="exo20" href="javascript:ajouteexo(1234,20,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1234&#39;,&#39;imageenonce1234&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1234"></a><div class="inner" id="innerenonce1234"><ol class="enumeratechiffre">
<li> Une urne contient 12 boules numérotées de 1 à 12. On en tire une hasard, et on considère les événements
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-181-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;quot;tirage d&#39;un nombre pair&#39;&#39;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2458" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2459" class="mjx-mrow"><span id="MJXc-Node-2460" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-2461" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-2462" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-2463" class="mjx-mrow"><span id="MJXc-Node-2464" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.522em;">"tirage d'un nombre pair''</span></span></span></span><span id="MJXc-Node-2465" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mtext>"tirage d'un nombre pair''</mtext></mrow><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-181">A=\textrm{"tirage d'un nombre pair''},</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-182-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;quot;tirage d&#39;un multiple de 3&#39;&#39;&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2466" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2467" class="mjx-mrow"><span id="MJXc-Node-2468" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span><span id="MJXc-Node-2469" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-2470" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-2471" class="mjx-mrow"><span id="MJXc-Node-2472" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.522em;">"tirage d'un multiple de 3''</span></span></span></span><span id="MJXc-Node-2473" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>B</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mtext>"tirage d'un multiple de 3''</mtext></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-182">B=\textrm{"tirage d'un multiple de 3''}.</script>
Les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-183-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2474" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2475" class="mjx-mrow"><span id="MJXc-Node-2476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-183">A</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-184-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2477" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2478" class="mjx-mrow"><span id="MJXc-Node-2479" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-184">B</script> sont-ils indépendants?
</li><li> Reprendre la question avec une urne contenant 13 boules.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1234&#39;,&#39;imageindication1234&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1234"></a><div class="inner" id="innerindication1234" style="display:none;">Écrire en termes d'ensembles les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-185-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2480" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2481" class="mjx-mrow"><span id="MJXc-Node-2482" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2483" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2484" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-2485" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mtext>&nbsp;</mtext><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-185">A,\ B</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-186-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2486" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2487" class="mjx-mrow"><span id="MJXc-Node-2488" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2489" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2490" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∩</mo><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-186">A\cap B</script>. Puis calculer les probabilités.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1234&#39;,&#39;imagecorrige1234&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1234"></a><div class="inner" id="innercorrige1234" style="display:none;"><ol class="enumeratechiffre">
<li> On a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-187-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2491" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2492" class="mjx-mrow"><span id="MJXc-Node-2493" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2494" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2495" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-2496" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2497" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2498" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2499" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2500" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-2501" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2502" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-2503" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2504" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-2505" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2506" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-2507" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>10</mn><mo>,</mo><mn>12</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-187">A=\{2,4,6,8,10,12\}</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-188-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2508" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2509" class="mjx-mrow"><span id="MJXc-Node-2510" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2511" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2512" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-2513" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-2514" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2515" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-2516" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2517" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-2518" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2519" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-2520" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>B</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>12</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-188">B=\{3,6,9,12\}</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-189-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2521" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2522" class="mjx-mrow"><span id="MJXc-Node-2523" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2524" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2525" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2526" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2527" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-2528" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-2529" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2530" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">12</span></span><span id="MJXc-Node-2531" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span><span id="MJXc-Node-2532" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>6</mn><mo>,</mo><mn>12</mn><mo fence="false" stretchy="false">}</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-189">A\cap B=\{6,12\}.</script>
On a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-190-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2533" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2534" class="mjx-mrow"><span id="MJXc-Node-2535" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2536" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2537" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2538" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2539" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2540" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2541" class="mjx-texatom"><span id="MJXc-Node-2542" class="mjx-mrow"><span id="MJXc-Node-2543" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2544" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-190">P(A)=1/2</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-191-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2545" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2546" class="mjx-mrow"><span id="MJXc-Node-2547" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2548" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2549" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2550" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2551" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2552" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2553" class="mjx-texatom"><span id="MJXc-Node-2554" class="mjx-mrow"><span id="MJXc-Node-2555" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2556" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-191">P(B)=1/3</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-192-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2557" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2558" class="mjx-mrow"><span id="MJXc-Node-2559" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2560" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2561" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2562" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2563" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2564" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2565" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2566" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2567" class="mjx-texatom"><span id="MJXc-Node-2568" class="mjx-mrow"><span id="MJXc-Node-2569" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2570" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-2571" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2572" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2573" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2574" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2575" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2576" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2577" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2578" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2579" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2580" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-192">P(A\cap B)=1/6=P(A)P(B).</script> Les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-193-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2581" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2582" class="mjx-mrow"><span id="MJXc-Node-2583" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-193">A</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-194-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2584" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2585" class="mjx-mrow"><span id="MJXc-Node-2586" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-194">B</script> sont indépendants.
</li><li> Les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-195-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2587" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2588" class="mjx-mrow"><span id="MJXc-Node-2589" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-195">A</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-196-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2590" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2591" class="mjx-mrow"><span id="MJXc-Node-2592" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-196">B</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-197-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2593" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2594" class="mjx-mrow"><span id="MJXc-Node-2595" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2596" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2597" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∩</mo><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-197">A\cap B</script> s'écrivent encore exactement de la même façon. Mais cette fois, on a : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-198-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2598" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2599" class="mjx-mrow"><span id="MJXc-Node-2600" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2601" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2602" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2603" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2604" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2605" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-2606" class="mjx-texatom"><span id="MJXc-Node-2607" class="mjx-mrow"><span id="MJXc-Node-2608" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2609" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>13</mn></math></span></span><script type="math/tex" id="MathJax-Element-198">P(A)=6/13</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-199-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2610" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2611" class="mjx-mrow"><span id="MJXc-Node-2612" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2613" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2614" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2615" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2616" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2617" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-2618" class="mjx-texatom"><span id="MJXc-Node-2619" class="mjx-mrow"><span id="MJXc-Node-2620" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2621" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>13</mn></math></span></span><script type="math/tex" id="MathJax-Element-199">P(B)=4/13</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-200-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;mo&gt;&amp;#x2260;&lt;/mo&gt;&lt;mn&gt;24&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;169&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2622" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2623" class="mjx-mrow"><span id="MJXc-Node-2624" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2625" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2626" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2627" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2628" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2629" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2630" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2631" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-2632" class="mjx-texatom"><span id="MJXc-Node-2633" class="mjx-mrow"><span id="MJXc-Node-2634" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2635" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">13</span></span><span id="MJXc-Node-2636" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.528em;">≠</span></span><span id="MJXc-Node-2637" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">24</span></span><span id="MJXc-Node-2638" class="mjx-texatom"><span id="MJXc-Node-2639" class="mjx-mrow"><span id="MJXc-Node-2640" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-2641" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">169</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>13</mn><mo>≠</mo><mn>24</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>169</mn></math></span></span><script type="math/tex" id="MathJax-Element-200">P(A\cap B)=2/13\neq 24/169</script>. Les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-201-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2642" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2643" class="mjx-mrow"><span id="MJXc-Node-2644" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-201">A</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-202-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2645" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2646" class="mjx-mrow"><span id="MJXc-Node-2647" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-202">B</script> ne sont pas indépendants. C'est conforme à l'intuition. Il n'y a plus la même répartition de boules paires et de boules impaires, et dans les multiples de 3 compris entre 1 et 13, la répartition des nombres pairs et impairs est restée inchangée.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>21</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Indépendance deux à deux et indépendance mutuelle<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1235"> [Signaler une erreur]</a><a id="exo21" href="javascript:ajouteexo(1235,21,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1235&#39;,&#39;imageenonce1235&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1235"></a><div class="inner" id="innerenonce1235">Votre voisine a deux enfants dont vous ignorez le sexe. On considère les trois événement suivants :
<ul class="rien">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-203-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2648" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2649" class="mjx-mrow"><span id="MJXc-Node-2650" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-203">A</script>="les deux enfants sont de sexes différents"
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-204-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2651" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2652" class="mjx-mrow"><span id="MJXc-Node-2653" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-204">B</script>="l'ainé est une fille"
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-205-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2654" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2655" class="mjx-mrow"><span id="MJXc-Node-2656" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-205">C</script>="le cadet est un garçon".
</li></ul>
Montrer que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-206-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2657" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2658" class="mjx-mrow"><span id="MJXc-Node-2659" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-206">A</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-207-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2660" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2661" class="mjx-mrow"><span id="MJXc-Node-2662" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-207">B</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-208-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2663" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2664" class="mjx-mrow"><span id="MJXc-Node-2665" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-208">C</script> sont deux à deux indépendants, mais ne sont pas mutuellement indépendants. On suppose que la probabilité à la naissance d'avoir une fille (respectivement un garçon) est égale à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-209-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2666" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2667" class="mjx-mrow"><span id="MJXc-Node-2668" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-2669" class="mjx-texatom"><span id="MJXc-Node-2670" class="mjx-mrow"><span id="MJXc-Node-2671" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">/</span></span></span></span><span id="MJXc-Node-2672" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-209">1/2</script>.</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1235&#39;,&#39;imageindication1235&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1235"></a><div class="inner" id="innerindication1235" style="display:none;">Calculer la probabilité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-210-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2673" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2674" class="mjx-mrow"><span id="MJXc-Node-2675" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-210">A</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-211-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2676" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2677" class="mjx-mrow"><span id="MJXc-Node-2678" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-211">B</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-212-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2679" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2680" class="mjx-mrow"><span id="MJXc-Node-2681" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-212">C</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-213-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2682" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2683" class="mjx-mrow"><span id="MJXc-Node-2684" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2685" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2686" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∩</mo><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-213">A\cap B</script>,...</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1235&#39;,&#39;imagecorrige1235&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1235"></a><div class="inner" id="innercorrige1235" style="display:none;">Clairement, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-214-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2687" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2688" class="mjx-mrow"><span id="MJXc-Node-2689" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2690" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2691" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2692" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2693" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2694" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2695" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2696" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-2697" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2698" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2699" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-2700" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-2701" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-214">P(B)=P(C)=\frac 12</script>. Les quatre possibilités pour les deux enfants, supposées équiprobables, sont (F,G), (F,F), (G,G), (G,F). On en déduit que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-215-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2702" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2703" class="mjx-mrow"><span id="MJXc-Node-2704" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2705" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2706" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2707" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2708" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2709" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-2710" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-2711" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-215">P(A)=\frac 12</script>.<br>
Ensuite <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-216-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2712" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2713" class="mjx-mrow"><span id="MJXc-Node-2714" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2715" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2716" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∩</mo><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-216">A\cap B</script> correspond à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-217-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2717" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2718" class="mjx-mrow"><span id="MJXc-Node-2719" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2720" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span><span id="MJXc-Node-2721" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2722" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">G</span></span><span id="MJXc-Node-2723" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-217">(F,G)</script> et donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-218-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2724" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2725" class="mjx-mrow"><span id="MJXc-Node-2726" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2727" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2728" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2729" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2730" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2731" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2732" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2733" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-2734" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.622em;"><span id="MJXc-Node-2735" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.372em; vertical-align: -0.44em;"></span></span><span id="MJXc-Node-2736" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2737" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2738" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2739" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2740" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2741" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2742" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2743" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2744" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-218">P(A\cap B)=\frac 14=P(A)P(B)</script>. On en déduit que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-219-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2745" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2746" class="mjx-mrow"><span id="MJXc-Node-2747" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-219">A</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-220-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2748" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2749" class="mjx-mrow"><span id="MJXc-Node-2750" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-220">B</script> sont indépendants. On prouve de la même façon que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-221-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2751" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2752" class="mjx-mrow"><span id="MJXc-Node-2753" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-221">A</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-222-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2754" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2755" class="mjx-mrow"><span id="MJXc-Node-2756" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-222">C</script> sont indépendants, et que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-223-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2757" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2758" class="mjx-mrow"><span id="MJXc-Node-2759" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-223">B</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-224-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2760" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2761" class="mjx-mrow"><span id="MJXc-Node-2762" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-224">C</script> sont indépendants.<br>
Enfin, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-225-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2763" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2764" class="mjx-mrow"><span id="MJXc-Node-2765" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2766" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2767" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2768" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2769" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-2770" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2771" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2772" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2773" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∩</mo><mi>B</mi><mo>∩</mo><mi>C</mi><mo>=</mo><mi>B</mi><mo>∩</mo><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-225">A\cap B\cap C=B\cap C</script> et donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-226-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2260;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2774" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2775" class="mjx-mrow"><span id="MJXc-Node-2776" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2777" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2778" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2779" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2780" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2781" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2782" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-2783" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2784" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2785" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-2786" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.622em;"><span id="MJXc-Node-2787" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.372em; vertical-align: -0.44em;"></span></span><span id="MJXc-Node-2788" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.528em;">≠</span></span><span id="MJXc-Node-2789" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2790" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2791" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-2792" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2793" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2794" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2795" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-2796" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2797" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2798" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2799" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-2800" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>≠</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-226">P(A\cap B\cap C)=\frac 14\neq P(A)P(B)P(C)</script>. Les trois événements ne sont pas mutuellement indépendants.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>22</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Probabilité d'une réunion et indépendance<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1238"> [Signaler une erreur]</a><a id="exo22" href="javascript:ajouteexo(1238,22,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1238&#39;,&#39;imageenonce1238&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1238"></a><div class="inner" id="innerenonce1238">Soient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-227-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2801" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2802" class="mjx-mrow"><span id="MJXc-Node-2803" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2804" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2805" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span id="MJXc-Node-2806" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-2807" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">…</span></span><span id="MJXc-Node-2808" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-2809" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-2810" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2811" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>A</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-227">A_1, \dots,A_n</script> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-228-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2812" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2813" class="mjx-mrow"><span id="MJXc-Node-2814" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-228">n</script> événements d’un espace probabilisé <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-229-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x03A9;&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2815" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2816" class="mjx-mrow"><span id="MJXc-Node-2817" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-2818" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">Ω</span></span><span id="MJXc-Node-2819" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-2820" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2821" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-229">(\Omega,P)</script>. On les suppose mutuellement indépendants et de probabilités respectives <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-230-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2822" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2823" class="mjx-mrow"><span id="MJXc-Node-2824" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2825" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2826" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span></span></span><span id="MJXc-Node-2827" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-2828" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2829" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-2830" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2831" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2832" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span></span></span><span id="MJXc-Node-2833" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>i</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-230">p_i = P(A_i)</script>.
Donner une expression simple de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-231-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2834" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2835" class="mjx-mrow"><span id="MJXc-Node-2836" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2837" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-2838" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2839" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2840" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span id="MJXc-Node-2841" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∪</span></span><span id="MJXc-Node-2842" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.328em;">⋯</span></span><span id="MJXc-Node-2843" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∪</span></span><span id="MJXc-Node-2844" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-2845" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2846" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-2847" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-231">P(A_1\cup\dots\cup A_n)</script> en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-232-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2848" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2849" class="mjx-mrow"><span id="MJXc-Node-2850" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2851" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2852" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span id="MJXc-Node-2853" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-2854" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">…</span></span><span id="MJXc-Node-2855" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-2856" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-2857" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2858" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>p</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-232">p_1,\dots,p_n</script>.<br>
Application : on suppose qu'une personne est soumise à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-233-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2859" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2860" class="mjx-mrow"><span id="MJXc-Node-2861" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-233">n</script> expériences indépendantes les unes des autres et qu'à chaque expérience, elle ait une probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-234-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2862" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2863" class="mjx-mrow"><span id="MJXc-Node-2864" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-234">p</script> d'avoir un accident. Quelle est la probabilité qu'elle ait au moins un accident?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1238&#39;,&#39;imageindication1238&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1238"></a><div class="inner" id="innerindication1238" style="display:none;">Passer par le complémentaire.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1238&#39;,&#39;imagecorrige1238&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1238"></a><div class="inner" id="innercorrige1238" style="display:none;">Si les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-235-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2865" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2866" class="mjx-mrow"><span id="MJXc-Node-2867" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2868" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2869" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-2870" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2871" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-2872" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2873" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-2874" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2875" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>A</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-235">A_1,\dots,A_n</script> sont mutuellement indépendants, les événements complémentaires
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-236-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2876" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2877" class="mjx-mrow"><span id="MJXc-Node-2878" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-2882" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.037em 0px -0.176em; letter-spacing: -0.253em;">¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-2879" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2880" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2881" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-2883" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2884" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-2885" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-2886" class="mjx-munderover MJXc-space1"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-2890" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.052em 0px -0.191em; letter-spacing: -0.282em;">¯¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-2887" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2888" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2889" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>A</mi><mn>1</mn></msub><mo accent="false">¯</mo></mover><mo>,</mo><mo>…</mo><mo>,</mo><mover><msub><mi>A</mi><mi>n</mi></msub><mo accent="false">¯</mo></mover></math></span></span><script type="math/tex" id="MathJax-Element-236">\overline{A_1},\dots,\overline{A_n}</script> le sont aussi. On en déduit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-237-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo&gt;&amp;#x22EF;&lt;/mo&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x220F;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/munderover&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x220F;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/munderover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-2891" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2892" class="mjx-mrow"><span id="MJXc-Node-2893" class="mjx-mtable" style="vertical-align: -3.603em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-2894" class="mjx-mtr" style="height: 1.534em;"><span id="MJXc-Node-2895" class="mjx-mtd" style="padding: 0px; text-align: right; width: 7.272em;"><span id="MJXc-Node-2896" class="mjx-mrow" style="margin-top: 0.109em;"><span id="MJXc-Node-2897" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2898" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2899" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2900" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2901" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-2902" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-2903" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-2904" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-2905" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-2906" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2907" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-2908" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2909" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-2910" class="mjx-mrow" style="margin-top: 0.109em;"><span id="MJXc-Node-2911" class="mjx-mi"></span><span id="MJXc-Node-2912" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2913" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 8.773em;"><span id="MJXc-Node-2914" class="mjx-mrow"><span id="MJXc-Node-2915" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2916" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2917" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2918" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2919" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-2923" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.037em 0px -0.176em; letter-spacing: -0.253em;">¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-2920" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2921" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2922" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-2924" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2925" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋯</span></span><span id="MJXc-Node-2926" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-2927" class="mjx-munderover MJXc-space2"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-2931" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.052em 0px -0.191em; letter-spacing: -0.282em;">¯¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-2928" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2929" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2930" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span><span id="MJXc-Node-2932" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-2933" class="mjx-mtr" style="height: 3.161em;"><span id="MJXc-Node-2934" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-2935" class="mjx-mrow" style="margin-top: 0.605em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2936" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-2937" class="mjx-mrow" style="margin-top: 0.605em;"><span id="MJXc-Node-2938" class="mjx-mi"></span><span id="MJXc-Node-2939" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2940" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-2941" class="mjx-mrow"><span id="MJXc-Node-2942" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2943" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2944" class="mjx-munderover MJXc-space2"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 0.604em;"><span id="MJXc-Node-2951" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-op"><span id="MJXc-Node-2945" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∏</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.092em;"><span id="MJXc-Node-2946" class="mjx-texatom" style=""><span id="MJXc-Node-2947" class="mjx-mrow"><span id="MJXc-Node-2948" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-2949" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2950" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-2952" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-2953" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2954" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-2958" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.056em 0px -0.195em; letter-spacing: -0.292em;">¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-2955" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2956" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2957" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span></span><span id="MJXc-Node-2959" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-2960" class="mjx-mtr" style="height: 3.011em;"><span id="MJXc-Node-2961" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-2962" class="mjx-mrow" style="margin-top: 0.605em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2963" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-2964" class="mjx-mrow" style="margin-top: 0.605em;"><span id="MJXc-Node-2965" class="mjx-mi"></span><span id="MJXc-Node-2966" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-2967" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-2968" class="mjx-mrow"><span id="MJXc-Node-2969" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2970" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2971" class="mjx-munderover MJXc-space2"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 0.604em;"><span id="MJXc-Node-2978" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-op"><span id="MJXc-Node-2972" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.703em; padding-bottom: 0.703em;">∏</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.092em;"><span id="MJXc-Node-2973" class="mjx-texatom" style=""><span id="MJXc-Node-2974" class="mjx-mrow"><span id="MJXc-Node-2975" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span><span id="MJXc-Node-2976" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-2977" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-2979" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2980" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2981" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2982" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-2983" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-2984" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span id="MJXc-Node-2985" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-2986" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mover><msub><mi>A</mi><mn>1</mn></msub><mo accent="false">¯</mo></mover><mo>∩</mo><mo>⋯</mo><mo>∩</mo><mover><msub><mi>A</mi><mi>n</mi></msub><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mn>1</mn><mo>−</mo><munderover><mo>∏</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><mover><msub><mi>A</mi><mi>i</mi></msub><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mn>1</mn><mo>−</mo><munderover><mo>∏</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>p</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-237">\begin{eqnarray*}
P(A_1\cup\dots\cup A_n)&=&1-P(\overline{A_1}\cap\dots\cap \overline{A_n})\\
&=&1-\prod_{i=1}^n P(\overline{A_i})\\
&=&1-\prod_{i=1}^n (1-p_i).
\end{eqnarray*}</script>
Dans le cas particulier proposé, la probabilité d'avoir au moins un accident vaut donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-238-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-2987" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2988" class="mjx-mrow"><span id="MJXc-Node-2989" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2990" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2991" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-2992" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-2993" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-2994" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-2995" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-2996" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-2997" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mi>n</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-238">1-(1-p)^n</script>.
</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>23</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Circuit électrique<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1237"> [Signaler une erreur]</a><a id="exo23" href="javascript:ajouteexo(1237,23,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1237&#39;,&#39;imageenonce1237&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1237"></a><div class="inner" id="innerenonce1237"><ol class="enumeratechiffre">
<li> Soient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-239-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-2998" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2999" class="mjx-mrow"><span id="MJXc-Node-3000" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-3001" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-3002" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-3003" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span><span id="MJXc-Node-3004" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-3005" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-3006" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mtext>&nbsp;</mtext><mi>B</mi><mo>,</mo><mtext>&nbsp;</mtext><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-239">A,\ B,\ C</script> trois événements. Montrer que :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-240-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3007" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3008" class="mjx-mrow"><span id="MJXc-Node-3009" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3010" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3011" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-3012" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∪</span></span><span id="MJXc-Node-3013" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span><span id="MJXc-Node-3014" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∪</span></span><span id="MJXc-Node-3015" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3016" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-3017" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-3018" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3019" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3020" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-3021" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-3022" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-3023" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3024" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3025" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span><span id="MJXc-Node-3026" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-3027" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-3028" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3029" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3030" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3031" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-3032" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-3033" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3034" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3035" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-3036" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∩</span></span><span id="MJXc-Node-3037" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span><span id="MJXc-Node-3038" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-3039" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-3040" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3041" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3042" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-3043" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∩</span></span><span id="MJXc-Node-3044" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3045" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-3046" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-3047" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3048" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3049" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span><span id="MJXc-Node-3050" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∩</span></span><span id="MJXc-Node-3051" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3052" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-3053" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-3054" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3055" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-3056" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-3057" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∩</span></span><span id="MJXc-Node-3058" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span><span id="MJXc-Node-3059" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">∩</span></span><span id="MJXc-Node-3060" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3061" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-3062" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo>∪</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-240">P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C).</script>
</li><li> On dispose de 3 composants électriques <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-241-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3063" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3064" class="mjx-mrow"><span id="MJXc-Node-3065" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-3066" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3067" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span id="MJXc-Node-3068" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-3069" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-3070" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-3071" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3072" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><mtext>&nbsp;</mtext><msub><mi>C</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-241">C_1,\ C_2</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-242-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3073" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3074" class="mjx-mrow"><span id="MJXc-Node-3075" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-3076" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3077" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-242">C_3</script> dont la probabilité de fonctionnement est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-243-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3078" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3079" class="mjx-mrow"><span id="MJXc-Node-3080" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3081" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3082" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-243">p_i</script>, et de fonctionnement totalement indépendant les uns des autres. Donner la probabilité de fonctionnement du circuit
<ol class="enumeratechiffre">
<li> si les composants sont disposés en série.
</li><li> si les composants sont disposés en parallèle.
</li><li> si le circuit est mixte : <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-244-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3083" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3084" class="mjx-mrow"><span id="MJXc-Node-3085" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-3086" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3087" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-244">C_1</script> est disposé en série avec le sous-circuit constitué de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-245-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3088" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3089" class="mjx-mrow"><span id="MJXc-Node-3090" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-3091" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3092" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-245">C_2</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-246-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3093" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3094" class="mjx-mrow"><span id="MJXc-Node-3095" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-3096" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3097" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-246">C_3</script> en parallèle.
</li></ol>
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1237&#39;,&#39;imageindication1237&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1237"></a><div class="inner" id="innerindication1237" style="display:none;"><ol class="enumeratechiffre">
<li> Appliquer trois fois la formule <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-247-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3098" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3099" class="mjx-mrow"><span id="MJXc-Node-3100" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3101" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3102" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span><span id="MJXc-Node-3103" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3104" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">G</span></span><span id="MJXc-Node-3105" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3106" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3107" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span id="MJXc-Node-3108" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span id="MJXc-Node-3109" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo>∪</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mo>.</mo><mo>.</mo><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-247">P(F\cup G)=...</script>.
</li><li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-248-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3110" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3111" class="mjx-mrow"><span id="MJXc-Node-3112" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3113" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3114" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-248">F_i</script> l'événement : ``le circuit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-249-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3115" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3116" class="mjx-mrow"><span id="MJXc-Node-3117" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-3118" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3119" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-249">C_i</script> fonctionne''. Il faut calculer pour le premier cas <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-250-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3120" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3121" class="mjx-mrow"><span id="MJXc-Node-3122" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3123" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3124" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3125" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3126" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3127" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3128" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3129" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3130" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3131" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3132" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3133" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3134" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3135" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∩</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-250">P(F_1\cap F_2\cap F_3)</script>, pour le second <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-251-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3136" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3137" class="mjx-mrow"><span id="MJXc-Node-3138" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3139" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3140" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3141" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3142" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3143" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3144" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3145" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3146" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3147" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3148" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3149" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3150" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3151" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-251">P(F_1\cup F_2\cup F_3)</script>, et pour le troisième <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-252-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3152" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3153" class="mjx-mrow"><span id="MJXc-Node-3154" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3155" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3156" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3157" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3158" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3159" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3160" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3161" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3162" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3163" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3164" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3165" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3166" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3167" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3168" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3169" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∩</mo><mo stretchy="false">(</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-252">P(F_1\cap(F_2\cup F_3))</script>.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1237&#39;,&#39;imagecorrige1237&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1237"></a><div class="inner" id="innercorrige1237" style="display:none;"><ol class="enumeratechiffre">
<li> On procède en deux temps. D'une part :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-253-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3170" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3171" class="mjx-mrow"><span id="MJXc-Node-3172" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3173" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-3174" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3175" class="mjx-mrow"><span id="MJXc-Node-3176" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3177" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3178" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3179" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3180" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3181" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3182" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span id="MJXc-Node-3183" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span id="MJXc-Node-3184" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3185" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3186" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3187" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3188" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3189" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3190" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3191" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3192" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3193" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3194" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3195" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3196" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3197" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3198" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3199" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3200" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3201" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3202" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3203" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3204" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3205" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3206" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3207" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mrow><mo>(</mo><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>C</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-253">P\left((A\cup B)\cup C\right)=P(A\cup B)+P(C)-P((A\cup B)\cap C).</script>
Mais,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-254-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3208" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3209" class="mjx-mrow"><span id="MJXc-Node-3210" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3211" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3212" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3213" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3214" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3215" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3216" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3217" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3218" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3219" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3220" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3221" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3222" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3223" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3224" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3225" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3226" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3227" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3228" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3229" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3230" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3231" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3232" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3233" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3234" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3235" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3236" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3237" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3238" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3239" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3240" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3241" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3242" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3243" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3244" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3245" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3246" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3247" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3248" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3249" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3250" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3251" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3252" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3253" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3254" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3255" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3256" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-3257" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3258" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mi>B</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>∩</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-254">P((A\cup B)\cap C)=P( (A\cap C)\cup (B\cap C))=P(A\cap C)+P(B\cap C)-P(A\cap B\cap C),</script>
et
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-255-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3259" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3260" class="mjx-mrow"><span id="MJXc-Node-3261" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3262" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3263" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3264" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3265" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3266" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3267" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3268" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3269" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3270" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3271" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3272" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3273" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3274" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3275" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3276" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3277" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3278" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3279" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3280" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3281" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3282" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3283" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3284" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-255">P(A\cup B)=P(A)+P(B)-P(A\cap B).</script>
On appelle aussi ceci la formule du crible de Poincaré, elle se généralise avec plusieurs événements par récurrence.
</li><li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-256-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3285" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3286" class="mjx-mrow"><span id="MJXc-Node-3287" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3288" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3289" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-256">F_i</script> l'événement : ``le composant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-257-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3290" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3291" class="mjx-mrow"><span id="MJXc-Node-3292" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-3293" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3294" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-257">C_i</script> fonctionne''. Par hypothèse, les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-258-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3295" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3296" class="mjx-mrow"><span id="MJXc-Node-3297" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3298" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3299" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-258">F_i</script> sont mutuellement indépendants. Il faut calculer pour le premier cas <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-259-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3300" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3301" class="mjx-mrow"><span id="MJXc-Node-3302" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3303" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3304" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3305" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3306" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3307" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3308" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3309" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3310" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3311" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3312" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3313" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3314" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3315" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∩</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-259">P(F_1\cap F_2\cap F_3)</script> (le circuit formé par les trois composants disposés en série fonctionne si et seulement si les trois composants fonctionnent), pour le second <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-260-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3316" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3317" class="mjx-mrow"><span id="MJXc-Node-3318" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3319" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3320" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3321" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3322" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3323" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3324" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3325" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3326" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3327" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3328" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3329" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3330" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3331" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-260">P(F_1\cup F_2\cup F_3)</script> (le circuit formé par les trois composants disposés en parallèle fonctionne si et seulement si un des trois composants fonctionne), et pour le troisième <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-261-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3332" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3333" class="mjx-mrow"><span id="MJXc-Node-3334" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3335" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3336" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3337" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3338" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3339" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3340" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3341" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3342" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3343" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3344" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3345" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3346" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3347" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3348" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3349" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∩</mo><mo stretchy="false">(</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-261">P(F_1\cap(F_2\cup F_3))</script>. On a :
<ol class="enumeratechiffre">
<li> Par indépendance des événements :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-262-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3350" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3351" class="mjx-mrow"><span id="MJXc-Node-3352" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3353" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3354" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3355" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3356" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3357" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3358" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3359" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3360" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3361" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3362" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3363" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3364" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3365" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3366" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3367" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3368" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3369" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3370" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3371" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3372" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3373" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3374" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3375" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3376" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∩</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>2</mn></msub><msub><mi>p</mi><mn>3</mn></msub><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-262">P(F_1\cap F_2\cap F_3)=p_1p_2p_3.</script>
</li><li> D'après la formule précédente, et par indépendance des événements :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-263-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3377" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3378" class="mjx-mrow"><span id="MJXc-Node-3379" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3380" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3381" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3382" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3383" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3384" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3385" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3386" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3387" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3388" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3389" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3390" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3391" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3392" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3393" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3394" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3395" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3396" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3397" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3398" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3399" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3400" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3401" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3402" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3403" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3404" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3405" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3406" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3407" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3408" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3409" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3410" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3411" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3412" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3413" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3414" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3415" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3416" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3417" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3418" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3419" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3420" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3421" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3422" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3423" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3424" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3425" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3426" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3427" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3428" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3429" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3430" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3431" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3432" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3433" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3434" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3435" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3436" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><mo>+</mo><msub><mi>p</mi><mn>2</mn></msub><mo>+</mo><msub><mi>p</mi><mn>3</mn></msub><mo>−</mo><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>2</mn></msub><mo>−</mo><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>3</mn></msub><mo>−</mo><msub><mi>p</mi><mn>2</mn></msub><msub><mi>p</mi><mn>3</mn></msub><mo>+</mo><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>2</mn></msub><msub><mi>p</mi><mn>3</mn></msub><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-263">P(F_1\cup F_2\cup F_3)=p_1+p_2+p_3-p_1p_2-p_1p_3-p_2p_3+p_1p_2p_3.</script>
</li><li> L'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-264-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3437" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3438" class="mjx-mrow"><span id="MJXc-Node-3439" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3440" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3441" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3442" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3443" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3444" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3445" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>2</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-264">F_2\cup F_3</script> est indépendant de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-265-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3446" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3447" class="mjx-mrow"><span id="MJXc-Node-3448" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3449" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3450" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-265">F_1</script>. On a donc :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-266-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3451" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3452" class="mjx-mrow"><span id="MJXc-Node-3453" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3454" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3455" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3456" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3457" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3458" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3459" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3460" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3461" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3462" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3463" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3464" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3465" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3466" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3467" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3468" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3469" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3470" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3471" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3472" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3473" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3474" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3475" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3477" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3478" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3479" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3480" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3481" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3482" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3483" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3484" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3485" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3486" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3487" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3488" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3489" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3490" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3491" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3492" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3493" class="mjx-texatom"><span id="MJXc-Node-3494" class="mjx-mrow"><span id="MJXc-Node-3495" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span><span id="MJXc-Node-3496" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3497" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3498" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3499" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3500" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3501" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3502" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3503" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3504" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3505" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3506" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3507" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3508" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3509" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3510" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3511" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3512" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3513" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3514" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3515" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3516" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3517" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3518" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3519" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3520" class="mjx-texatom"><span id="MJXc-Node-3521" class="mjx-mrow"><span id="MJXc-Node-3522" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∩</mo><mo stretchy="false">(</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∩</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-266">P(F_1\cap(F_2\cup F_3))=P(F_1)P(F_2\cup F_3)=P(F_1)\big(P(F_2)+P(F_3)-P(F_2\cap F_3)\big)</script>
soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-267-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x222A;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3523" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3524" class="mjx-mrow"><span id="MJXc-Node-3525" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3526" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3527" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3528" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3529" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3530" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3531" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3532" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3533" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3534" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3535" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∪</span></span><span id="MJXc-Node-3536" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-3537" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3538" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3539" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3540" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3541" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3542" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-3543" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3544" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3545" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-3546" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3547" class="mjx-mrow"><span id="MJXc-Node-3548" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3549" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3550" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3551" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3552" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3553" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3554" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3555" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-3556" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3557" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3558" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3559" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3560" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3561" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span><span id="MJXc-Node-3562" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span id="MJXc-Node-3563" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo>∩</mo><mo stretchy="false">(</mo><msub><mi>F</mi><mn>2</mn></msub><mo>∪</mo><msub><mi>F</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><msub><mi>p</mi><mn>2</mn></msub><mo>+</mo><msub><mi>p</mi><mn>3</mn></msub><mo>−</mo><msub><mi>p</mi><mn>2</mn></msub><msub><mi>p</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-267">P(F_1\cap(F_2\cup F_3))=p_1\left(p_2+p_3-p_2p_3\right).</script>
</li></ol></li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>24</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Tirages successifs dans une urne<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1243"> [Signaler une erreur]</a><a id="exo24" href="javascript:ajouteexo(1243,24,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1243&#39;,&#39;imageenonce1243&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1243"></a><div class="inner" id="innerenonce1243">On considère une urne contenant 4 boules blanches et 3 boules noires. On tire
une à une et sans remise 3 boules de l'urne. Quelle est la probabilité pour que la première boule tirée
soit blanche, la seconde blanche et la troisième noire?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1243&#39;,&#39;imageindication1243&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1243"></a><div class="inner" id="innerindication1243" style="display:none;">Lire le titre de l'exercice.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1243&#39;,&#39;imagecorrige1243&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1243"></a><div class="inner" id="innercorrige1243" style="display:none;">On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-268-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3564" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3565" class="mjx-mrow"><span id="MJXc-Node-3566" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3567" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3568" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-268">B_i</script> (resp. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-269-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3569" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3570" class="mjx-mrow"><span id="MJXc-Node-3571" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-3572" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3573" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-269">N_i</script>) l'événement : ``La <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-270-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3574" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3575" class="mjx-mrow"><span id="MJXc-Node-3576" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">i</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></span></span><script type="math/tex" id="MathJax-Element-270">i</script>-ème boule tirée est blanche (resp. noire)''.
On cherche à calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-271-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3577" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3578" class="mjx-mrow"><span id="MJXc-Node-3579" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3580" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3581" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3582" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3583" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3584" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3585" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3586" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3587" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3588" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3589" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-3590" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3591" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3592" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>B</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>B</mi><mn>2</mn></msub><mo>∩</mo><msub><mi>N</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-271">P(B_1\cap B_2\cap N_3)</script>, ce que l'on va faire en utilisant la formule des
probabilités composées :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-272-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3593" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3594" class="mjx-mrow"><span id="MJXc-Node-3595" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3596" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3597" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3598" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3599" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3600" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3601" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3602" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3603" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3604" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3605" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-3606" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3607" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3608" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3609" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3610" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3611" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3612" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3613" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3614" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3615" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3616" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3617" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3618" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3619" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3620" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3621" class="mjx-texatom"><span id="MJXc-Node-3622" class="mjx-mrow"><span id="MJXc-Node-3623" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-3624" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3625" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3626" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3627" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3628" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3629" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3630" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-3631" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3632" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3633" class="mjx-texatom"><span id="MJXc-Node-3634" class="mjx-mrow"><span id="MJXc-Node-3635" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-3636" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3637" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3638" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3639" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3640" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3641" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3642" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3643" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3644" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>B</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>B</mi><mn>2</mn></msub><mo>∩</mo><msub><mi>N</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>B</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>B</mi><mn>2</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>B</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>N</mi><mn>3</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>B</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>B</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-272">P(B_1\cap B_2\cap N_3)=P(B_1)P(B_2|B_1)P(N_3|B_1\cap B_2).</script>
Chacune des probabilités qui apparaît est facile à calculer, car
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-273-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3645" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3646" class="mjx-mrow"><span id="MJXc-Node-3647" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3648" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3649" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3650" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3651" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3652" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3653" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3654" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span><span id="MJXc-Node-3655" class="mjx-texatom"><span id="MJXc-Node-3656" class="mjx-mrow"><span id="MJXc-Node-3657" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3658" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>B</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>7</mn></math></span></span><script type="math/tex" id="MathJax-Element-273">P(B_1)=4/7</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-274-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3659" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3660" class="mjx-mrow"><span id="MJXc-Node-3661" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3662" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3663" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3664" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3665" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3666" class="mjx-texatom"><span id="MJXc-Node-3667" class="mjx-mrow"><span id="MJXc-Node-3668" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-3669" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3670" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3671" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3672" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3673" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3674" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-3675" class="mjx-texatom"><span id="MJXc-Node-3676" class="mjx-mrow"><span id="MJXc-Node-3677" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3678" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>B</mi><mn>2</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>B</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-274">P(B_2|B_1)=3/6</script> (il reste 6 boules dont 3 blanches)
et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-275-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3679" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3680" class="mjx-mrow"><span id="MJXc-Node-3681" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3682" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3683" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-3684" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3685" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3686" class="mjx-texatom"><span id="MJXc-Node-3687" class="mjx-mrow"><span id="MJXc-Node-3688" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-3689" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3690" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3691" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3692" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3693" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3694" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3695" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3696" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3697" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3698" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-3699" class="mjx-texatom"><span id="MJXc-Node-3700" class="mjx-mrow"><span id="MJXc-Node-3701" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3702" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>N</mi><mn>3</mn></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>B</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>B</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>5</mn></math></span></span><script type="math/tex" id="MathJax-Element-275">P(N_3|B_1\cap B_2)=3/5</script>. Finalement, on obtient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-276-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mn&gt;35&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3703" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3704" class="mjx-mrow"><span id="MJXc-Node-3705" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3706" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3707" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-3708" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3709" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-3710" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3711" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-3712" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3713" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-3714" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3715" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.085em;"><span id="MJXc-Node-3716" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3717" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span id="MJXc-Node-3718" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3719" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3720" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.34em;"><span id="MJXc-Node-3721" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.633em;"><span id="MJXc-Node-3722" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">35</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.395em; vertical-align: -0.448em;"></span></span><span id="MJXc-Node-3723" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>B</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>B</mi><mn>2</mn></msub><mo>∩</mo><msub><mi>N</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>6</mn><mn>35</mn></mfrac><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-276">P(B_1\cap B_2\cap N_3)=\frac 6{35}.</script></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>25</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Deux ateliers<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2410"> [Signaler une erreur]</a><a id="exo25" href="javascript:ajouteexo(2410,25,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2410&#39;,&#39;imageenonce2410&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2410"></a><div class="inner" id="innerenonce2410">Dans une entreprise deux ateliers fabriquent les mêmes pièces. L'atelier 1 fabrique en une journée deux fois plus de pièces que l'atelier 2.
Le pourcentage de pièces défectueuses est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-277-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0025;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3724" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3725" class="mjx-mrow"><span id="MJXc-Node-3726" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span><span id="MJXc-Node-3727" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.393em;">%</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-277">3\%</script> pour l'atelier 1 et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-278-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0025;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-3728" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3729" class="mjx-mrow"><span id="MJXc-Node-3730" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span><span id="MJXc-Node-3731" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.393em;">%</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-278">4\%</script> pour l'atelier 2. On prélève une pièce au hasard dans l'ensemble de la production d'une journée.
Déterminer
<ol class="enumeratechiffre">
<li> la probabilité que cette pièce provienne de l'atelier 1;
</li><li> la probabilité que cette pièce provienne de l'atelier 1 et est défectueuse;
</li><li> la probabilité que cette pièce provienne de l'atelier 1 sachant qu'elle est défectueuse.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2410&#39;,&#39;imageindication2410&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2410"></a><div class="inner" id="innerindication2410" style="display:none;"></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2410&#39;,&#39;imagecorrige2410&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2410"></a><div class="inner" id="innercorrige2410" style="display:none;">Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-279-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3732" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3733" class="mjx-mrow"><span id="MJXc-Node-3734" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-279">A</script> l'événement ``la pièce provient de l'atelier 1'', <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-280-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3735" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3736" class="mjx-mrow"><span id="MJXc-Node-3737" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-280">B</script> l'événement ``la pièce provient de l'atelier 2'' et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-281-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3738" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3739" class="mjx-mrow"><span id="MJXc-Node-3740" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-281">D</script> l'événement ``la pièce est défectueuse''.
<ol class="enumeratechiffre">
<li> L'énoncé nous dit que les 2/3 des pièces produites proviennent de l'atelier 1. Donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-282-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3741" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3742" class="mjx-mrow"><span id="MJXc-Node-3743" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3744" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3745" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3746" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3747" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3748" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-3749" class="mjx-texatom"><span id="MJXc-Node-3750" class="mjx-mrow"><span id="MJXc-Node-3751" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-3752" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-282">P(A)=2/3</script>.
</li><li> On cherche <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-283-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;03&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;50&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3753" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3754" class="mjx-mrow"><span id="MJXc-Node-3755" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3756" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3757" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3758" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3759" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-3760" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3761" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3762" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3763" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.241em; padding-right: 0.071em;"><span id="MJXc-Node-3764" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span id="MJXc-Node-3765" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3766" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-3767" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3768" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3769" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3770" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3771" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3772" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3773" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-3774" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-3775" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">03</span></span><span id="MJXc-Node-3776" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3777" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-3778" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.632em;"><span id="MJXc-Node-3779" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.379em; vertical-align: -0.447em;"></span></span><span id="MJXc-Node-3780" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3781" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.318em;"><span id="MJXc-Node-3782" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.633em;"><span id="MJXc-Node-3783" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">50</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.38em; vertical-align: -0.448em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mi>A</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>03</mn><mo>×</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>50</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-283">P(A\cap D)=P_A(D)P(A)=0,03\times \frac 23=\frac 1{50}</script>.
</li><li> On démontre de même que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-284-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;75&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3784" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3785" class="mjx-mrow"><span id="MJXc-Node-3786" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3787" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3788" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3789" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3790" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-3791" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3792" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3793" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.849em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.2em; top: -1.318em;"><span id="MJXc-Node-3794" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.2em; bottom: -0.643em;"><span id="MJXc-Node-3795" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">75</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.849em;"></span></span><span class="mjx-vsize" style="height: 1.387em; vertical-align: -0.455em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo>∩</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>75</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-284">P(B\cap D)=\frac 1{75}</script> et donc que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-285-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3796" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3797" class="mjx-mrow"><span id="MJXc-Node-3798" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3799" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3800" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-3801" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3802" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3803" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3804" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3805" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3806" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3807" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-3808" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3809" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-3810" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3811" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3812" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3813" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3814" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-3815" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3816" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3817" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-3818" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-3819" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3820" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>D</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo>∩</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>30</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-285">P(D)=P(A\cap D)+P(B\cap D)=\frac 1{30}.</script>
Ainsi, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-286-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3821" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3822" class="mjx-mrow"><span id="MJXc-Node-3823" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-3824" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-3825" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span></span><span id="MJXc-Node-3826" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3827" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3828" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3829" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3830" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.418em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.418em; top: -1.542em;"><span id="MJXc-Node-3831" class="mjx-mrow"><span id="MJXc-Node-3832" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3833" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3834" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3835" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-3836" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-3837" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 4.418em; bottom: -1.042em;"><span id="MJXc-Node-3838" class="mjx-mrow"><span id="MJXc-Node-3839" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3840" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3841" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-3842" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.418em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-3843" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3844" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.367em;"><span id="MJXc-Node-3845" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-3846" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.099em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3847" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>P</mi><mi>D</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-286">P_D(A)=\frac{P(A\cap D)}{P(D)}=\frac 35.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>26</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - A partir de dénombrement<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1244"> [Signaler une erreur]</a><a id="exo26" href="javascript:ajouteexo(1244,26,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1244&#39;,&#39;imageenonce1244&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1244"></a><div class="inner" id="innerenonce1244">Une urne contient 8 boules blanches et 2 boules noires, indiscernables au toucher. On tire sans remise et successivement 3 boules de cette urne.
<ol class="enumeratechiffre">
<li> Quelle est la probabilité qu'au moins une boule noire figure dans le tirage?
</li><li> Sachant qu'au moins une boule noire figure dans le tirage, quelle est la probabilité que la première boule tirée soit noire?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1244&#39;,&#39;imageindication1244&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1244"></a><div class="inner" id="innerindication1244" style="display:none;"><ol class="enumeratechiffre">
<li> Dénombrer!
</li><li> Dénombrer les tirages tels que la première boule est noire, et appliquer la définition des probabilités conditionnelles.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1244&#39;,&#39;imagecorrige1244&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1244"></a><div class="inner" id="innercorrige1244" style="display:none;"><ol class="enumeratechiffre">
<li> Distinguons les boules et ordonnons les tirages. Il y a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-287-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3848" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3849" class="mjx-mrow"><span id="MJXc-Node-3850" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-3851" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3852" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-3853" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3854" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn></math></span></span><script type="math/tex" id="MathJax-Element-287">10\times 9\times 8</script> tirages possibles. Calculons maintenant le nombre de tirages comprenant au moins une boule noire. Il y a deux possibilités :
<ul class="rien">
<li> Le tirage ne comporte qu'une seule boule noire. Il y a 3 façons de choisir la position de la boule noire (lors du premier tirage, du deuxième, etc...), 2 choix pour cette boule, et ensuite <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-288-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3855" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3856" class="mjx-mrow"><span id="MJXc-Node-3857" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-3858" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3859" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><mn>7</mn></math></span></span><script type="math/tex" id="MathJax-Element-288">8\times 7</script> choix pour les deux boules blanches. En tout, il y a donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-289-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3860" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3861" class="mjx-mrow"><span id="MJXc-Node-3862" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-3863" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3864" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-3865" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3866" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-3867" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3868" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>7</mn></math></span></span><script type="math/tex" id="MathJax-Element-289">3\times 2\times 8\times 7</script> tirages correspondants.
</li><li> Le tirage comporte les deux boules noires. On choisit d'abord les deux tirages où on a pris les boules noires : on choisit 2 places parmi 3, soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-290-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-OPEN&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;(&lt;/mo&gt;&lt;/mrow&gt;&lt;mfrac linethickness=&quot;0&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow class=&quot;MJX-TeXAtom-CLOSE&quot;&gt;&lt;mo maxsize=&quot;1.2em&quot; minsize=&quot;1.2em&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3869" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3870" class="mjx-mrow"><span id="MJXc-Node-3871" class="mjx-mrow"><span id="MJXc-Node-3872" class="mjx-TeXmathchoice"><span id="MJXc-Node-3873" class="mjx-texatom"><span id="MJXc-Node-3874" class="mjx-mrow"><span id="MJXc-Node-3875" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">(</span></span></span></span></span><span id="MJXc-Node-3876" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.354em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.5em; top: -1.318em;"><span id="MJXc-Node-3877" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.5em; bottom: -0.513em;"><span id="MJXc-Node-3878" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-vsize" style="height: 1.294em; vertical-align: -0.363em;"></span></span><span id="MJXc-Node-3879" class="mjx-TeXmathchoice"><span id="MJXc-Node-3880" class="mjx-texatom"><span id="MJXc-Node-3881" class="mjx-mrow"><span id="MJXc-Node-3882" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">)</span></span></span></span></span></span><span id="MJXc-Node-3883" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3884" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mfrac linethickness="0"><mn>3</mn><mn>2</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>=</mo><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-290">\binom 32=3</script>. Ce choix fait, il y a 2 choix pour les boules noires, et 8 choix pour les boules blanches. Il y a donc en tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-291-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3885" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3886" class="mjx-mrow"><span id="MJXc-Node-3887" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-3888" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3889" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-3890" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3891" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>8</mn></math></span></span><script type="math/tex" id="MathJax-Element-291">3\times 2\times 8</script> tels tirages.
</li></ul>
Le nombre total de tirages est donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-292-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3892" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3893" class="mjx-mrow"><span id="MJXc-Node-3894" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-3895" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3896" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-3897" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3898" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>8</mn></math></span></span><script type="math/tex" id="MathJax-Element-292">6\times 8\times 8</script> et si on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-293-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3899" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3900" class="mjx-mrow"><span id="MJXc-Node-3901" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-293">B</script> l'événement "Au moins une boule noire figure dans le tirage", alors la probabilité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-294-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3902" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3903" class="mjx-mrow"><span id="MJXc-Node-3904" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-294">B</script> est égale à
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-295-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3905" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3906" class="mjx-mrow"><span id="MJXc-Node-3907" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3908" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3909" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-3910" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3911" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3912" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.645em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.645em; top: -1.368em;"><span id="MJXc-Node-3913" class="mjx-mrow"><span id="MJXc-Node-3914" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-3915" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3916" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-3917" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3918" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="mjx-denominator" style="width: 4.645em; bottom: -0.733em;"><span id="MJXc-Node-3919" class="mjx-mrow"><span id="MJXc-Node-3920" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-3921" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3922" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-3923" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3924" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.645em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3925" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3926" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-3927" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-3928" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">15</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3929" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>6</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>8</mn></mrow><mrow><mn>10</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn></mrow></mfrac><mo>=</mo><mfrac><mn>8</mn><mn>15</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-295">P(B)=\frac{6\times 8\times 8}{10\times 9\times 8}=\frac{8}{15}.</script>
Remarquons que, comme souvent dans ce type d'exercice, il est plus facile de déterminer la probabilité du complémentaire : en effet, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-296-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3930" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3931" class="mjx-mrow"><span id="MJXc-Node-3932" class="mjx-texatom"><span id="MJXc-Node-3933" class="mjx-mrow"><span id="MJXc-Node-3934" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.213em;"><span id="MJXc-Node-3936" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-3935" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">¯</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-296">\bar B</script> est l'événement "on ne tire que des boules blanches". Le nombre de tirages correspondant est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-297-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3937" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3938" class="mjx-mrow"><span id="MJXc-Node-3939" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-3940" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3941" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-3942" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3943" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><mn>7</mn><mo>×</mo><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-297">8\times 7\times 6</script> et donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-298-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3944" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3945" class="mjx-mrow"><span id="MJXc-Node-3946" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3947" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3948" class="mjx-texatom"><span id="MJXc-Node-3949" class="mjx-mrow"><span id="MJXc-Node-3950" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.213em;"><span id="MJXc-Node-3952" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-3951" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span></span></span><span id="MJXc-Node-3953" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3954" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3955" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.645em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.645em; top: -1.378em;"><span id="MJXc-Node-3956" class="mjx-mrow"><span id="MJXc-Node-3957" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span><span id="MJXc-Node-3958" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3959" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-3960" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3961" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="mjx-denominator" style="width: 4.645em; bottom: -0.733em;"><span id="MJXc-Node-3962" class="mjx-mrow"><span id="MJXc-Node-3963" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-3964" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3965" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-3966" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3967" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.645em;"></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3968" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3969" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.922em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.922em; top: -1.378em;"><span id="MJXc-Node-3970" class="mjx-mrow"><span id="MJXc-Node-3971" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-3972" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3973" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="mjx-denominator" style="width: 2.922em; bottom: -0.733em;"><span id="MJXc-Node-3974" class="mjx-mrow"><span id="MJXc-Node-3975" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-3976" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3977" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.922em;"></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-3978" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3979" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.378em;"><span id="MJXc-Node-3980" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-3981" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">15</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.11em; vertical-align: -0.733em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>8</mn><mo>×</mo><mn>7</mn><mo>×</mo><mn>6</mn></mrow><mrow><mn>10</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>7</mn><mo>×</mo><mn>6</mn></mrow><mrow><mn>10</mn><mo>×</mo><mn>9</mn></mrow></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>15</mn></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-298">P(\bar B)=\frac{8\times 7\times 6}{10\times 9\times 8}=\frac{7\times 6}{10\times 9}=\frac{7}{15}</script>
ce qui est bien la valeur attendue (ouf)!
</li><li> Le dénombrement du nombre de tirages tels que la première boule soit noire est plus simple. Il y a exactement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-299-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3982" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3983" class="mjx-mrow"><span id="MJXc-Node-3984" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-3985" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3986" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-3987" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-3988" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn></math></span></span><script type="math/tex" id="MathJax-Element-299">2\times 9\times 8</script> tirages. Si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-300-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-3989" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3990" class="mjx-mrow"><span id="MJXc-Node-3991" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-300">A</script> est l'événement : "La première boule tirée est noire", alors
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-301-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-3992" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-3993" class="mjx-mrow"><span id="MJXc-Node-3994" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-3995" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-3996" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-3997" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-3998" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-3999" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.645em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.645em; top: -1.368em;"><span id="MJXc-Node-4000" class="mjx-mrow"><span id="MJXc-Node-4001" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-4002" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-4003" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-4004" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-4005" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="mjx-denominator" style="width: 4.645em; bottom: -0.733em;"><span id="MJXc-Node-4006" class="mjx-mrow"><span id="MJXc-Node-4007" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span><span id="MJXc-Node-4008" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-4009" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-4010" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-4011" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.645em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4012" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4013" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-4014" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-4015" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-4016" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn></mrow><mrow><mn>10</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-301">P(A)=\frac{2\times 9\times 8}{10\times 9\times 8}=\frac 15.</script>
On cherche <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-302-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4017" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4018" class="mjx-mrow"><span id="MJXc-Node-4019" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4020" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4021" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-4022" class="mjx-texatom"><span id="MJXc-Node-4023" class="mjx-mrow"><span id="MJXc-Node-4024" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-4025" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-4026" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-302">P(A|B)</script>. Par définition,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-303-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-4027" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4028" class="mjx-mrow"><span id="MJXc-Node-4029" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4030" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4031" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-4032" class="mjx-texatom"><span id="MJXc-Node-4033" class="mjx-mrow"><span id="MJXc-Node-4034" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-4035" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-4036" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4037" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4038" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.349em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.349em; top: -1.542em;"><span id="MJXc-Node-4039" class="mjx-mrow"><span id="MJXc-Node-4040" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4041" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4042" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-4043" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-4044" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-4045" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 4.349em; bottom: -1.042em;"><span id="MJXc-Node-4046" class="mjx-mrow"><span id="MJXc-Node-4047" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4048" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4049" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-4050" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.349em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-4051" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4052" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.488em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.488em; top: -1.542em;"><span id="MJXc-Node-4053" class="mjx-mrow"><span id="MJXc-Node-4054" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4055" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4056" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-4057" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 2.488em; bottom: -1.042em;"><span id="MJXc-Node-4058" class="mjx-mrow"><span id="MJXc-Node-4059" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4060" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4061" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-4062" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.488em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-4063" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4064" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.367em;"><span id="MJXc-Node-4065" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-4066" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">8</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.099em; vertical-align: -0.733em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>8</mn></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-303">P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{P(A)}{P(B)}=\frac 38</script>
puisque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-304-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2282;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4067" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4068" class="mjx-mrow"><span id="MJXc-Node-4069" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-4070" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">⊂</span></span><span id="MJXc-Node-4071" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>⊂</mo><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-304">A\subset B</script>.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>27</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - <a href="http://www.bibmath.net/ressources/signalerreur.php?numero=3092"> [Signaler une erreur]</a><a id="exo27" href="javascript:ajouteexo(3092,27,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce3092&#39;,&#39;imageenonce3092&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce3092"></a><div class="inner" id="innerenonce3092">Un fumeur, après avoir lu une série de statistiques
effrayantes sur les risques de cancer, décide d'arrêter de fumer; toujours d'après des
statistiques, on estime les probabilités suivantes : si cette personne
n'a pas fumé le <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-305-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4072" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4073" class="mjx-mrow"><span id="MJXc-Node-4074" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-305">n</script>-ième jour, alors la probabilité
pour qu'elle ne fume pas le jour suivant est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-306-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4075" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4076" class="mjx-mrow" style="width: 1.278em;"><span id="MJXc-Node-4077" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-4078" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4079" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4080" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-306">0,\!3</script>;
mais si elle a fumé le <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-307-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4081" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4082" class="mjx-mrow"><span id="MJXc-Node-4083" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-307">n</script>-ième jour, alors la probabilité
pour qu'elle ne fume pas le jour suivant est <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-308-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4084" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4085" class="mjx-mrow" style="width: 1.278em;"><span id="MJXc-Node-4086" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-4087" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4088" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4089" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">9</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>9</mn></math></span></span><script type="math/tex" id="MathJax-Element-308">0,\!9</script>.
Pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-309-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4090" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4091" class="mjx-mrow"><span id="MJXc-Node-4092" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4093" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≥</span></span><span id="MJXc-Node-4094" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-309">n\geq 0</script>, on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-310-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4095" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4096" class="mjx-mrow"><span id="MJXc-Node-4097" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4098" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4099" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-310">F_n</script> l'événement ``la personne fume le <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-311-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4100" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4101" class="mjx-mrow"><span id="MJXc-Node-4102" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-311">n</script>-ième jour'' et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-312-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4103" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4104" class="mjx-mrow"><span id="MJXc-Node-4105" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4106" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4107" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-312">p_n</script> la probabilité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-313-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4108" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4109" class="mjx-mrow"><span id="MJXc-Node-4110" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4111" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4112" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-313">F_n</script>.
En particulier on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-314-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4113" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4114" class="mjx-mrow"><span id="MJXc-Node-4115" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4116" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4117" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span><span id="MJXc-Node-4118" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4119" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-314">p_0=1</script>.
<ol class="enumeratechiffre">
<li> Démontrer que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-315-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4120" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4121" class="mjx-mrow" style="width: 8.719em;"><span id="MJXc-Node-4122" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4123" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4124" class="mjx-texatom" style=""><span id="MJXc-Node-4125" class="mjx-mrow"><span id="MJXc-Node-4126" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4127" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4128" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span><span id="MJXc-Node-4129" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4130" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-4131" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-4132" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4133" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4134" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">6</span></span><span id="MJXc-Node-4135" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4136" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4137" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4138" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4139" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-4140" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4141" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4142" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>6</mn><msub><mi>p</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>7</mn></math></span></span><script type="math/tex" id="MathJax-Element-315">p_{n+1}=-0,\!6p_n+0,\!7</script>.
</li><li> La personne va-t-elle s'arrêter de fumer?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication3092&#39;,&#39;imageindication3092&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication3092"></a><div class="inner" id="innerindication3092" style="display:none;"><ol class="enumeratechiffre">
<li> Appliquer la formule des probabilités totales.
</li><li> C'est une suite arithmético-géométrique.
</li></ol>
</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige3092&#39;,&#39;imagecorrige3092&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige3092"></a><div class="inner" id="innercorrige3092" style="display:none;"><ol class="enumeratechiffre">
<li> D'après la formule des probabilités totales,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-316-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-4143" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4144" class="mjx-mrow"><span id="MJXc-Node-4145" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4146" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4147" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4148" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4149" class="mjx-texatom" style=""><span id="MJXc-Node-4150" class="mjx-mrow"><span id="MJXc-Node-4151" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4152" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4153" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4154" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4155" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4156" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-4157" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4158" class="mjx-texatom" style=""><span id="MJXc-Node-4159" class="mjx-mrow"><span id="MJXc-Node-4160" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4161" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4162" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-4163" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4164" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4165" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4166" class="mjx-texatom" style=""><span id="MJXc-Node-4167" class="mjx-mrow"><span id="MJXc-Node-4168" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4169" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4170" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4171" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4172" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4173" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4174" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4175" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4176" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4177" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4178" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4179" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-4180" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.573em; padding-right: 0.071em;"><span id="MJXc-Node-4181" class="mjx-texatom" style=""><span id="MJXc-Node-4182" class="mjx-mrow"><span id="MJXc-Node-4183" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-4187" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.043em 0px -0.182em; letter-spacing: -0.266em;">¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4184" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4185" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4186" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-4188" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4189" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4190" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4191" class="mjx-texatom" style=""><span id="MJXc-Node-4192" class="mjx-mrow"><span id="MJXc-Node-4193" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4194" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4195" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4196" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4197" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4198" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4199" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-4203" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.043em 0px -0.182em; letter-spacing: -0.266em;">¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4200" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4201" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4202" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span><span id="MJXc-Node-4204" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4205" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>F</mi><mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>F</mi><mi>n</mi></msub><mo accent="false">¯</mo></mover></mrow></msub><mo stretchy="false">(</mo><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mover><msub><mi>F</mi><mi>n</mi></msub><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-316">P(F_{n+1})=P_{F_n}(F_{n+1})P(F_n)+P_{\overline{F_n}}(F_{n+1})P(\overline{F_n}).</script>
L'énoncé nous dit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-317-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;et&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-4206" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4207" class="mjx-mrow" style="width: 15.039em;"><span id="MJXc-Node-4208" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-4209" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4210" class="mjx-texatom" style=""><span id="MJXc-Node-4211" class="mjx-mrow"><span id="MJXc-Node-4212" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4213" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4214" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-4215" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4216" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-4224" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.037em 0px -0.176em; letter-spacing: -0.254em;">¯¯¯¯¯¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4217" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4218" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4219" class="mjx-texatom" style=""><span id="MJXc-Node-4220" class="mjx-mrow"><span id="MJXc-Node-4221" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4222" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4223" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span></span><span id="MJXc-Node-4225" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4226" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4227" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4228" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4229" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4230" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">9</span></span><span id="MJXc-Node-4231" class="mjx-texatom"><span id="MJXc-Node-4232" class="mjx-mrow"><span id="MJXc-Node-4233" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">&nbsp;et&nbsp;</span></span></span></span><span id="MJXc-Node-4234" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-4235" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.573em; padding-right: 0.071em;"><span id="MJXc-Node-4236" class="mjx-texatom" style=""><span id="MJXc-Node-4237" class="mjx-mrow"><span id="MJXc-Node-4238" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-4242" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.043em 0px -0.182em; letter-spacing: -0.266em;">¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4239" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4240" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4241" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-4243" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4244" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-4252" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.037em 0px -0.176em; letter-spacing: -0.254em;">¯¯¯¯¯¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4245" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4246" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4247" class="mjx-texatom" style=""><span id="MJXc-Node-4248" class="mjx-mrow"><span id="MJXc-Node-4249" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4250" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4251" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span></span><span id="MJXc-Node-4253" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4254" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4255" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4256" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4257" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4258" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>F</mi><mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><mover><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>9</mn><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;et&nbsp;</mtext></mrow><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>F</mi><mi>n</mi></msub><mo accent="false">¯</mo></mover></mrow></msub><mo stretchy="false">(</mo><mover><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>3</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-317">P_{F_n}(\overline{F_{n+1}})=0,\!9\textrm{ et }P_{\overline{F_n}}(\overline{F_{n+1}})=0,\!3</script>
d'ou
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-318-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;et&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;7.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-4259" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4260" class="mjx-mrow" style="width: 15.317em;"><span id="MJXc-Node-4261" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-4262" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4263" class="mjx-texatom" style=""><span id="MJXc-Node-4264" class="mjx-mrow"><span id="MJXc-Node-4265" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4266" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4267" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-4268" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4269" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4270" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4271" class="mjx-texatom" style=""><span id="MJXc-Node-4272" class="mjx-mrow"><span id="MJXc-Node-4273" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4274" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4275" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4276" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4277" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4278" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4279" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4280" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4281" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4282" class="mjx-texatom"><span id="MJXc-Node-4283" class="mjx-mrow"><span id="MJXc-Node-4284" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">&nbsp;et&nbsp;</span></span></span></span><span id="MJXc-Node-4285" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-4286" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.573em; padding-right: 0.071em;"><span id="MJXc-Node-4287" class="mjx-texatom" style=""><span id="MJXc-Node-4288" class="mjx-mrow"><span id="MJXc-Node-4289" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-4295" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.043em 0px -0.182em; letter-spacing: -0.266em;">¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-4290" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4291" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4292" class="mjx-texatom" style=""><span id="MJXc-Node-4293" class="mjx-mrow"><span id="MJXc-Node-4294" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-4296" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4297" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.106em;"><span id="MJXc-Node-4298" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4299" class="mjx-texatom" style=""><span id="MJXc-Node-4300" class="mjx-mrow"><span id="MJXc-Node-4301" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4302" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4303" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4304" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4305" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4306" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4307" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4308" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4309" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>F</mi><mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;et&nbsp;</mtext></mrow><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mover><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub><mo accent="false">¯</mo></mover></mrow></msub><mo stretchy="false">(</mo><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>7.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-318">P_{F_n}(F_{n+1})=0,\!1\textrm{ et }P_{\overline{F_{n}}}(F_{n+1})=0,\!7.</script>
On a donc
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-319-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;7.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-4310" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4311" class="mjx-mrow" style="width: 18.535em;"><span id="MJXc-Node-4312" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4313" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4314" class="mjx-texatom" style=""><span id="MJXc-Node-4315" class="mjx-mrow"><span id="MJXc-Node-4316" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4317" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4318" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4319" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4320" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4321" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4322" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4323" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4324" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4325" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4326" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4327" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4328" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4329" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4330" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4331" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-4332" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4333" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4334" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4335" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-4336" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4337" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4338" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4339" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4340" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4341" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4342" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4343" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4344" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-4345" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4346" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4347" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4348" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4349" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4350" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4351" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4352" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>1</mn><msub><mi>p</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>7</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>p</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>6</mn><msub><mi>p</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>7.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-319">p_{n+1}=0,\!1 p_n+0,\!7(1-p_n)=-0,\!6p_n+0,\!7.</script>
</li><li> On a une suite arithmético-géométrique. L'équation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-320-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x2113;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mi&gt;&amp;#x2113;&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4353" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4354" class="mjx-mrow" style="width: 6.724em;"><span id="MJXc-Node-4355" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ℓ</span></span><span id="MJXc-Node-4356" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4357" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4358" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4359" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4360" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4361" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-4362" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ℓ</span></span><span id="MJXc-Node-4363" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4364" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4365" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4366" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4367" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℓ</mi><mo>=</mo><mo>−</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>6</mn><mi>ℓ</mi><mo>+</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>7</mn></math></span></span><script type="math/tex" id="MathJax-Element-320">\ell=-0,\!6\ell+0,\!7</script>
donne <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-321-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x2113;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4368" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4369" class="mjx-mrow"><span id="MJXc-Node-4370" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ℓ</span></span><span id="MJXc-Node-4371" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4372" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-4373" class="mjx-texatom"><span id="MJXc-Node-4374" class="mjx-mrow"><span id="MJXc-Node-4375" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4376" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">16</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℓ</mi><mo>=</mo><mn>7</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>16</mn></math></span></span><script type="math/tex" id="MathJax-Element-321">\ell=7/16</script>. La suite <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-322-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4377" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4378" class="mjx-mrow"><span id="MJXc-Node-4379" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4380" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-4381" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4382" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>q</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-322">(q_n)</script> définie par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-323-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x2113;&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4384" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4385" class="mjx-mrow"><span id="MJXc-Node-4386" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-4387" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4388" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4389" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4390" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-4391" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4392" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4393" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4394" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">ℓ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>n</mi></msub><mo>=</mo><msub><mi>p</mi><mi>n</mi></msub><mo>−</mo><mi>ℓ</mi></math></span></span><script type="math/tex" id="MathJax-Element-323">q_n=p_n-\ell</script> est une suite géométrique de raison <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-324-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;negativethinmathspace&quot; /&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4395" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4396" class="mjx-mrow" style="width: 2.056em;"><span id="MJXc-Node-4397" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-4398" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-4399" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4400" class="mjx-mspace" style="margin-right: -0.167em; width: 0px; height: 0px;"></span><span id="MJXc-Node-4401" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>0</mn><mo>,</mo><mspace width="negativethinmathspace"></mspace><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-324">-0,\!6</script>, elle
converge vers <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-325-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4402" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4403" class="mjx-mrow"><span id="MJXc-Node-4404" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-325">0</script>, et donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-326-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4405" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4406" class="mjx-mrow"><span id="MJXc-Node-4407" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4408" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4409" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4410" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4411" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>p</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-326">(p_n)</script> tend vers <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-327-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4412" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4413" class="mjx-mrow"><span id="MJXc-Node-4414" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-4415" class="mjx-texatom"><span id="MJXc-Node-4416" class="mjx-mrow"><span id="MJXc-Node-4417" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4418" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">16</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>16</mn></math></span></span><script type="math/tex" id="MathJax-Element-327">7/16</script>. La probabilité que la personne fume le <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-328-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4419" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4420" class="mjx-mrow"><span id="MJXc-Node-4421" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-328">n</script>-ième jour
tend vers <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-329-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4422" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4423" class="mjx-mrow"><span id="MJXc-Node-4424" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">7</span></span><span id="MJXc-Node-4425" class="mjx-texatom"><span id="MJXc-Node-4426" class="mjx-mrow"><span id="MJXc-Node-4427" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4428" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">16</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>16</mn></math></span></span><script type="math/tex" id="MathJax-Element-329">7/16</script>. Elle ne va pas s'arrêter de fumer!
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>28</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Déplacement d'un pion sur un triangle<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=3091"> [Signaler une erreur]</a><a id="exo28" href="javascript:ajouteexo(3091,28,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce3091&#39;,&#39;imageenonce3091&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce3091"></a><div class="inner" id="innerenonce3091">On considère trois points distincts du plan nommés <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-330-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4429" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4430" class="mjx-mrow"><span id="MJXc-Node-4431" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-330">A</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-331-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4432" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4433" class="mjx-mrow"><span id="MJXc-Node-4434" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-331">B</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-332-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4435" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4436" class="mjx-mrow"><span id="MJXc-Node-4437" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-332">C</script>.
Nous allons étudier le déplacement aléatoire d’un pion se déplaçant sur ces trois points. A l’étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-333-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4438" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4439" class="mjx-mrow"><span id="MJXc-Node-4440" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4441" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4442" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-333">n=0</script>, on suppose que le pion se trouve sur le point <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-334-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4443" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4444" class="mjx-mrow"><span id="MJXc-Node-4445" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-334">A</script>.
Ensuite, le mouvement aléatoire du pion respecte les deux règles suivantes :
<ul class="rien">
<li> le mouvement du pion de l’étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-335-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4446" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4447" class="mjx-mrow"><span id="MJXc-Node-4448" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-335">n</script> à l’étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-336-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4449" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4450" class="mjx-mrow"><span id="MJXc-Node-4451" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4452" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4453" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-336">n+1</script> ne dépend que de la position du pion à l’étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-337-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4454" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4455" class="mjx-mrow"><span id="MJXc-Node-4456" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-337">n</script>;
</li><li> pour passer de l'étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-338-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4457" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4458" class="mjx-mrow"><span id="MJXc-Node-4459" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-338">n</script> à l'étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-339-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4460" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4461" class="mjx-mrow"><span id="MJXc-Node-4462" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4463" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4464" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-339">n+1</script>, on suppose que le pion a une chance sur deux de rester sur place,
sinon il se déplace de manière équiprobable vers l’un des deux autres points.
</li></ul>
Pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-340-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4465" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4466" class="mjx-mrow"><span id="MJXc-Node-4467" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4468" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-4469" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-4470" class="mjx-mrow"><span id="MJXc-Node-4471" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-340">n\in\mathbb N</script>, on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-341-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4472" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4473" class="mjx-mrow"><span id="MJXc-Node-4474" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4475" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4476" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-341">A_n</script> l’évènement "le pion se trouve en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-342-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4477" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4478" class="mjx-mrow"><span id="MJXc-Node-4479" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-342">A</script> à l’étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-343-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4480" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4481" class="mjx-mrow"><span id="MJXc-Node-4482" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-343">n</script>", <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-344-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4483" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4484" class="mjx-mrow"><span id="MJXc-Node-4485" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4486" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4487" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-344">B_n</script> l’évènement "le pion se trouve en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-345-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4488" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4489" class="mjx-mrow"><span id="MJXc-Node-4490" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-345">B</script> à l’étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-346-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4491" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4492" class="mjx-mrow"><span id="MJXc-Node-4493" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-346">n</script>"
et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-347-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4494" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4495" class="mjx-mrow"><span id="MJXc-Node-4496" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-4497" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4498" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-347">C_n</script> l’évènement "le pion se trouve en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-348-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4499" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4500" class="mjx-mrow"><span id="MJXc-Node-4501" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-348">C</script> à l’étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-349-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4502" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4503" class="mjx-mrow"><span id="MJXc-Node-4504" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-349">n</script>". On note également, pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-350-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4505" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4506" class="mjx-mrow"><span id="MJXc-Node-4507" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4508" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-4509" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-4510" class="mjx-mrow"><span id="MJXc-Node-4511" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-350">n\in\mathbb N</script>,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-351-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;et&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4512" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4513" class="mjx-mrow"><span id="MJXc-Node-4514" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4515" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4516" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4517" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4518" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4519" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4520" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4521" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4522" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4523" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-4524" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4525" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-4526" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4527" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4528" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4529" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4530" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4531" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4532" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4533" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4534" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4535" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-4536" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4537" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-4538" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4539" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4540" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4541" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4542" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4543" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4544" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-4545" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4546" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4547" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span><span id="MJXc-Node-4548" class="mjx-texatom"><span id="MJXc-Node-4549" class="mjx-mrow"><span id="MJXc-Node-4550" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.328em;">&nbsp;et&nbsp;</span></span></span></span><span id="MJXc-Node-4551" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.186em;"><span id="MJXc-Node-4552" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.186em;">V</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4553" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4554" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4555" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-4556" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎝</span></span></span><span id="MJXc-Node-4557" class="mjx-mtable" style="vertical-align: -1.65em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-4558" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-4559" class="mjx-mtd" style="padding: 0px; width: 0.989em;"><span id="MJXc-Node-4560" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4561" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4562" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4563" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-4564" class="mjx-mtr" style="height: 1.4em;"><span id="MJXc-Node-4565" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-4566" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4567" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4568" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4569" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-4570" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-4571" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-4572" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4573" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4574" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4575" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-4576" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎠</span></span></span></span><span id="MJXc-Node-4577" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mtext>&nbsp;</mtext><msub><mi>b</mi><mi>n</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>B</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mtext>&nbsp;</mtext><msub><mi>c</mi><mi>n</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;et&nbsp;</mtext></mrow><msub><mi>V</mi><mi>n</mi></msub><mo>=</mo><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>a</mi><mi>n</mi></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>n</mi></msub></mtd></mtr><mtr><mtd><msub><mi>c</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-351">a_n=P(A_n),\ b_n=P(B_n),\ c_n=P(C_n)\textrm{ et } V_n=\left(\begin{array}{c}a_n\\b_n\\c_n\end{array}\right).</script>
<ol class="enumeratechiffre">
<li> Calculer les nombres <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-352-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4578" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4579" class="mjx-mrow"><span id="MJXc-Node-4580" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4581" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4582" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-352">a_n</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-353-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4583" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4584" class="mjx-mrow"><span id="MJXc-Node-4585" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4586" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4587" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-353">b_n</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-354-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4588" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4589" class="mjx-mrow"><span id="MJXc-Node-4590" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4591" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4592" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-354">c_n</script> pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-355-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4593" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4594" class="mjx-mrow"><span id="MJXc-Node-4595" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4596" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4597" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-4598" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-4599" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-355">n=0,1</script>.
</li><li> Pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-356-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4600" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4601" class="mjx-mrow"><span id="MJXc-Node-4602" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4603" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-4604" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-4605" class="mjx-mrow"><span id="MJXc-Node-4606" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-356">n\in\mathbb N</script>, exprimer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-357-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4607" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4608" class="mjx-mrow"><span id="MJXc-Node-4609" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4610" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4611" class="mjx-texatom" style=""><span id="MJXc-Node-4612" class="mjx-mrow"><span id="MJXc-Node-4613" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4614" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4615" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-357">a_{n+1}</script> en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-358-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4616" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4617" class="mjx-mrow"><span id="MJXc-Node-4618" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4619" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4620" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-358">a_n</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-359-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4621" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4622" class="mjx-mrow"><span id="MJXc-Node-4623" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4624" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4625" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-359">b_n</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-360-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4626" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4627" class="mjx-mrow"><span id="MJXc-Node-4628" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4629" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4630" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-360">c_n</script>. Faire de même pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-361-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4631" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4632" class="mjx-mrow"><span id="MJXc-Node-4633" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4634" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4635" class="mjx-texatom" style=""><span id="MJXc-Node-4636" class="mjx-mrow"><span id="MJXc-Node-4637" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4638" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4639" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-361">b_{n+1}</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-362-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4640" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4641" class="mjx-mrow"><span id="MJXc-Node-4642" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4643" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4644" class="mjx-texatom" style=""><span id="MJXc-Node-4645" class="mjx-mrow"><span id="MJXc-Node-4646" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4647" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4648" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-362">c_{n+1}</script>.
</li><li> Donner une matrice <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-363-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4649" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4650" class="mjx-mrow"><span id="MJXc-Node-4651" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-363">M</script> telle que, pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-364-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4652" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4653" class="mjx-mrow"><span id="MJXc-Node-4654" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4655" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-4656" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-4657" class="mjx-mrow"><span id="MJXc-Node-4658" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-364">n\in\mathbb N</script>, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-365-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4659" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4660" class="mjx-mrow"><span id="MJXc-Node-4661" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.186em;"><span id="MJXc-Node-4662" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.186em;">V</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4663" class="mjx-texatom" style=""><span id="MJXc-Node-4664" class="mjx-mrow"><span id="MJXc-Node-4665" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4666" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4667" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span><span id="MJXc-Node-4668" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4669" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-4670" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.186em;"><span id="MJXc-Node-4671" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.186em;">V</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4672" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>M</mi><msub><mi>V</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-365">V_{n+1}=MV_n</script>.
</li><li> On admet que, pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-366-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4673" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4674" class="mjx-mrow"><span id="MJXc-Node-4675" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4676" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-4677" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-4678" class="mjx-mrow"><span id="MJXc-Node-4679" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-366">n\in\mathbb N</script>, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-367-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msup&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x22C5;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable columnalign=&quot;center center center&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4680" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4681" class="mjx-mrow"><span id="MJXc-Node-4682" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.081em;"><span id="MJXc-Node-4683" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.081em;">M</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.22em; padding-right: 0.071em;"><span id="MJXc-Node-4684" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4685" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-4686" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.382em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.382em; top: -1.368em;"><span id="MJXc-Node-4687" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span><span class="mjx-denominator" style="width: 2.382em; bottom: -0.819em;"><span id="MJXc-Node-4688" class="mjx-mrow"><span id="MJXc-Node-4689" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span><span id="MJXc-Node-4690" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.328em;">⋅</span></span><span id="MJXc-Node-4691" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-4692" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4693" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.291em; width: 2.382em;"></span></span><span class="mjx-vsize" style="height: 2.187em; vertical-align: -0.819em;"></span></span><span id="MJXc-Node-4694" class="mjx-mrow"><span id="MJXc-Node-4695" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎝</span></span></span><span id="MJXc-Node-4696" class="mjx-mtable" style="vertical-align: -1.65em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-4697" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-4698" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 2.682em;"><span id="MJXc-Node-4699" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4700" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4701" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4702" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4703" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4704" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4705" class="mjx-mtd" style="padding: 0px 0.5em; width: 2.682em;"><span id="MJXc-Node-4706" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4707" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4708" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4709" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4710" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-4711" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4712" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 2.682em;"><span id="MJXc-Node-4713" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4714" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4715" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4716" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4717" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-4718" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-4719" class="mjx-mtr" style="height: 1.4em;"><span id="MJXc-Node-4720" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-4721" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4722" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4723" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4724" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4725" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-4726" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4727" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-4728" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4729" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4730" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4731" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4732" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4733" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4734" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-4735" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4736" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4737" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4738" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4739" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-4740" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-4741" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-4742" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-4743" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4744" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4745" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4746" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4747" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-4748" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4749" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-4750" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4751" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4752" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4753" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4754" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-4755" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-4756" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-4757" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-4758" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4759" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-4760" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4761" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-4762" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-4763" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎠</span></span></span></span><span id="MJXc-Node-4764" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>M</mi><mi>n</mi></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mo>⋅</mo><msup><mn>4</mn><mi>n</mi></msup></mrow></mfrac><mrow><mo>(</mo><mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msup><mn>4</mn><mi>n</mi></msup><mo>+</mo><mn>2</mn></mtd><mtd><msup><mn>4</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn></mtd><mtd><msup><mn>4</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd><msup><mn>4</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn></mtd><mtd><msup><mn>4</mn><mi>n</mi></msup><mo>+</mo><mn>2</mn></mtd><mtd><msup><mn>4</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd><msup><mn>4</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn></mtd><mtd><msup><mn>4</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn></mtd><mtd><msup><mn>4</mn><mi>n</mi></msup><mo>+</mo><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-367">M^n=\frac{1}{3\cdot 4^n}\left(\begin{array}{ccc}
4^n+2&4^n-1&4^n-1\\
4^n-1&4^n+2&4^n-1\\
4^n-1&4^n-1&4^n+2
\end{array} \right).</script>
En déduire une expression de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-368-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4765" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4766" class="mjx-mrow"><span id="MJXc-Node-4767" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4768" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4769" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-368">a_n</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-369-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4770" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4771" class="mjx-mrow"><span id="MJXc-Node-4772" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4773" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4774" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-369">b_n</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-370-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4775" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4776" class="mjx-mrow"><span id="MJXc-Node-4777" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4778" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4779" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-370">c_n</script> pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-371-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4780" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4781" class="mjx-mrow"><span id="MJXc-Node-4782" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-4783" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-4784" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-4785" class="mjx-mrow"><span id="MJXc-Node-4786" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-371">n\in\mathbb N</script>.
</li><li> Déterminer les limites respectives des suites <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-372-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4787" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4788" class="mjx-mrow"><span id="MJXc-Node-4789" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4790" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4791" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4792" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4793" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-372">(a_n)</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-373-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4794" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4795" class="mjx-mrow"><span id="MJXc-Node-4796" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4797" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4798" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4799" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4800" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-373">(b_n)</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-374-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4801" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4802" class="mjx-mrow"><span id="MJXc-Node-4803" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-4804" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4805" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4806" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-4807" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>c</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-374">(c_n)</script>. Interpréter le résultat.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication3091&#39;,&#39;imageindication3091&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication3091"></a><div class="inner" id="innerindication3091" style="display:none;"><ol class="enumeratechiffre">
<li> Utiliser que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-375-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4808" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4809" class="mjx-mrow"><span id="MJXc-Node-4810" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4811" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4812" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4813" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4814" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-4815" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4816" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mn>1</mn></msub><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-375">b_1=c_1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-376-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4817" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4818" class="mjx-mrow"><span id="MJXc-Node-4819" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4820" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4821" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4822" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4823" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-4824" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4825" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4826" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4827" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-4828" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4829" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4830" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4831" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-376">a_1+b_1+c_1=1</script>.
</li><li> Appliquer la formule des probabilités totales.
</li><li>
</li><li>
</li><li>
</li></ol>
</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige3091&#39;,&#39;imagecorrige3091&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige3091"></a><div class="inner" id="innercorrige3091" style="display:none;"><ol class="enumeratechiffre">
<li> Puisqu'en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-377-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4832" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4833" class="mjx-mrow"><span id="MJXc-Node-4834" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4835" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4836" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-377">n=0</script> le pion est en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-378-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4837" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4838" class="mjx-mrow"><span id="MJXc-Node-4839" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-378">A</script>, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-379-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4840" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4841" class="mjx-mrow"><span id="MJXc-Node-4842" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4843" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4844" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-4845" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4846" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-379">a_0=1</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-380-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4847" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4848" class="mjx-mrow"><span id="MJXc-Node-4849" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4850" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4851" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-4852" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4853" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-380">b_0=0</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-381-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4854" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4855" class="mjx-mrow"><span id="MJXc-Node-4856" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4857" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4858" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-4859" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4860" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-381">c_0=0</script>.
A l'étape <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-382-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4861" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4862" class="mjx-mrow"><span id="MJXc-Node-4863" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4864" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4865" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-382">n=1</script>, d'après les informations de l'énoncé, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-383-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4866" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4867" class="mjx-mrow"><span id="MJXc-Node-4868" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4869" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4870" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4871" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4872" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4873" class="mjx-texatom"><span id="MJXc-Node-4874" class="mjx-mrow"><span id="MJXc-Node-4875" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4876" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-383">a_1=1/2</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-384-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4877" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4878" class="mjx-mrow"><span id="MJXc-Node-4879" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4880" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4881" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4882" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4883" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-4884" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4885" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mn>1</mn></msub><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-384">b_1=c_1</script>. Puisque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-385-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4886" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4887" class="mjx-mrow"><span id="MJXc-Node-4888" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4889" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4890" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4891" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4892" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-4893" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4894" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4895" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4896" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-4897" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4898" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4899" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4900" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-385">a_1+b_1+c_1=1</script>, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-386-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4901" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4902" class="mjx-mrow"><span id="MJXc-Node-4903" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4904" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4905" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4906" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4907" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-4908" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4909" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-4910" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4911" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-4912" class="mjx-texatom"><span id="MJXc-Node-4913" class="mjx-mrow"><span id="MJXc-Node-4914" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-4915" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mn>1</mn></msub><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-386">b_1=c_1=1/4</script>.
</li><li> Les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-387-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4916" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4917" class="mjx-mrow"><span id="MJXc-Node-4918" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4919" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4920" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4921" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-4922" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-4923" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4924" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4925" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>n</mi></msub><mo>,</mo><mtext>&nbsp;</mtext><msub><mi>B</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-387">A_n,\ B_n</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-388-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-4926" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4927" class="mjx-mrow"><span id="MJXc-Node-4928" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-4929" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4930" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-388">C_n</script> forment un système complet d'événements. D'après la formule des probabilités totales,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-389-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-4931" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-4932" class="mjx-mrow"><span id="MJXc-Node-4933" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4934" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4935" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4936" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4937" class="mjx-texatom" style=""><span id="MJXc-Node-4938" class="mjx-mrow"><span id="MJXc-Node-4939" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4940" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4941" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4942" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4943" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-4944" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-4945" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.241em; padding-right: 0.071em;"><span id="MJXc-Node-4946" class="mjx-texatom" style=""><span id="MJXc-Node-4947" class="mjx-mrow"><span id="MJXc-Node-4948" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4949" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4950" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-4951" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4952" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4953" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4954" class="mjx-texatom" style=""><span id="MJXc-Node-4955" class="mjx-mrow"><span id="MJXc-Node-4956" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4957" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4958" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4959" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4960" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4961" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4962" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4963" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4964" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4965" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4966" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4967" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-4968" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4969" class="mjx-texatom" style=""><span id="MJXc-Node-4970" class="mjx-mrow"><span id="MJXc-Node-4971" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4972" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4973" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-4974" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4975" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4976" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4977" class="mjx-texatom" style=""><span id="MJXc-Node-4978" class="mjx-mrow"><span id="MJXc-Node-4979" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-4980" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4981" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-4982" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4983" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-4984" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4985" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4986" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4987" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-4988" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-4989" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-4990" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-4991" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-4992" class="mjx-texatom" style=""><span id="MJXc-Node-4993" class="mjx-mrow"><span id="MJXc-Node-4994" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-4995" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-4996" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-4997" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-4998" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4999" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5000" class="mjx-texatom" style=""><span id="MJXc-Node-5001" class="mjx-mrow"><span id="MJXc-Node-5002" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5003" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5004" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5005" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5006" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5007" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5008" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-5009" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5010" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5011" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5012" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>A</mi><mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>B</mi><mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>B</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>C</mi><mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-389">P(A_{n+1})=P_{A_n}(A_{n+1})P(A_n)+P_{B_n}(A_{n+1})P(B_n)+P_{C_n}(A_{n+1})P(C_n).</script>
Comme à la question précédente, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-390-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5013" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5014" class="mjx-mrow"><span id="MJXc-Node-5015" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-5016" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.241em; padding-right: 0.071em;"><span id="MJXc-Node-5017" class="mjx-texatom" style=""><span id="MJXc-Node-5018" class="mjx-mrow"><span id="MJXc-Node-5019" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5020" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5021" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-5022" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5023" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5024" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5025" class="mjx-texatom" style=""><span id="MJXc-Node-5026" class="mjx-mrow"><span id="MJXc-Node-5027" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5028" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5029" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5030" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5031" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5032" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5033" class="mjx-texatom"><span id="MJXc-Node-5034" class="mjx-mrow"><span id="MJXc-Node-5035" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5036" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>A</mi><mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-390">P_{A_n}(A_{n+1})=1/2</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-391-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5037" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5038" class="mjx-mrow"><span id="MJXc-Node-5039" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-5040" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5041" class="mjx-texatom" style=""><span id="MJXc-Node-5042" class="mjx-mrow"><span id="MJXc-Node-5043" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5044" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5045" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-5046" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5047" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5048" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5049" class="mjx-texatom" style=""><span id="MJXc-Node-5050" class="mjx-mrow"><span id="MJXc-Node-5051" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5052" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5053" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5054" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5055" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5056" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5057" class="mjx-texatom"><span id="MJXc-Node-5058" class="mjx-mrow"><span id="MJXc-Node-5059" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5060" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>B</mi><mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-391">P_{B_n}(A_{n+1})=1/4</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-392-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5061" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5062" class="mjx-mrow"><span id="MJXc-Node-5063" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-5064" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-5065" class="mjx-texatom" style=""><span id="MJXc-Node-5066" class="mjx-mrow"><span id="MJXc-Node-5067" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-5068" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5069" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span></span></span><span id="MJXc-Node-5070" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5071" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5072" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5073" class="mjx-texatom" style=""><span id="MJXc-Node-5074" class="mjx-mrow"><span id="MJXc-Node-5075" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5076" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5077" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5078" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5079" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5080" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5081" class="mjx-texatom"><span id="MJXc-Node-5082" class="mjx-mrow"><span id="MJXc-Node-5083" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5084" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>C</mi><mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-392">P_{C_n}(A_{n+1})=1/4</script>. On en déduit que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-393-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5085" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5086" class="mjx-mrow"><span id="MJXc-Node-5087" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5088" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5089" class="mjx-texatom" style=""><span id="MJXc-Node-5090" class="mjx-mrow"><span id="MJXc-Node-5091" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5092" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5093" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5094" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5095" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5096" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-5097" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-5098" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5099" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5100" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5101" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5102" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5103" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.719em;"><span id="MJXc-Node-5104" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-5105" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5106" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5107" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5108" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5109" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5110" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.719em;"><span id="MJXc-Node-5111" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-5112" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5113" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5114" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5115" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>b</mi><mi>n</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>c</mi><mi>n</mi></msub><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-393">a_{n+1}=\frac 12 a_n+\frac 14b_n+\frac 14 c_n.</script>
En raisonnant de la même façon, ou par symétrie,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-394-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5116" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5117" class="mjx-mrow"><span id="MJXc-Node-5118" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5119" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5120" class="mjx-texatom" style=""><span id="MJXc-Node-5121" class="mjx-mrow"><span id="MJXc-Node-5122" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5123" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5124" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5125" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5126" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5127" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.719em;"><span id="MJXc-Node-5128" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-5129" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5130" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5131" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5132" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5133" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5134" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-5135" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-5136" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5137" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5138" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5139" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5140" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5141" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.719em;"><span id="MJXc-Node-5142" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-5143" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5144" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5145" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>b</mi><mi>n</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>c</mi><mi>n</mi></msub></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-394">b_{n+1}=\frac 14 a_n+\frac 12b_n+\frac 14 c_n</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-395-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5146" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5147" class="mjx-mrow"><span id="MJXc-Node-5148" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5149" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5150" class="mjx-texatom" style=""><span id="MJXc-Node-5151" class="mjx-mrow"><span id="MJXc-Node-5152" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-5153" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5154" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5155" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5156" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5157" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.719em;"><span id="MJXc-Node-5158" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-5159" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5160" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5161" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5162" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5163" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5164" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.719em;"><span id="MJXc-Node-5165" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-5166" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5167" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5168" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5169" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5170" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5171" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-5172" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-5173" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5174" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5175" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5176" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>c</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>b</mi><mi>n</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>c</mi><mi>n</mi></msub><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-395">c_{n+1}=\frac 14 a_n+\frac 14b_n+\frac 12 c_n.</script>
</li><li> D'après la question précédente, la matrice
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-396-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable columnalign=&quot;center center center&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5177" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5178" class="mjx-mrow"><span id="MJXc-Node-5179" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-5180" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5181" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-5182" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.719em;"><span id="MJXc-Node-5183" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.087em; vertical-align: -0.719em;"></span></span><span id="MJXc-Node-5184" class="mjx-mrow"><span id="MJXc-Node-5185" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎝</span></span></span><span id="MJXc-Node-5186" class="mjx-mtable" style="vertical-align: -1.65em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-5187" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-5188" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span id="MJXc-Node-5189" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5190" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5191" class="mjx-mtd" style="padding: 0px 0.5em; width: 0.5em;"><span id="MJXc-Node-5192" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5193" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5194" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 0.5em;"><span id="MJXc-Node-5195" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5196" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5197" class="mjx-mtr" style="height: 1.4em;"><span id="MJXc-Node-5198" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-5199" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5200" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5201" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-5202" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5203" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5204" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-5205" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5206" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5207" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-5208" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-5209" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5210" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5211" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-5212" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5213" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5214" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-5215" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-5216" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-5217" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎠</span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>M</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mrow><mo>(</mo><mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-396">M=\frac 14\left(\begin{array}{ccc}
2&1&1\\
1&2&1\\
1&1&2
\end{array}\right)</script>
convient.
</li><li> On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-397-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5218" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5219" class="mjx-mrow"><span id="MJXc-Node-5220" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.186em;"><span id="MJXc-Node-5221" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.186em;">V</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5222" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5223" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5224" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.081em;"><span id="MJXc-Node-5225" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.22em; padding-right: 0.071em;"><span id="MJXc-Node-5226" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5227" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.186em;"><span id="MJXc-Node-5228" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.186em;">V</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5229" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>n</mi></msub><mo>=</mo><msup><mi>M</mi><mi>n</mi></msup><msub><mi>V</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-397">V_n=M^n V_0</script>, ce qui donne
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-398-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x22C5;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x22C5;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;&amp;#x22C5;&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5230" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5231" class="mjx-mrow"><span id="MJXc-Node-5232" class="mjx-mrow"><span id="MJXc-Node-5233" class="mjx-mo" style="vertical-align: -1.528em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.703em; padding-bottom: 0.295em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.214em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪
⎪
⎪
⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.214em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪
⎪
⎪
⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.231em; padding-bottom: 1.17em;">⎩</span></span></span><span id="MJXc-Node-5234" class="mjx-mtable" style="vertical-align: -2.452em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-5235" class="mjx-mtr" style="height: 1.734em;"><span id="MJXc-Node-5236" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: right; width: 0.989em;"><span id="MJXc-Node-5237" class="mjx-mrow" style="margin-top: 0.038em;"><span id="MJXc-Node-5238" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5239" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5240" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5241" class="mjx-mtd" style="padding: 0px 0.5em; width: 0.778em;"><span id="MJXc-Node-5242" class="mjx-mrow" style="margin-top: 0.038em;"><span id="MJXc-Node-5243" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5244" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 2.242em;"><span id="MJXc-Node-5245" class="mjx-mrow"><span id="MJXc-Node-5246" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.724em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.438em; top: -1.469em;"><span id="MJXc-Node-5247" class="mjx-mrow" style=""><span id="MJXc-Node-5248" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5249" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5250" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5251" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5252" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.438em; bottom: -0.701em;"><span id="MJXc-Node-5253" class="mjx-mrow" style=""><span id="MJXc-Node-5254" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-5255" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋅</span></span><span id="MJXc-Node-5256" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5257" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5258" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.724em;"></span></span><span class="mjx-vsize" style="height: 1.534em; vertical-align: -0.496em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5259" class="mjx-mtr" style="height: 1.934em;"><span id="MJXc-Node-5260" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span id="MJXc-Node-5261" class="mjx-mrow" style="margin-top: 0.038em;"><span id="MJXc-Node-5262" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5263" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5264" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5265" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-5266" class="mjx-mrow" style="margin-top: 0.038em;"><span id="MJXc-Node-5267" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5268" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-5269" class="mjx-mrow"><span id="MJXc-Node-5270" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.724em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.438em; top: -1.469em;"><span id="MJXc-Node-5271" class="mjx-mrow" style=""><span id="MJXc-Node-5272" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5273" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5274" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5275" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5276" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.438em; bottom: -0.701em;"><span id="MJXc-Node-5277" class="mjx-mrow" style=""><span id="MJXc-Node-5278" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-5279" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋅</span></span><span id="MJXc-Node-5280" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5281" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5282" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.724em;"></span></span><span class="mjx-vsize" style="height: 1.534em; vertical-align: -0.496em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5283" class="mjx-mtr" style="height: 1.734em;"><span id="MJXc-Node-5284" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span id="MJXc-Node-5285" class="mjx-mrow" style="margin-top: 0.038em;"><span id="MJXc-Node-5286" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5287" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5288" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5289" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-5290" class="mjx-mrow" style="margin-top: 0.038em;"><span id="MJXc-Node-5291" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5292" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-5293" class="mjx-mrow"><span id="MJXc-Node-5294" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.724em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.438em; top: -1.469em;"><span id="MJXc-Node-5295" class="mjx-mrow" style=""><span id="MJXc-Node-5296" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5297" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5298" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5299" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5300" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.438em; bottom: -0.701em;"><span id="MJXc-Node-5301" class="mjx-mrow" style=""><span id="MJXc-Node-5302" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-5303" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.295em;">⋅</span></span><span id="MJXc-Node-5304" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5305" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.607em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5306" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.724em;"></span></span><span class="mjx-vsize" style="height: 1.534em; vertical-align: -0.496em;"></span></span><span id="MJXc-Node-5307" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-5308" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>a</mi><mi>n</mi></msub></mtd><mtd><mo>=</mo></mtd><mtd><mfrac><mrow><msup><mn>4</mn><mi>n</mi></msup><mo>+</mo><mn>2</mn></mrow><mrow><mn>3</mn><mo>⋅</mo><msup><mn>4</mn><mi>n</mi></msup></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>n</mi></msub></mtd><mtd><mo>=</mo></mtd><mtd><mfrac><mrow><msup><mn>4</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn><mo>⋅</mo><msup><mn>4</mn><mi>n</mi></msup></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mi>c</mi><mi>n</mi></msub></mtd><mtd><mo>=</mo></mtd><mtd><mfrac><mrow><msup><mn>4</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn><mo>⋅</mo><msup><mn>4</mn><mi>n</mi></msup></mrow></mfrac><mo>.</mo></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-398">\left\{\begin{array}{rcl}
a_n&=&\frac{4^n+2}{3\cdot 4^n}\\
b_n&=&\frac{4^n-1}{3\cdot 4^n}\\
c_n&=&\frac{4^n-1}{3\cdot 4^n}.\\
\end{array}
\right.</script>
On remarque qu'on a bien <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-399-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5309" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5310" class="mjx-mrow"><span id="MJXc-Node-5311" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5312" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5313" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5314" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5315" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-5316" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5317" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5318" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5319" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-5320" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5321" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5322" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5323" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><msub><mi>b</mi><mi>n</mi></msub><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-399">a_n+b_n+c_n=1</script>.
</li><li> Les suites <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-400-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5324" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5325" class="mjx-mrow"><span id="MJXc-Node-5326" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5327" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5328" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5329" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5330" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-400">(a_n)</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-401-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5331" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5332" class="mjx-mrow"><span id="MJXc-Node-5333" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5334" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5335" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5336" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5337" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-401">(b_n)</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-402-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5338" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5339" class="mjx-mrow"><span id="MJXc-Node-5340" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5341" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5342" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5343" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-5344" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>c</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-402">(c_n)</script> convergent toutes les trois vers 1/3. Au bout d'un très grand nombre de déplacements,
les probabilités que le pion soit sur chacun des 3 points sont presque égales.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>29</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Sauts de puce<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1250"> [Signaler une erreur]</a><a id="exo29" href="javascript:ajouteexo(1250,29,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1250&#39;,&#39;imageenonce1250&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1250"></a><div class="inner" id="innerenonce1250">Une particule se trouve à l'instant 0 au point d'abscisse <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-403-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5345" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5346" class="mjx-mrow"><span id="MJXc-Node-5347" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-403">a</script> (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-404-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5348" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5349" class="mjx-mrow"><span id="MJXc-Node-5350" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-404">a</script> entier), sur un segment gradué de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-405-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5351" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5352" class="mjx-mrow"><span id="MJXc-Node-5353" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-405">0</script> à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-406-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5354" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5355" class="mjx-mrow"><span id="MJXc-Node-5356" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-406">N</script> (on suppose donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-407-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2264;&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5357" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5358" class="mjx-mrow"><span id="MJXc-Node-5359" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-5360" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≤</span></span><span id="MJXc-Node-5361" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-5362" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≤</span></span><span id="MJXc-Node-5363" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-407">0\leq a\leq N</script>).
A chaque instant, elle fait un bond de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-408-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5364" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5365" class="mjx-mrow"><span id="MJXc-Node-5366" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5367" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-408">+1</script> avec la probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-409-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5368" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5369" class="mjx-mrow"><span id="MJXc-Node-5370" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-409">p</script> (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-410-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5371" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5372" class="mjx-mrow"><span id="MJXc-Node-5373" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-5374" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&lt;</span></span><span id="MJXc-Node-5375" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span><span id="MJXc-Node-5376" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&lt;</span></span><span id="MJXc-Node-5377" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5378" class="mjx-texatom"><span id="MJXc-Node-5379" class="mjx-mrow"><span id="MJXc-Node-5380" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">/</span></span></span></span><span id="MJXc-Node-5381" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-410">0<p<1/2</script>), ou un bond de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-411-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5382" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5383" class="mjx-mrow"><span id="MJXc-Node-5384" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-5385" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-411">-1</script> avec la probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-412-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5386" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5387" class="mjx-mrow"><span id="MJXc-Node-5388" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-5389" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5390" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5391" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-5392" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-412">q=1-p</script>.
Autrement dit, si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-413-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5393" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5394" class="mjx-mrow"><span id="MJXc-Node-5395" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5396" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5397" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-413">x_n</script> est l'abscisse de la particule à l'instant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-414-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5398" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5399" class="mjx-mrow"><span id="MJXc-Node-5400" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-414">n</script>, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-415-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow&gt;&lt;mtext&gt;avec probabilit&amp;#xE9;&amp;#xA0;&lt;/mtext&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow&gt;&lt;mtext&gt;avec probabilit&amp;#xE9;&amp;#xA0;&lt;/mtext&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mtext&gt;.&lt;/mtext&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5401" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5402" class="mjx-mrow"><span id="MJXc-Node-5403" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5404" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5405" class="mjx-texatom" style=""><span id="MJXc-Node-5406" class="mjx-mrow"><span id="MJXc-Node-5407" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-5408" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5409" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span><span id="MJXc-Node-5410" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5411" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-5412" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.236em; padding-bottom: 1.236em;">{</span></span><span id="MJXc-Node-5413" class="mjx-mtable" style="vertical-align: -1em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-5414" class="mjx-mtr" style="height: 1.25em;"><span id="MJXc-Node-5415" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 2.754em;"><span id="MJXc-Node-5416" class="mjx-mrow" style="margin-top: -0.175em;"><span id="MJXc-Node-5417" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5418" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5419" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-5420" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5421" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5422" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 9.696em;"><span id="MJXc-Node-5423" class="mjx-mrow" style="margin-top: -0.175em;"><span id="MJXc-Node-5424" class="mjx-texatom"><span id="MJXc-Node-5425" class="mjx-mrow"><span id="MJXc-Node-5426" class="mjx-mrow"><span id="MJXc-Node-5427" class="mjx-mtext"><span class="mjx-char" style="padding-top: 0.522em; padding-bottom: 0.198em;"><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">avec probabilit</span><span class="mjx-charbox MJXc-TeX-unknown-R" style="padding-bottom: 0.273em; width: 0.493em;">é</span><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">&nbsp;</span></span></span><span id="MJXc-Node-5428" class="mjx-texatom"><span id="MJXc-Node-5429" class="mjx-mrow"><span id="MJXc-Node-5430" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-5431" class="mjx-mtr" style="height: 1.25em;"><span id="MJXc-Node-5432" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span id="MJXc-Node-5433" class="mjx-mrow" style="margin-top: -0.175em;"><span id="MJXc-Node-5434" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5435" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5436" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-5437" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-5438" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-5439" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span id="MJXc-Node-5440" class="mjx-mrow" style="margin-top: -0.175em;"><span id="MJXc-Node-5441" class="mjx-texatom"><span id="MJXc-Node-5442" class="mjx-mrow"><span id="MJXc-Node-5443" class="mjx-mrow"><span id="MJXc-Node-5444" class="mjx-mtext"><span class="mjx-char" style="padding-top: 0.522em; padding-bottom: 0.198em;"><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">avec probabilit</span><span class="mjx-charbox MJXc-TeX-unknown-R" style="padding-bottom: 0.273em; width: 0.493em;">é</span><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">&nbsp;</span></span></span><span id="MJXc-Node-5445" class="mjx-texatom"><span id="MJXc-Node-5446" class="mjx-mrow"><span id="MJXc-Node-5447" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-5448" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-5449" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span id="MJXc-Node-5450" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">.</span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-5451" class="mjx-mo" style="width: 0.12em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mn>1</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mrow><mtext>avec probabilité&nbsp;</mtext><mrow class="MJX-TeXAtom-ORD"><mi>p</mi></mrow></mrow></mrow></mtd></mtr><mtr><mtd><msub><mi>x</mi><mi>n</mi></msub><mo>−</mo><mn>1</mn></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mrow><mtext>avec probabilité&nbsp;</mtext><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mo>−</mo><mi>p</mi></mrow><mtext>.</mtext></mrow></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-415">x_{n+1}=\left\{\begin{array}{ll}
x_n+1&\textrm{avec probabilité $p$}\\
x_n-1&\textrm{avec probabilité $1-p$.}
\end{array}\right.</script>
Le processus se termine lorsque la particule atteint une des extrémités du segment (i.e. s'il existe <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-416-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5452" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5453" class="mjx-mrow"><span id="MJXc-Node-5454" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5455" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5456" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-416">x_n</script> avec <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-417-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5457" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5458" class="mjx-mrow"><span id="MJXc-Node-5459" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5460" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5461" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-5462" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5463" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-417">x_n=0</script> ou <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-418-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5464" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5465" class="mjx-mrow"><span id="MJXc-Node-5466" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5467" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5468" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-5469" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5470" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-418">x_n=N</script>).
<ol class="enumeratechiffre">
<li> Écrire une fonction python <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-419-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mtext mathvariant=&quot;monospace&quot;&gt;puce(a,N,p)&lt;/mtext&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5471" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5472" class="mjx-mrow"><span id="MJXc-Node-5473" class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.458em; padding-bottom: 0.458em;">puce(a,N,p)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="monospace">puce(a,N,p)</mtext></math></span></span><script type="math/tex" id="MathJax-Element-419">\verb+puce(a,N,p)+</script> qui simule cette marche aléatoire et qui retourne l'endroit où la puce sort du segment (<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-420-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5474" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5475" class="mjx-mrow"><span id="MJXc-Node-5476" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-420">0</script> ou <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-421-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5477" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5478" class="mjx-mrow"><span id="MJXc-Node-5479" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-421">N</script>) et le nombre de pas nécessaires pour que le processus s'arrête.
</li><li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-422-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5480" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5481" class="mjx-mrow"><span id="MJXc-Node-5482" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5483" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5484" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>a</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-422">u_a</script> la probabilité pour que la particule partant de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-423-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5485" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5486" class="mjx-mrow"><span id="MJXc-Node-5487" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-423">a</script>, le processus s'arrête en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-424-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5488" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5489" class="mjx-mrow"><span id="MJXc-Node-5490" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-424">0</script>.
<ol class="enumeratechiffre">
<li> Que vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-425-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5491" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5492" class="mjx-mrow"><span id="MJXc-Node-5493" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5494" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5495" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-425">u_0</script>? <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-426-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5496" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5497" class="mjx-mrow"><span id="MJXc-Node-5498" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5499" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5500" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>N</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-426">u_N</script>?
</li><li> Montrer que si <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-427-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5501" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5502" class="mjx-mrow"><span id="MJXc-Node-5503" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span id="MJXc-Node-5504" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&lt;</span></span><span id="MJXc-Node-5505" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-5506" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">&lt;</span></span><span id="MJXc-Node-5507" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>a</mi><mo>&lt;</mo><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-427">0<a<N</script>, alors <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-428-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5508" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5509" class="mjx-mrow"><span id="MJXc-Node-5510" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5511" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5512" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span id="MJXc-Node-5513" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-5514" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-5515" class="mjx-mrow"><span id="MJXc-Node-5516" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span><span id="MJXc-Node-5517" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5518" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5519" class="mjx-texatom" style=""><span id="MJXc-Node-5520" class="mjx-mrow"><span id="MJXc-Node-5521" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-5522" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5523" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span></span></span><span id="MJXc-Node-5524" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5525" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-5526" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5527" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5528" class="mjx-texatom" style=""><span id="MJXc-Node-5529" class="mjx-mrow"><span id="MJXc-Node-5530" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span><span id="MJXc-Node-5531" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-5532" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>a</mi></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><mi>q</mi><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-428">u_a={pu_{a+1}}+qu_{a-1}</script>.
</li><li> En déduire l'expression exacte de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-429-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5533" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5534" class="mjx-mrow"><span id="MJXc-Node-5535" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5536" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5537" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>a</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-429">u_a</script>.
</li></ol>
</li><li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-430-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5538" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5539" class="mjx-mrow"><span id="MJXc-Node-5540" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5541" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">v</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5542" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>a</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-430">v_a</script> la probabilité pour que la particule partant de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-431-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5543" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5544" class="mjx-mrow"><span id="MJXc-Node-5545" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-431">a</script>, le processus s'arrête en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-432-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5546" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5547" class="mjx-mrow"><span id="MJXc-Node-5548" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-432">N</script>. Reprendre les questions précédentes avec <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-433-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5549" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5550" class="mjx-mrow"><span id="MJXc-Node-5551" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5552" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">v</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5553" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>a</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-433">v_a</script> au lieu de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-434-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5554" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5555" class="mjx-mrow"><span id="MJXc-Node-5556" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5557" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5558" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>a</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-434">u_a</script>.
</li><li> Calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-435-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-5559" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5560" class="mjx-mrow"><span id="MJXc-Node-5561" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5562" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5563" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span><span id="MJXc-Node-5564" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-5565" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-5566" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">v</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5567" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">a</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>a</mi></msub><mo>+</mo><msub><mi>v</mi><mi>a</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-435">u_a+v_a</script>. Qu'en déduisez-vous?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1250&#39;,&#39;imageindication1250&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1250"></a><div class="inner" id="innerindication1250" style="display:none;"><ol class="enumeratechiffre">
<li>
</li><li>
<ol class="enumeratechiffre">
<li> Il suffit de décrypter l'énoncé.
</li><li> Si la particule atteint le point <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-436-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5568" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5569" class="mjx-mrow"><span id="MJXc-Node-5570" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5571" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5572" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-436">a+1</script>, la probabilité que le processus s'arrête en 0 vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-437-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5573" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5574" class="mjx-mrow"><span id="MJXc-Node-5575" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5576" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5577" class="mjx-texatom" style=""><span id="MJXc-Node-5578" class="mjx-mrow"><span id="MJXc-Node-5579" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5580" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5581" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-437">u_{a+1}</script>. Appliquer la formule des probabilités totales.
</li><li> Réviser les suites récurrentes d'ordre 2.
</li></ol>
</li><li> Les mêmes indications que celles de la question précédente conviennent!
</li><li> Que dire d'un événement de probabilité 1?
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1250&#39;,&#39;imagecorrige1250&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1250"></a><div class="inner" id="innercorrige1250" style="display:none;"><ol class="enumeratechiffre">
<li> Voici un exemple de fonction.<br>
<p></p><div class="codebox"><div style="font-family: &#39;Courier New&#39;;font-size:10pt" dir="ltr">
<br>import random
<br>
<br>def puce(a,N,p):
<br>&nbsp;&nbsp;&nbsp;&nbsp;pas=0
<br>&nbsp;&nbsp;&nbsp;&nbsp;position=a
<br>&nbsp;&nbsp;&nbsp;&nbsp;while ( (position&gt;0) and (position&lt;N) ):
<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if (random.random()&lt;p):
<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;position=position+1
<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;else:
<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;position=position-1
<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;pas=pas+1
<br>&nbsp;&nbsp;&nbsp;&nbsp;return (position,pas)
<br></div></div><p>
</p></li><li>
<ol class="enumeratechiffre">
<li> Par définition, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-438-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5582" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5583" class="mjx-mrow"><span id="MJXc-Node-5584" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5585" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5586" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-5587" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5588" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-438">u_0=1</script> (le processus commence en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-439-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5589" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5590" class="mjx-mrow"><span id="MJXc-Node-5591" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-439">0</script>, il s'arrête immédiatement, en 0), et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-440-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5592" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5593" class="mjx-mrow"><span id="MJXc-Node-5594" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5595" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5596" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-5597" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5598" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>N</mi></msub><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-440">u_N=0</script> (le processus commence en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-441-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5599" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5600" class="mjx-mrow"><span id="MJXc-Node-5601" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-441">N</script>, il s'arrête aussitôt, en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-442-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5602" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5603" class="mjx-mrow"><span id="MJXc-Node-5604" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-442">N</script>).
</li><li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-443-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5605" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5606" class="mjx-mrow"><span id="MJXc-Node-5607" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-443">A</script> l'événement : "Partant de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-444-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5608" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5609" class="mjx-mrow"><span id="MJXc-Node-5610" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-444">a</script>, le processus s'arrête en 0", <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-445-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5611" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5612" class="mjx-mrow"><span id="MJXc-Node-5613" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-445">B</script> l'événement : "Partant de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-446-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5614" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5615" class="mjx-mrow"><span id="MJXc-Node-5616" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-446">a</script> à l'instant 0, à l'instant 1, la particule est en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-447-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5617" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5618" class="mjx-mrow"><span id="MJXc-Node-5619" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5620" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5621" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-447">a+1</script>", et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-448-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5622" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5623" class="mjx-mrow"><span id="MJXc-Node-5624" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-448">C</script> l'événement : "Partant de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-449-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5625" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5626" class="mjx-mrow"><span id="MJXc-Node-5627" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-449">a</script> à l'instant 0, à l'instant 1, la particule est en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-450-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5628" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5629" class="mjx-mrow"><span id="MJXc-Node-5630" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5631" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5632" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-450">a-1</script>", de sorte que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-451-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5633" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5634" class="mjx-mrow"><span id="MJXc-Node-5635" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-5636" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5637" class="mjx-texatom MJXc-space3"><span id="MJXc-Node-5638" class="mjx-mrow"><span id="MJXc-Node-5639" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.213em;"><span id="MJXc-Node-5641" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-5640" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>B</mi><mo stretchy="false">¯</mo></mover></mrow></math></span></span><script type="math/tex" id="MathJax-Element-451">C=\bar B</script>. Par la formule des probabilités totales :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-452-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5642" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5643" class="mjx-mrow"><span id="MJXc-Node-5644" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5645" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5646" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-5647" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5648" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5649" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5650" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5651" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-5652" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5653" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5654" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5655" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-5656" class="mjx-texatom"><span id="MJXc-Node-5657" class="mjx-mrow"><span id="MJXc-Node-5658" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-5659" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-5660" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5661" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5662" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5663" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5664" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-5665" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5666" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5667" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5668" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-5669" class="mjx-texatom"><span id="MJXc-Node-5670" class="mjx-mrow"><span id="MJXc-Node-5671" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-5672" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-5673" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5674" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>C</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-452">P(A)=P(B)P(A|B)+P(C)P(A|C).</script>
Maintenant, puisqu'on part à l'instant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-453-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5675" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5676" class="mjx-mrow"><span id="MJXc-Node-5677" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span><span id="MJXc-Node-5678" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5679" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-453">t=0</script> de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-454-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5680" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5681" class="mjx-mrow"><span id="MJXc-Node-5682" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-454">a</script>, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-455-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5683" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5684" class="mjx-mrow"><span id="MJXc-Node-5685" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5686" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5687" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-5688" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5689" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5690" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-455">P(B)=p</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-456-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5691" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5692" class="mjx-mrow"><span id="MJXc-Node-5693" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5694" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5695" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-5696" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5697" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5698" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>q</mi></math></span></span><script type="math/tex" id="MathJax-Element-456">P(C)=q</script>. D'autre part, si la particule est à l'instant 1 en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-457-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5699" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5700" class="mjx-mrow"><span id="MJXc-Node-5701" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5702" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5703" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-457">a+1</script>, la probabilité que le processus s'arrête en 0 vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-458-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5704" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5705" class="mjx-mrow"><span id="MJXc-Node-5706" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5707" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5708" class="mjx-texatom" style=""><span id="MJXc-Node-5709" class="mjx-mrow"><span id="MJXc-Node-5710" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5711" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5712" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-458">u_{a+1}</script>. On a donc :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-459-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5713" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5714" class="mjx-mrow"><span id="MJXc-Node-5715" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5716" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5717" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-5718" class="mjx-texatom"><span id="MJXc-Node-5719" class="mjx-mrow"><span id="MJXc-Node-5720" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-5721" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-5722" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5723" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5724" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-5725" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5726" class="mjx-texatom" style=""><span id="MJXc-Node-5727" class="mjx-mrow"><span id="MJXc-Node-5728" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5729" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5730" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-459">P(A|B)=u_{a+1}</script>, et de même <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-460-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5731" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5732" class="mjx-mrow"><span id="MJXc-Node-5733" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-5734" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5735" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-5736" class="mjx-texatom"><span id="MJXc-Node-5737" class="mjx-mrow"><span id="MJXc-Node-5738" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-5739" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-5740" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-5741" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5742" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-5743" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5744" class="mjx-texatom" style=""><span id="MJXc-Node-5745" class="mjx-mrow"><span id="MJXc-Node-5746" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5747" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5748" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-460">P(A|C)=u_{a-1}</script>. On en déduit la formule de récurrence :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-461-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5749" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5750" class="mjx-mrow"><span id="MJXc-Node-5751" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5752" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5753" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-5754" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5755" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-5756" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5757" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5758" class="mjx-texatom" style=""><span id="MJXc-Node-5759" class="mjx-mrow"><span id="MJXc-Node-5760" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5761" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5762" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5763" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5764" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-5765" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5766" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5767" class="mjx-texatom" style=""><span id="MJXc-Node-5768" class="mjx-mrow"><span id="MJXc-Node-5769" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5770" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5771" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5772" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>u</mi><mi>a</mi></msub><mo>=</mo><mi>p</mi><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>q</mi><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-461">u_a=pu_{a+1}+qu_{a-1}.</script>
</li><li> Pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-462-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5773" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5774" class="mjx-mrow"><span id="MJXc-Node-5775" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-462">a</script> allant de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-463-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5776" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5777" class="mjx-mrow"><span id="MJXc-Node-5778" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-463">1</script> à <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-464-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5779" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5780" class="mjx-mrow"><span id="MJXc-Node-5781" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-5782" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5783" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-464">N-1</script>, la suite <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-465-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5784" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5785" class="mjx-mrow"><span id="MJXc-Node-5786" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-5787" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5788" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5789" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-5790" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>u</mi><mi>a</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-465">(u_a)</script> vérifie la formule de récurrence :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-466-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5791" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5792" class="mjx-mrow"><span id="MJXc-Node-5793" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5794" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5795" class="mjx-texatom" style=""><span id="MJXc-Node-5796" class="mjx-mrow"><span id="MJXc-Node-5797" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5798" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5799" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5800" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5801" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.368em;"><span id="MJXc-Node-5802" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-5803" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.272em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-5804" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5805" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5806" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-5807" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5808" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-5809" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-5810" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-5811" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5812" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5813" class="mjx-texatom" style=""><span id="MJXc-Node-5814" class="mjx-mrow"><span id="MJXc-Node-5815" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5816" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5817" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5818" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><msub><mi>u</mi><mi>a</mi></msub><mo>−</mo><mfrac><mi>q</mi><mi>p</mi></mfrac><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-466">u_{a+1}=\frac{1}{p}u_a-\frac{q}{p}u_{a-1}.</script>
L'équation caractéristique de cette récurrence est :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-467-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mfrac&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5819" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5820" class="mjx-mrow"><span id="MJXc-Node-5821" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5822" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5823" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-5824" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5825" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.368em;"><span id="MJXc-Node-5826" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-5827" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.272em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-5828" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span><span id="MJXc-Node-5829" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5830" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-5831" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-5832" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-5833" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5834" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>r</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mi>r</mi><mo>+</mo><mfrac><mi>q</mi><mi>p</mi></mfrac><mo>=</mo><mn>0.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-467">r^2-\frac{1}{p}r+\frac{q}{p}=0.</script>
Cette équation du second degré admet deux solutions distinctes,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-468-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;&amp;#xA0;et&amp;#xA0;&lt;/mtext&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5835" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5836" class="mjx-mrow"><span id="MJXc-Node-5837" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5838" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5839" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-5840" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5841" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5842" class="mjx-texatom"><span id="MJXc-Node-5843" class="mjx-mrow"><span id="MJXc-Node-5844" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">&nbsp;et&nbsp;</span></span></span></span><span id="MJXc-Node-5845" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5846" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">r</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5847" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span id="MJXc-Node-5848" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5849" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-5850" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-5851" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>r</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mtext>&nbsp;et&nbsp;</mtext></mrow><msub><mi>r</mi><mn>2</mn></msub><mo>=</mo><mfrac><mi>q</mi><mi>p</mi></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-468">r_1=1\textrm{ et }r_2=\frac{q}{p}</script>
(remarquer ici l'utilisation de l'hypothèse <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-469-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&amp;#x2260;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5852" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5853" class="mjx-mrow"><span id="MJXc-Node-5854" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-5855" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.528em;">≠</span></span><span id="MJXc-Node-5856" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-5857" class="mjx-texatom"><span id="MJXc-Node-5858" class="mjx-mrow"><span id="MJXc-Node-5859" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-5860" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≠</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-469">p\neq 1/2</script> qui permet d'affirmer que les deux racines sont distinctes).
Il existe donc des réels <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-470-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5861" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5862" class="mjx-mrow"><span id="MJXc-Node-5863" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-470">\lambda</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-471-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03BC;&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5864" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5865" class="mjx-mrow"><span id="MJXc-Node-5866" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">μ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-471">\mu</script> tels que, pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-472-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5867" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5868" class="mjx-mrow"><span id="MJXc-Node-5869" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-472">a</script> dans <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-473-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5870" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5871" class="mjx-mrow"><span id="MJXc-Node-5872" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-5873" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-5874" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-5875" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">…</span></span><span id="MJXc-Node-5876" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-5877" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-5878" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-473">\{0,\dots,N\}</script>, on ait :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-474-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;&amp;#x03BB;&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03BC;&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5879" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5880" class="mjx-mrow"><span id="MJXc-Node-5881" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5882" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5883" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-5884" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5885" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">λ</span></span><span id="MJXc-Node-5886" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5887" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">μ</span></span><span id="MJXc-Node-5888" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5889" class="mjx-mrow"><span id="MJXc-Node-5890" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-5891" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-5892" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-5893" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-5894" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5895" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-5896" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>u</mi><mi>a</mi></msub><mo>=</mo><mi>λ</mi><mo>+</mo><mi>μ</mi><msup><mrow><mo>(</mo><mfrac><mi>q</mi><mi>p</mi></mfrac><mo>)</mo></mrow><mi>a</mi></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-474">u_a=\lambda+\mu\left(\frac{q}{p}\right)^a.</script>
Utilisant que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-475-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5897" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5898" class="mjx-mrow"><span id="MJXc-Node-5899" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5900" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5901" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-5902" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5903" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-475">u_0=1</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-476-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-5904" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5905" class="mjx-mrow"><span id="MJXc-Node-5906" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5907" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5908" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-5909" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5910" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>N</mi></msub><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-476">u_N=0</script>, on obtient :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-477-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5911" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5912" class="mjx-mrow"><span id="MJXc-Node-5913" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5914" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5915" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-5916" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5917" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.766em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.766em; top: -1.575em;"><span id="MJXc-Node-5918" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-5919" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.096em; padding-right: 0.071em;"><span id="MJXc-Node-5920" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span><span class="mjx-denominator" style="width: 3.766em; bottom: -1.001em;"><span id="MJXc-Node-5921" class="mjx-mrow"><span id="MJXc-Node-5922" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-5923" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0.096em; padding-right: 0.071em;"><span id="MJXc-Node-5924" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-5925" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5926" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-5927" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5928" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.766em;"></span></span><span class="mjx-vsize" style="height: 2.576em; vertical-align: -1.001em;"></span></span><span id="MJXc-Node-5929" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5930" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 3.766em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.766em; top: -1.575em;"><span id="MJXc-Node-5931" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5932" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5933" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span><span class="mjx-denominator" style="width: 3.766em; bottom: -1.001em;"><span id="MJXc-Node-5934" class="mjx-mrow"><span id="MJXc-Node-5935" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5936" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5937" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-5938" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5939" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-5940" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0.096em; padding-right: 0.071em;"><span id="MJXc-Node-5941" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.766em;"></span></span><span class="mjx-vsize" style="height: 2.576em; vertical-align: -1.001em;"></span></span><span id="MJXc-Node-5942" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5943" class="mjx-mrow"><span id="MJXc-Node-5944" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-5945" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-5946" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-5947" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-5948" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5949" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-5950" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>u</mi><mi>a</mi></msub><mo>=</mo><mfrac><msup><mi>q</mi><mi>N</mi></msup><mrow><msup><mi>q</mi><mi>N</mi></msup><mo>−</mo><msup><mi>p</mi><mi>N</mi></msup></mrow></mfrac><mo>+</mo><mfrac><msup><mi>p</mi><mi>N</mi></msup><mrow><msup><mi>p</mi><mi>N</mi></msup><mo>−</mo><msup><mi>q</mi><mi>N</mi></msup></mrow></mfrac><msup><mrow><mo>(</mo><mfrac><mi>q</mi><mi>p</mi></mfrac><mo>)</mo></mrow><mi>a</mi></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-477">u_a=\frac{q^N}{q^N-p^N}+\frac{p^N}{p^N-q^N}\left(\frac{q}{p}\right)^a.</script>
</li></ol>
</li><li> Le même raisonnement prouve que :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-478-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5951" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5952" class="mjx-mrow"><span id="MJXc-Node-5953" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5954" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">v</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5955" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-5956" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5957" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-5958" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5959" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">v</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5960" class="mjx-texatom" style=""><span id="MJXc-Node-5961" class="mjx-mrow"><span id="MJXc-Node-5962" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5963" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5964" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5965" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5966" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span><span id="MJXc-Node-5967" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5968" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">v</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5969" class="mjx-texatom" style=""><span id="MJXc-Node-5970" class="mjx-mrow"><span id="MJXc-Node-5971" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span><span id="MJXc-Node-5972" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5973" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-5974" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>v</mi><mi>a</mi></msub><mo>=</mo><mi>p</mi><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>q</mi><msub><mi>v</mi><mrow class="MJX-TeXAtom-ORD"><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-478">v_a=pv_{a+1}+qv_{a-1}.</script>
La résolution de cette récurrence donne :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-479-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-5975" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5976" class="mjx-mrow"><span id="MJXc-Node-5977" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5978" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">v</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5979" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-5980" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-5981" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.766em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.766em; top: -1.575em;"><span id="MJXc-Node-5982" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5983" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5984" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span><span class="mjx-denominator" style="width: 3.766em; bottom: -1.001em;"><span id="MJXc-Node-5985" class="mjx-mrow"><span id="MJXc-Node-5986" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5987" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5988" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-5989" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-5990" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-5991" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0.096em; padding-right: 0.071em;"><span id="MJXc-Node-5992" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.766em;"></span></span><span class="mjx-vsize" style="height: 2.576em; vertical-align: -1.001em;"></span></span><span id="MJXc-Node-5993" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-5994" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 3.766em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.766em; top: -1.575em;"><span id="MJXc-Node-5995" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-5996" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-5997" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span><span class="mjx-denominator" style="width: 3.766em; bottom: -1.001em;"><span id="MJXc-Node-5998" class="mjx-mrow"><span id="MJXc-Node-5999" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.014em;"><span id="MJXc-Node-6000" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0.096em; padding-right: 0.071em;"><span id="MJXc-Node-6001" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span><span id="MJXc-Node-6002" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6003" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-6004" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6005" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.085em;">N</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 3.766em;"></span></span><span class="mjx-vsize" style="height: 2.576em; vertical-align: -1.001em;"></span></span><span id="MJXc-Node-6006" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6007" class="mjx-mrow"><span id="MJXc-Node-6008" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">(</span></span><span id="MJXc-Node-6009" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.703em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.703em; top: -1.178em;"><span id="MJXc-Node-6010" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em; padding-right: 0.014em;">q</span></span></span><span class="mjx-denominator" style="width: 0.703em; bottom: -0.905em;"><span id="MJXc-Node-6011" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.703em;"></span></span><span class="mjx-vsize" style="height: 2.083em; vertical-align: -0.905em;"></span></span><span id="MJXc-Node-6012" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.228em; padding-bottom: 1.228em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6013" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-6014" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>v</mi><mi>a</mi></msub><mo>=</mo><mfrac><msup><mi>p</mi><mi>N</mi></msup><mrow><msup><mi>p</mi><mi>N</mi></msup><mo>−</mo><msup><mi>q</mi><mi>N</mi></msup></mrow></mfrac><mo>+</mo><mfrac><msup><mi>p</mi><mi>N</mi></msup><mrow><msup><mi>q</mi><mi>N</mi></msup><mo>−</mo><msup><mi>p</mi><mi>N</mi></msup></mrow></mfrac><msup><mrow><mo>(</mo><mfrac><mi>q</mi><mi>p</mi></mfrac><mo>)</mo></mrow><mi>a</mi></msup><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-479">v_a=\frac{p^N}{p^N-q^N}+\frac{p^N}{q^N-p^N}\left(\frac{q}{p}\right)^a.</script>
</li><li> On vérifie aisément que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-480-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6015" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6016" class="mjx-mrow"><span id="MJXc-Node-6017" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6018" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">u</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6019" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-6020" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6021" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-6022" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">v</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6023" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">a</span></span></span></span><span id="MJXc-Node-6024" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6025" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>a</mi></msub><mo>+</mo><msub><mi>v</mi><mi>a</mi></msub><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-480">u_a+v_a=1</script>. Ceci signifie que, presque sûrement, le processus va s'arrêter.
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>30</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - La puce<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2189"> [Signaler une erreur]</a><a id="exo30" href="javascript:ajouteexo(2189,30,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2189&#39;,&#39;imageenonce2189&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2189"></a><div class="inner" id="innerenonce2189">Le sommets d'un triangle équilatéral sont numérotés 1, 2 et 3. Une puce saute de sommet en sommet de la façon suivante : si à l'instant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-481-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6026" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6027" class="mjx-mrow"><span id="MJXc-Node-6028" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-481">n</script> elle se trouve sur un sommet donné, elle saute à l'instant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-482-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6029" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6030" class="mjx-mrow"><span id="MJXc-Node-6031" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-6032" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-6033" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-482">n+1</script> vers l'un des deux sommets voisins avec probabilité 1/2. On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-483-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6034" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6035" class="mjx-mrow"><span id="MJXc-Node-6036" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6037" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6038" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-483">X_n</script> la position de la puce à l'instant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-484-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6039" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6040" class="mjx-mrow"><span id="MJXc-Node-6041" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-484">n</script>.
<ol class="enumeratechiffre">
<li> On suppose dans cette question seulement que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-485-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6042" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6043" class="mjx-mrow"><span id="MJXc-Node-6044" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6045" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6046" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-485">X_0</script> suit une loi uniforme sur l'ensemble <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-486-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6047" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6048" class="mjx-mrow"><span id="MJXc-Node-6049" class="mjx-mrow"><span id="MJXc-Node-6050" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">{</span></span><span id="MJXc-Node-6051" class="mjx-mrow"><span id="MJXc-Node-6052" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-6053" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-6054" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span id="MJXc-Node-6055" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.522em;">,</span></span><span id="MJXc-Node-6056" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span></span><span id="MJXc-Node-6057" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">}</span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>}</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-486">\left\{1,2,3\right\}</script>. Décrire la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-487-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6058" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6059" class="mjx-mrow"><span id="MJXc-Node-6060" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6061" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6062" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-487">X_1</script> puis la loi de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-488-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6063" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6064" class="mjx-mrow"><span id="MJXc-Node-6065" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6066" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6067" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-488">X_n</script> pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-489-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6068" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6069" class="mjx-mrow"><span id="MJXc-Node-6070" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-6071" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≥</span></span><span id="MJXc-Node-6072" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-489">n\ge 1</script>.<br>
On s'intéresse maintenant au cas général. On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-490-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6073" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6074" class="mjx-mrow"><span id="MJXc-Node-6075" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6076" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6077" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-6078" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-6079" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-6080" class="mjx-mo" style="vertical-align: -1.094em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.407em; margin-bottom: -0.1em; margin-top: 0px;">⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎝</span></span></span><span id="MJXc-Node-6081" class="mjx-mtable" style="vertical-align: -1.763em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-6082" class="mjx-mtr" style="height: 1.275em;"><span id="MJXc-Node-6083" class="mjx-mtd" style="padding: 0px; width: 4.428em;"><span id="MJXc-Node-6084" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6085" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6086" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-6087" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6088" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6089" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-6090" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-6091" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-6092" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6093" class="mjx-mtr" style="height: 1.475em;"><span id="MJXc-Node-6094" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-6095" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6096" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6097" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-6098" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6099" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6100" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-6101" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-6102" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span id="MJXc-Node-6103" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6104" class="mjx-mtr" style="height: 1.275em;"><span id="MJXc-Node-6105" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-6106" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6107" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6108" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">[</span></span><span id="MJXc-Node-6109" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6110" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6111" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-6112" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-6113" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">3</span></span><span id="MJXc-Node-6114" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">]</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-6115" class="mjx-mo" style="vertical-align: -1.094em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.407em; margin-bottom: -0.1em; margin-top: 0px;">⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎠</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>n</mi></msub><mo>=</mo><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>P</mi><mo stretchy="false">[</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">[</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mi>P</mi><mo stretchy="false">[</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mo stretchy="false">]</mo></mtd></mtr></mtable><mo>)</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-490">U_n=\begin{pmatrix}P[X_n=1]\\P[X_n=2]\\P[X_n=3]\end{pmatrix}</script>.
</li><li> Montrer qu'il existe une matrice <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-491-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6116" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6117" class="mjx-mrow"><span id="MJXc-Node-6118" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-491">A</script> que l'on déterminera telle que pour tout <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-492-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6119" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6120" class="mjx-mrow"><span id="MJXc-Node-6121" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-6122" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≥</span></span><span id="MJXc-Node-6123" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-492">n\ge 0</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-493-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6124" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6125" class="mjx-mrow"><span id="MJXc-Node-6126" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6127" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6128" class="mjx-texatom" style=""><span id="MJXc-Node-6129" class="mjx-mrow"><span id="MJXc-Node-6130" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-6131" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">+</span></span><span id="MJXc-Node-6132" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span></span></span></span></span><span id="MJXc-Node-6133" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-6134" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-6135" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6136" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6137" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>A</mi><msub><mi>U</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-493">U_{n+1}=AU_n</script>.
</li><li> Montrer qu'il existe une matrice orthogonale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-494-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6138" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6139" class="mjx-mrow"><span id="MJXc-Node-6140" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></span></span><script type="math/tex" id="MathJax-Element-494">P</script>, qu'on ne cherchera pas à calculer, telle que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-495-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; text-align: center; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msup&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msup&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6141" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6142" class="mjx-mrow"><span id="MJXc-Node-6143" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-6144" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-6145" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6146" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-6147" class="mjx-mo" style="vertical-align: -1.056em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.332em; margin-bottom: -0.1em; margin-top: 0px;">⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎝</span></span></span><span id="MJXc-Node-6148" class="mjx-mtable" style="vertical-align: -1.725em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-6149" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-6150" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span id="MJXc-Node-6151" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6152" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6153" class="mjx-mtd" style="padding: 0px 0.5em; width: 2.278em;"><span id="MJXc-Node-6154" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6155" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6156" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 2.278em;"><span id="MJXc-Node-6157" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6158" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6159" class="mjx-mtr" style="height: 1.475em;"><span id="MJXc-Node-6160" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-6161" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6162" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6163" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-6164" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6165" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-6166" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-6167" class="mjx-texatom"><span id="MJXc-Node-6168" class="mjx-mrow"><span id="MJXc-Node-6169" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">/</span></span></span></span><span id="MJXc-Node-6170" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6171" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-6172" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6173" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6174" class="mjx-mtr" style="height: 1.275em;"><span id="MJXc-Node-6175" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-6176" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6177" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6178" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-6179" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6180" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6181" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-6182" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6183" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.393em;">−</span></span><span id="MJXc-Node-6184" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-6185" class="mjx-texatom"><span id="MJXc-Node-6186" class="mjx-mrow"><span id="MJXc-Node-6187" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">/</span></span></span></span><span id="MJXc-Node-6188" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-6189" class="mjx-mo" style="vertical-align: -1.056em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.332em; margin-bottom: -0.1em; margin-top: 0px;">⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.912em; padding-bottom: 0.912em;">⎠</span></span></span></span><span id="MJXc-Node-6190" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-6191" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.271em; padding-right: 0.071em;"><span id="MJXc-Node-6192" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.393em; padding-bottom: 0.263em;">t</span></span></span></span><span id="MJXc-Node-6193" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.256em; padding-bottom: 0.328em;">&nbsp;</span></span><span id="MJXc-Node-6194" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.191em; padding-bottom: 0.328em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo>=</mo><mi>P</mi><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><msup><mi>P</mi><mi>t</mi></msup><mtext>&nbsp;</mtext><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-495">A=P\begin{pmatrix}1&0&0\\0&-1/2&0\\0&0&-1/2\end{pmatrix}P^t\ .</script>
</li><li> En déduire que la suite <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-496-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2265;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6195" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6196" class="mjx-mrow"><span id="MJXc-Node-6197" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-6198" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6199" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6200" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span id="MJXc-Node-6201" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6202" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6203" class="mjx-texatom" style=""><span id="MJXc-Node-6204" class="mjx-mrow"><span id="MJXc-Node-6205" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-6206" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.458em;">≥</span></span><span id="MJXc-Node-6207" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>U</mi><mi>n</mi></msub><msub><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-496">(U_n)_{n\ge 0}</script> converge vers un vecteur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-497-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6208" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6209" class="mjx-mrow"><span id="MJXc-Node-6210" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6211" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6212" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.133em; padding-bottom: 0.328em;">∞</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-497">U_\infty</script> vérifiant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-498-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6213" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6214" class="mjx-mrow"><span id="MJXc-Node-6215" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6216" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6217" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.133em; padding-bottom: 0.328em;">∞</span></span></span></span><span id="MJXc-Node-6218" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.068em; padding-bottom: 0.328em;">=</span></span><span id="MJXc-Node-6219" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-6220" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6221" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6222" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.133em; padding-bottom: 0.328em;">∞</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub><mo>=</mo><mi>A</mi><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-498">U_\infty=AU_\infty</script>.
</li><li> Que vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-499-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6223" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6224" class="mjx-mrow"><span id="MJXc-Node-6225" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6226" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.328em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6227" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.133em; padding-bottom: 0.328em;">∞</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-499">U_\infty</script> ?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2189&#39;,&#39;imageindication2189&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2189"></a><div class="inner" id="innerindication2189" style="display:none;"><ol class="enumeratechiffre">
<li> Formule des probabilités totales.
</li><li> Itou.
</li><li> Réduction des matrices symétriques réelles.
</li><li> Calculer <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-500-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6228" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6229" class="mjx-mrow"><span id="MJXc-Node-6230" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6231" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6232" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>A</mi><mi>n</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-500">A^n</script>.
</li><li> Résoudre l'équation précédente.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2189&#39;,&#39;imagecorrige2189&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2189"></a><div class="inner" id="innercorrige2189" style="display:none;"><ol class="enumeratechiffre">
<li> D'après la formule des probabilités totales, on a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-501-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-6233" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6234" class="mjx-mrow"><span id="MJXc-Node-6235" class="mjx-mtable" style="vertical-align: -1.488em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-6236" class="mjx-mtr" style="height: 1.225em;"><span id="MJXc-Node-6237" class="mjx-mtd" style="padding: 0px; text-align: right; width: 4.579em;"><span id="MJXc-Node-6238" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6239" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6240" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6241" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6242" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6243" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-6244" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6245" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6246" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6247" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-6248" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6249" class="mjx-mi"></span><span id="MJXc-Node-6250" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6251" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 39.907em;"><span id="MJXc-Node-6252" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6253" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6254" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6255" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6256" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6257" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-6258" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6259" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6260" class="mjx-texatom"><span id="MJXc-Node-6261" class="mjx-mrow"><span id="MJXc-Node-6262" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6263" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6264" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6265" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-6266" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6267" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6268" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6269" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6270" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6271" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6272" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6273" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-6274" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6275" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6276" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6277" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6278" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6279" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6280" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6281" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6282" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-6283" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6284" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6285" class="mjx-texatom"><span id="MJXc-Node-6286" class="mjx-mrow"><span id="MJXc-Node-6287" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6288" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6289" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6290" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-6291" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6292" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6293" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6294" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6295" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6296" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6297" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6298" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-6299" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6300" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6301" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6302" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6303" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6304" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6305" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6306" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6307" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-6308" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6309" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6310" class="mjx-texatom"><span id="MJXc-Node-6311" class="mjx-mrow"><span id="MJXc-Node-6312" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6313" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6314" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6315" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-6316" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6317" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-6318" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6319" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6320" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6321" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6322" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6323" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-6324" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6325" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-6326" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6327" class="mjx-mtr" style="height: 2.25em;"><span id="MJXc-Node-6328" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-6329" class="mjx-mrow" style="margin-top: 0.368em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6330" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-6331" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-6332" class="mjx-mi"></span><span id="MJXc-Node-6333" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6334" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-6335" class="mjx-mrow"><span id="MJXc-Node-6336" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-6337" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6338" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6339" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6340" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6341" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-6342" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6343" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-6344" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-6345" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6346" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6347" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6348" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6349" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-6350" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6351" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-6352" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-6353" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6354" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6355" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.733em;"><span id="MJXc-Node-6356" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-6357" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>0</mn></msub><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mn>0</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-501">\begin{eqnarray*}
P(X_1=1)&=&P(X_1=1|X_0=1)P(X_0=1)+P(X_1=1|X_0=2)P(X_0=2)+P(X_1=1|X_0=3)P(X_0=3)\\
&=&0+\frac 12\times\frac 13+\frac 12\times\frac 13=\frac 13.
\end{eqnarray*}</script>
De la même façon on trouve que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-502-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6358" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6359" class="mjx-mrow"><span id="MJXc-Node-6360" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6361" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6362" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6363" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6364" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-6365" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6366" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6367" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6368" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6369" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6370" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6371" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6372" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6373" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span id="MJXc-Node-6374" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6375" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-6376" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6377" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6378" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6379" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.632em;"><span id="MJXc-Node-6380" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.379em; vertical-align: -0.447em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-502">P(X_1=2)=P(X_1=3)=\frac 13</script>. La variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-503-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6381" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6382" class="mjx-mrow"><span id="MJXc-Node-6383" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6384" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6385" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-503">X_1</script> suit donc une loi uniforme sur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-504-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo fence=&quot;false&quot; stretchy=&quot;false&quot;&gt;}&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6386" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6387" class="mjx-mrow"><span id="MJXc-Node-6388" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">{</span></span><span id="MJXc-Node-6389" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6390" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-6391" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6392" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-6393" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-6394" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">}</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo fence="false" stretchy="false">}</mo></math></span></span><script type="math/tex" id="MathJax-Element-504">\{1,2,3\}</script>. Par récurrence immédiate, la variable aléatoire <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-505-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6395" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6396" class="mjx-mrow"><span id="MJXc-Node-6397" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6398" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6399" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-505">X_n</script> suivra aussi pour tout entier <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-506-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6400" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6401" class="mjx-mrow"><span id="MJXc-Node-6402" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-506">n</script> la même loi.
</li><li> On applique encore la formule des probabilités totales. On a
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-507-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-6403" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6404" class="mjx-mrow"><span id="MJXc-Node-6405" class="mjx-mtable" style="vertical-align: -1.477em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-6406" class="mjx-mtr" style="height: 1.225em;"><span id="MJXc-Node-6407" class="mjx-mtd" style="padding: 0px; text-align: right; width: 5.554em;"><span id="MJXc-Node-6408" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6409" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6410" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6411" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6412" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6413" class="mjx-texatom" style=""><span id="MJXc-Node-6414" class="mjx-mrow"><span id="MJXc-Node-6415" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-6416" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6417" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-6418" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6419" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6420" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6421" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-6422" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6423" class="mjx-mi"></span><span id="MJXc-Node-6424" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6425" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 43.255em;"><span id="MJXc-Node-6426" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6427" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6428" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6429" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6430" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6431" class="mjx-texatom" style=""><span id="MJXc-Node-6432" class="mjx-mrow"><span id="MJXc-Node-6433" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-6434" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6435" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-6436" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6437" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6438" class="mjx-texatom"><span id="MJXc-Node-6439" class="mjx-mrow"><span id="MJXc-Node-6440" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6441" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6442" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6443" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6444" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6445" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6446" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6447" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6448" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6449" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6450" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6451" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6452" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6453" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6454" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6455" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6456" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6457" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6458" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6459" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6460" class="mjx-texatom" style=""><span id="MJXc-Node-6461" class="mjx-mrow"><span id="MJXc-Node-6462" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-6463" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6464" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-6465" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6466" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6467" class="mjx-texatom"><span id="MJXc-Node-6468" class="mjx-mrow"><span id="MJXc-Node-6469" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6470" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6471" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6472" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6473" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6474" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6475" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6477" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6478" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6479" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6480" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6481" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6482" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6483" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6484" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6485" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6486" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6487" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6488" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6489" class="mjx-texatom" style=""><span id="MJXc-Node-6490" class="mjx-mrow"><span id="MJXc-Node-6491" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-6492" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6493" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-6494" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6495" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6496" class="mjx-texatom"><span id="MJXc-Node-6497" class="mjx-mrow"><span id="MJXc-Node-6498" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-6499" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6500" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6501" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6502" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6503" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-6504" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6505" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6506" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6507" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6508" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6509" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6510" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6511" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-6512" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6513" class="mjx-mtr" style="height: 2.228em;"><span id="MJXc-Node-6514" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-6515" class="mjx-mrow" style="margin-top: 0.368em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6516" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-6517" class="mjx-mrow" style="margin-top: 0.368em;"><span id="MJXc-Node-6518" class="mjx-mi"></span><span id="MJXc-Node-6519" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6520" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-6521" class="mjx-mrow"><span id="MJXc-Node-6522" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-6523" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6524" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6525" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6526" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6527" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6528" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6529" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6530" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6531" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6532" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6533" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6534" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6535" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6536" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6537" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6538" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6539" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6540" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6541" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6542" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6543" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6544" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-6545" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-6546" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-6547" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6548" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6549" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6550" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6551" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6552" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6553" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-6554" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6555" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mn>0</mn><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-507">\begin{eqnarray*}
P(X_{n+1}=1)&=&P(X_{n+1}=1|X_n=1)P(X_n=1)+P(X_{n+1}=1|X_n=2)P(X_n=2)+P(X_{n+1}=1|X_n=3)P(X_n=3)\\
&=&0P(X_n=1)+\frac 12 P(X_n=2)+\frac 12 P(X_n=3).
\end{eqnarray*}</script>
En effectuant le même calcul pour <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-508-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6556" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6557" class="mjx-mrow"><span id="MJXc-Node-6558" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6559" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6560" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6561" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6562" class="mjx-texatom" style=""><span id="MJXc-Node-6563" class="mjx-mrow"><span id="MJXc-Node-6564" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-6565" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6566" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-6567" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6568" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-6569" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-508">P(X_{n+1}=2)</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-509-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6570" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6571" class="mjx-mrow"><span id="MJXc-Node-6572" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6573" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6574" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.024em;"><span id="MJXc-Node-6575" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.024em;">X</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6576" class="mjx-texatom" style=""><span id="MJXc-Node-6577" class="mjx-mrow"><span id="MJXc-Node-6578" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-6579" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6580" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-6581" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6582" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-6583" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>3</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-509">P(X_{n+1}=3)</script>, on trouve que la matrice
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-510-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable columnalign=&quot;center center center&quot; rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-6584" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6585" class="mjx-mrow"><span id="MJXc-Node-6586" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-6587" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6588" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-6589" class="mjx-mo" style="vertical-align: -1.528em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.425em; margin-bottom: -0.1em; margin-top: 0px;">⎜
⎜
⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎝</span></span></span><span id="MJXc-Node-6590" class="mjx-mtable" style="vertical-align: -2.197em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-6591" class="mjx-mtr" style="height: 1.564em;"><span id="MJXc-Node-6592" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.735em;"><span id="MJXc-Node-6593" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6594" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6595" class="mjx-mtd" style="padding: 0px 0.5em; width: 0.735em;"><span id="MJXc-Node-6596" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6597" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6598" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-6599" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6600" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 0.735em;"><span id="MJXc-Node-6601" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6602" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6603" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-6604" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6605" class="mjx-mtr" style="height: 1.764em;"><span id="MJXc-Node-6606" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-6607" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6608" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6609" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-6610" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6611" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-6612" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6613" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6614" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-6615" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6616" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6617" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-6618" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6619" class="mjx-mtr" style="height: 1.564em;"><span id="MJXc-Node-6620" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-6621" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6622" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6623" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-6624" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6625" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-6626" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6627" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6628" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-6629" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6630" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-6631" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6632" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-6633" class="mjx-mo" style="vertical-align: -1.528em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.425em; margin-bottom: -0.1em; margin-top: 0px;">⎟
⎟
⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎠</span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>A</mi><mo>=</mo><mrow><mo>(</mo><mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-510">A=\left(\begin{array}{ccc}
0&\frac 12&\frac 12\\
\frac 12&0&\frac 12\\
\frac 12&\frac 12&0
\end{array}\right)</script>
convient.
</li><li> La matrice <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-511-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6634" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6635" class="mjx-mrow"><span id="MJXc-Node-6636" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-511">A</script> est symétrique réelle. Elle est donc diagonalisable, et si on note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-512-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6637" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6638" class="mjx-mrow"><span id="MJXc-Node-6639" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-512">D</script> la matrice de ses valeurs propres, il existe une matrice orthogonale <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-513-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6640" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6641" class="mjx-mrow"><span id="MJXc-Node-6642" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></span></span><script type="math/tex" id="MathJax-Element-513">P</script> telle que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-514-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6643" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6644" class="mjx-mrow"><span id="MJXc-Node-6645" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-6646" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6647" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6648" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-6649" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-6650" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.271em; padding-right: 0.071em;"><span id="MJXc-Node-6651" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mi>D</mi><msup><mi>P</mi><mi>t</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-514">A=PDP^t</script>. Reste à démontrer que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-515-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6652" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6653" class="mjx-mrow"><span id="MJXc-Node-6654" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-515">1</script> est valeur propre de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-516-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6655" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6656" class="mjx-mrow"><span id="MJXc-Node-6657" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-516">A</script>, et que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-517-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6658" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6659" class="mjx-mrow"><span id="MJXc-Node-6660" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6661" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6662" class="mjx-texatom"><span id="MJXc-Node-6663" class="mjx-mrow"><span id="MJXc-Node-6664" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-6665" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-517">-1/2</script> est valeur propre double. On peut calculer
le polynôme caractéristique de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-518-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6666" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6667" class="mjx-mrow"><span id="MJXc-Node-6668" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-518">A</script>, ou remarquer que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-519-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6669" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6670" class="mjx-mrow"><span id="MJXc-Node-6671" class="mjx-mrow"><span id="MJXc-Node-6672" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎝</span></span></span><span id="MJXc-Node-6673" class="mjx-mtable" style="vertical-align: -1.65em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-6674" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-6675" class="mjx-mtd" style="padding: 0px; width: 0.5em;"><span id="MJXc-Node-6676" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6677" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6678" class="mjx-mtr" style="height: 1.4em;"><span id="MJXc-Node-6679" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-6680" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6681" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6682" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-6683" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-6684" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6685" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-6686" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎠</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-519">\begin{pmatrix}1\\1\\1\end{pmatrix}</script> est vecteur propre de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-520-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6687" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6688" class="mjx-mrow"><span id="MJXc-Node-6689" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-520">A</script> pour la valeur propre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-521-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6690" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6691" class="mjx-mrow"><span id="MJXc-Node-6692" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-521">1</script>, et que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-522-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;msub&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6693" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6694" class="mjx-mrow"><span id="MJXc-Node-6695" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-6696" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6697" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6698" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-6699" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span id="MJXc-Node-6700" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-6701" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6702" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>I</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-522">A+\frac 12I_3</script> est de rang 1, ce qui assure que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-523-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6703" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6704" class="mjx-mrow"><span id="MJXc-Node-6705" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-6706" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-6707" class="mjx-texatom"><span id="MJXc-Node-6708" class="mjx-mrow"><span id="MJXc-Node-6709" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-6710" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-523">-1/2</script> est valeur propre d'ordre 2.
</li><li> On a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-524-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6711" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6712" class="mjx-mrow"><span id="MJXc-Node-6713" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6714" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6715" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6716" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6717" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-6718" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6719" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6720" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6721" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6722" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-6723" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6724" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6725" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6726" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6727" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6728" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-6729" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.271em; padding-right: 0.071em;"><span id="MJXc-Node-6730" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span></span><span id="MJXc-Node-6731" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6732" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6733" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>n</mi></msub><mo>=</mo><msup><mi>A</mi><mi>n</mi></msup><msub><mi>U</mi><mn>0</mn></msub><mo>=</mo><mi>P</mi><msup><mi>D</mi><mi>n</mi></msup><msup><mi>P</mi><mi>t</mi></msup><msub><mi>U</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-524">U_n=A^n U_0=PD^n P^t U_0</script> où on a conservé les notations de la question précédente. Mais
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-525-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msup&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x2192;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-6734" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6735" class="mjx-mrow"><span id="MJXc-Node-6736" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-6737" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-6738" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6739" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.236em; padding-bottom: 0.353em;">→</span></span><span id="MJXc-Node-6740" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-6741" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6742" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-6743" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎝</span></span></span><span id="MJXc-Node-6744" class="mjx-mtable" style="vertical-align: -1.65em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-6745" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-6746" class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span id="MJXc-Node-6747" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6748" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6749" class="mjx-mtd" style="padding: 0px 0.5em; width: 0.5em;"><span id="MJXc-Node-6750" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6751" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6752" class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 0.5em;"><span id="MJXc-Node-6753" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6754" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6755" class="mjx-mtr" style="height: 1.4em;"><span id="MJXc-Node-6756" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-6757" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6758" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6759" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-6760" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6761" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6762" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-6763" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6764" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6765" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-6766" class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span id="MJXc-Node-6767" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6768" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6769" class="mjx-mtd" style="padding: 0.2em 0.5em 0px;"><span id="MJXc-Node-6770" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6771" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-6772" class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span id="MJXc-Node-6773" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6774" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-6775" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎠</span></span></span></span><span id="MJXc-Node-6776" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>D</mi><mi>n</mi></msup><mo stretchy="false">→</mo><mi>B</mi><mo>=</mo><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-525">D^n\to B=\begin{pmatrix}
1&0&0\\
0&0&0\\
0&0&0
\end{pmatrix}.</script>
Donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-526-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6777" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6778" class="mjx-mrow"><span id="MJXc-Node-6779" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6780" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6781" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6782" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-6783" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>U</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-526">(U_n)</script> converge vers <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-527-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msup&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6784" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6785" class="mjx-mrow"><span id="MJXc-Node-6786" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6787" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-6788" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-6789" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.271em; padding-right: 0.071em;"><span id="MJXc-Node-6790" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.411em; padding-bottom: 0.295em;">t</span></span></span></span><span id="MJXc-Node-6791" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6792" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6793" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span></span></span><span id="MJXc-Node-6794" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6795" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6796" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6797" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>B</mi><msup><mi>P</mi><mi>t</mi></msup><msub><mi>U</mi><mn>0</mn></msub><mo>=</mo><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-527">PBP^tU_0=U_\infty</script>. De la relation <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-528-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6798" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6799" class="mjx-mrow"><span id="MJXc-Node-6800" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6801" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6802" class="mjx-texatom" style=""><span id="MJXc-Node-6803" class="mjx-mrow"><span id="MJXc-Node-6804" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-6805" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-6806" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span><span id="MJXc-Node-6807" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6808" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-6809" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6810" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6811" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>A</mi><msub><mi>U</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-528">U_{n+1}=AU_n</script>, on tire en passant à la limite que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-529-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6812" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6813" class="mjx-mrow"><span id="MJXc-Node-6814" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6815" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6816" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span id="MJXc-Node-6817" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6818" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-6819" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6820" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6821" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span id="MJXc-Node-6822" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub><mo>=</mo><mi>A</mi><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-529">U_\infty=AU_\infty.</script>
</li><li> On sait que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-530-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6823" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6824" class="mjx-mrow"><span id="MJXc-Node-6825" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-530">1</script> est valeur propre de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-531-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6826" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6827" class="mjx-mrow"><span id="MJXc-Node-6828" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-531">A</script> d'ordre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-532-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6829" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6830" class="mjx-mrow"><span id="MJXc-Node-6831" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-532">1</script>, et on a déjà trouvé un vecteur propre précédemment, à savoir <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-533-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6832" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6833" class="mjx-mrow"><span id="MJXc-Node-6834" class="mjx-mrow"><span id="MJXc-Node-6835" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎝</span></span></span><span id="MJXc-Node-6836" class="mjx-mtable" style="vertical-align: -1.65em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-6837" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-6838" class="mjx-mtd" style="padding: 0px; width: 0.5em;"><span id="MJXc-Node-6839" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6840" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6841" class="mjx-mtr" style="height: 1.4em;"><span id="MJXc-Node-6842" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-6843" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6844" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6845" class="mjx-mtr" style="height: 1.2em;"><span id="MJXc-Node-6846" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-6847" class="mjx-mrow" style="margin-top: -0.2em;"><span id="MJXc-Node-6848" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-6849" class="mjx-mo" style="vertical-align: -0.981em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.182em; margin-bottom: -0.1em; margin-top: 0px;">⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎠</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-533">\begin{pmatrix}1\\1\\1\end{pmatrix}</script>. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-534-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6850" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6851" class="mjx-mrow"><span id="MJXc-Node-6852" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6853" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6854" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-534">U_\infty</script> est donc proportionnel à ce vecteur propre. Mais, puisque la somme des lignes de chaque vecteur <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-535-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6855" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6856" class="mjx-mrow"><span id="MJXc-Node-6857" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6858" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6859" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-535">U_n</script> fait <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-536-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6860" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6861" class="mjx-mrow"><span id="MJXc-Node-6862" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-536">1</script>, il en est de même de la somme des lignes des vecteurs de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-537-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6863" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6864" class="mjx-mrow"><span id="MJXc-Node-6865" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6866" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6867" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-537">U_\infty</script>. Ainsi, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-538-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x221E;&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6868" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6869" class="mjx-mrow"><span id="MJXc-Node-6870" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-6871" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-6872" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">∞</span></span></span></span><span id="MJXc-Node-6873" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6874" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-6875" class="mjx-mo" style="vertical-align: -1.55em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.44em; margin-bottom: -0.1em; margin-top: 0px;">⎜
⎜
⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎝</span></span></span><span id="MJXc-Node-6876" class="mjx-mtable" style="vertical-align: -2.219em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-6877" class="mjx-mtr" style="height: 1.579em;"><span id="MJXc-Node-6878" class="mjx-mtd" style="padding: 0px; width: 0.735em;"><span id="MJXc-Node-6879" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6880" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6881" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.632em;"><span id="MJXc-Node-6882" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.379em; vertical-align: -0.447em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6883" class="mjx-mtr" style="height: 1.779em;"><span id="MJXc-Node-6884" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-6885" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6886" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6887" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.632em;"><span id="MJXc-Node-6888" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.379em; vertical-align: -0.447em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-6889" class="mjx-mtr" style="height: 1.579em;"><span id="MJXc-Node-6890" class="mjx-mtd" style="padding: 0.2em 0px 0px;"><span id="MJXc-Node-6891" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-6892" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-6893" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.632em;"><span id="MJXc-Node-6894" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.379em; vertical-align: -0.447em;"></span></span><span class="mjx-strut"></span></span></span></span></span></span><span id="MJXc-Node-6895" class="mjx-mo" style="vertical-align: -1.55em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.44em; margin-bottom: -0.1em; margin-top: 0px;">⎟
⎟
⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">⎠</span></span></span></span><span id="MJXc-Node-6896" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi mathvariant="normal">∞</mi></msub><mo>=</mo><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr></mtable><mo>)</mo></mrow><mo>.</mo></math></span></span><script type="math/tex" id="MathJax-Element-538">U_\infty=\begin{pmatrix}\frac 13\\\frac 13\\ \frac 13\end{pmatrix}.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>31</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Clés USB<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1242"> [Signaler une erreur]</a><a id="exo31" href="javascript:ajouteexo(1242,31,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1242&#39;,&#39;imageenonce1242&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1242"></a><div class="inner" id="innerenonce1242">Le gérant d'un magasin d'informatique a reçu un lot de clés USB. <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-539-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0025;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6897" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6898" class="mjx-mrow"><span id="MJXc-Node-6899" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">5</span></span><span id="MJXc-Node-6900" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.393em;">%</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-539">5\%</script> des boites sont abîmées. Le gérant estime que :
<ul class="rien">
<li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-540-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0025;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6901" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6902" class="mjx-mrow"><span id="MJXc-Node-6903" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">60</span></span><span id="MJXc-Node-6904" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.393em;">%</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-540">60\%</script> des boites abîmées contiennent au moins une clé défectueuse.
</li><li> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-541-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;98&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0025;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6905" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6906" class="mjx-mrow"><span id="MJXc-Node-6907" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">98</span></span><span id="MJXc-Node-6908" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.393em;">%</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>98</mn><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-541">98\%</script> des boites non abîmées ne contiennent aucune clé défectueuse.
</li></ul>
Un client achète une boite du lot. On désigne par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-542-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6909" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6910" class="mjx-mrow"><span id="MJXc-Node-6911" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-542">A</script> l'événement : ``la boite est abîmée'' et par <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-543-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6912" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6913" class="mjx-mrow"><span id="MJXc-Node-6914" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">D</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-543">D</script> l'événement ``la boite achetée contient au moins une clé défectueuse''.
<ol class="enumeratechiffre">
<li> Donner les probabilités de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-544-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6915" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6916" class="mjx-mrow"><span id="MJXc-Node-6917" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6918" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-6919" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-6920" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-544">P(A)</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-545-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6921" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6922" class="mjx-mrow"><span id="MJXc-Node-6923" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6924" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-6925" class="mjx-texatom"><span id="MJXc-Node-6926" class="mjx-mrow"><span id="MJXc-Node-6927" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-6929" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.328em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-6928" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span></span></span></span><span id="MJXc-Node-6930" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-545">P(\bar A)</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-546-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6931" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6932" class="mjx-mrow"><span id="MJXc-Node-6933" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6934" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-6935" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">D</span></span><span id="MJXc-Node-6936" class="mjx-texatom"><span id="MJXc-Node-6937" class="mjx-mrow"><span id="MJXc-Node-6938" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span><span id="MJXc-Node-6939" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-6940" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-546">P(D|A)</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-547-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6941" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6942" class="mjx-mrow"><span id="MJXc-Node-6943" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6944" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-6945" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">D</span></span><span id="MJXc-Node-6946" class="mjx-texatom"><span id="MJXc-Node-6947" class="mjx-mrow"><span id="MJXc-Node-6948" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span><span id="MJXc-Node-6949" class="mjx-texatom"><span id="MJXc-Node-6950" class="mjx-mrow"><span id="MJXc-Node-6951" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-6953" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.328em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-6952" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span></span></span></span><span id="MJXc-Node-6954" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-547">P(D|\bar A)</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-548-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6955" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6956" class="mjx-mrow"><span id="MJXc-Node-6957" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6958" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-6959" class="mjx-texatom"><span id="MJXc-Node-6960" class="mjx-mrow"><span id="MJXc-Node-6961" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.22em;"><span id="MJXc-Node-6963" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.328em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-6962" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">D</span></span></span></span></span></span></span><span id="MJXc-Node-6964" class="mjx-texatom"><span id="MJXc-Node-6965" class="mjx-mrow"><span id="MJXc-Node-6966" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span><span id="MJXc-Node-6967" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span><span id="MJXc-Node-6968" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">¯</mo></mover></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-548">P(\bar D|A)</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-549-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6969" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6970" class="mjx-mrow"><span id="MJXc-Node-6971" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6972" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">(</span></span><span id="MJXc-Node-6973" class="mjx-texatom"><span id="MJXc-Node-6974" class="mjx-mrow"><span id="MJXc-Node-6975" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.22em;"><span id="MJXc-Node-6977" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.328em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-6976" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">D</span></span></span></span></span></span></span><span id="MJXc-Node-6978" class="mjx-texatom"><span id="MJXc-Node-6979" class="mjx-mrow"><span id="MJXc-Node-6980" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">|</span></span></span></span><span id="MJXc-Node-6981" class="mjx-texatom"><span id="MJXc-Node-6982" class="mjx-mrow"><span id="MJXc-Node-6983" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-6985" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.328em; padding-bottom: 0.328em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-6984" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.522em; padding-bottom: 0.263em;">A</span></span></span></span></span></span></span><span id="MJXc-Node-6986" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">¯</mo></mover></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-549">P(\bar D|\bar A)</script>. En déduire la probabilité de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-550-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-6987" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6988" class="mjx-mrow"><span id="MJXc-Node-6989" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.458em; padding-bottom: 0.263em;">D</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-550">D</script>.
</li><li> Le client constate qu'une des clés achetées est défectueuse. Quelle est
la probabilité pour qu'il ait acheté une boite abîmée?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1242&#39;,&#39;imageindication1242&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1242"></a><div class="inner" id="innerindication1242" style="display:none;"><ol class="enumeratechiffre">
<li> Traduire les informations de l'énoncé, utiliser la formule des probabilités totales.
</li><li> Utiliser la formule de Bayes.
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1242&#39;,&#39;imagecorrige1242&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1242"></a><div class="inner" id="innercorrige1242" style="display:none;"><ol class="enumeratechiffre">
<li> L'énoncé donne directement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-551-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;05&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-6990" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-6991" class="mjx-mrow"><span id="MJXc-Node-6992" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-6993" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-6994" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-6995" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-6996" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-6997" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-6998" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-6999" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">05</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>05</mn></math></span></span><script type="math/tex" id="MathJax-Element-551">P(A)=0,05</script>, d'où <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-552-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;95&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7000" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7001" class="mjx-mrow"><span id="MJXc-Node-7002" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7003" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7004" class="mjx-texatom"><span id="MJXc-Node-7005" class="mjx-mrow"><span id="MJXc-Node-7006" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-7008" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7007" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span></span></span><span id="MJXc-Node-7009" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7010" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7011" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-7012" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7013" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">95</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>95</mn></math></span></span><script type="math/tex" id="MathJax-Element-552">P(\bar A)=0,95</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-553-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7014" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7015" class="mjx-mrow"><span id="MJXc-Node-7016" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7017" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7018" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7019" class="mjx-texatom"><span id="MJXc-Node-7020" class="mjx-mrow"><span id="MJXc-Node-7021" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7022" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7023" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7024" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7025" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-7026" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7027" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-553">P(D|A)=0,6</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-554-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;98&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7028" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7029" class="mjx-mrow"><span id="MJXc-Node-7030" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7031" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7032" class="mjx-texatom"><span id="MJXc-Node-7033" class="mjx-mrow"><span id="MJXc-Node-7034" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.22em;"><span id="MJXc-Node-7036" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7035" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span></span></span></span></span><span id="MJXc-Node-7037" class="mjx-texatom"><span id="MJXc-Node-7038" class="mjx-mrow"><span id="MJXc-Node-7039" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7040" class="mjx-texatom"><span id="MJXc-Node-7041" class="mjx-mrow"><span id="MJXc-Node-7042" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-7044" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7043" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span></span></span><span id="MJXc-Node-7045" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7046" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7047" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-7048" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7049" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">98</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">¯</mo></mover></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>98</mn></math></span></span><script type="math/tex" id="MathJax-Element-554">P(\bar D|\bar A)=0,98</script>. On en déduit :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-555-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7050" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7051" class="mjx-mrow"><span id="MJXc-Node-7052" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7053" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7054" class="mjx-texatom"><span id="MJXc-Node-7055" class="mjx-mrow"><span id="MJXc-Node-7056" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.22em;"><span id="MJXc-Node-7058" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7057" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span></span></span></span></span><span id="MJXc-Node-7059" class="mjx-texatom"><span id="MJXc-Node-7060" class="mjx-mrow"><span id="MJXc-Node-7061" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7062" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7063" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7064" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7065" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7066" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7067" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7068" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7069" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7070" class="mjx-texatom"><span id="MJXc-Node-7071" class="mjx-mrow"><span id="MJXc-Node-7072" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7073" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7074" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7075" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7076" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-7077" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7078" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">¯</mo></mover></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>4</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-555">P(\bar D|A)=1-P(D|A)=0,4</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-556-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;02.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7079" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7080" class="mjx-mrow"><span id="MJXc-Node-7081" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7082" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7083" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7084" class="mjx-texatom"><span id="MJXc-Node-7085" class="mjx-mrow"><span id="MJXc-Node-7086" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7087" class="mjx-texatom"><span id="MJXc-Node-7088" class="mjx-mrow"><span id="MJXc-Node-7089" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-7091" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7090" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span></span></span><span id="MJXc-Node-7092" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7093" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7094" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7095" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7096" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7097" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7098" class="mjx-texatom"><span id="MJXc-Node-7099" class="mjx-mrow"><span id="MJXc-Node-7100" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.22em;"><span id="MJXc-Node-7102" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7101" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span></span></span></span></span><span id="MJXc-Node-7103" class="mjx-texatom"><span id="MJXc-Node-7104" class="mjx-mrow"><span id="MJXc-Node-7105" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7106" class="mjx-texatom"><span id="MJXc-Node-7107" class="mjx-mrow"><span id="MJXc-Node-7108" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-7110" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7109" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span></span></span><span id="MJXc-Node-7111" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7112" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7113" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-7114" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7115" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">02.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>D</mi><mo stretchy="false">¯</mo></mover></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>02.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-556">P(D|\bar A)=1-P(\bar D|\bar A)=0,02.</script>
D'après la formule des probabilités totales :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-557-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;49&lt;/mn&gt;&lt;mn&gt;1000&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7116" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7117" class="mjx-mrow"><span id="MJXc-Node-7118" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7119" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7120" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7121" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7122" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7123" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7124" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7125" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7126" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7127" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7128" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7129" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7130" class="mjx-texatom"><span id="MJXc-Node-7131" class="mjx-mrow"><span id="MJXc-Node-7132" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7133" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7134" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7135" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7136" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7137" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7138" class="mjx-texatom"><span id="MJXc-Node-7139" class="mjx-mrow"><span id="MJXc-Node-7140" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-7142" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7141" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span></span></span><span id="MJXc-Node-7143" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7144" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7145" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7146" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7147" class="mjx-texatom"><span id="MJXc-Node-7148" class="mjx-mrow"><span id="MJXc-Node-7149" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7150" class="mjx-texatom"><span id="MJXc-Node-7151" class="mjx-mrow"><span id="MJXc-Node-7152" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.264em;"><span id="MJXc-Node-7154" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7153" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span></span></span><span id="MJXc-Node-7155" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7156" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7157" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.2em; top: -1.379em;"><span id="MJXc-Node-7158" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">49</span></span></span><span class="mjx-denominator" style="width: 2.2em; bottom: -0.733em;"><span id="MJXc-Node-7159" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1000</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.2em;"></span></span><span class="mjx-vsize" style="height: 2.111em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-7160" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>49</mn><mn>1000</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-557">P(D)=P(A)P(D|A)+P(\bar A)P(D|\bar A)=\frac{49}{1000}.</script>
</li><li> On obtient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-558-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7161" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7162" class="mjx-mrow"><span id="MJXc-Node-7163" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7164" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7165" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7166" class="mjx-texatom"><span id="MJXc-Node-7167" class="mjx-mrow"><span id="MJXc-Node-7168" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7169" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7170" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>D</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-558">P(A|D)</script> grâce à la formule de Bayes :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-559-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;mn&gt;49&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7171" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7172" class="mjx-mrow"><span id="MJXc-Node-7173" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7174" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7175" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7176" class="mjx-texatom"><span id="MJXc-Node-7177" class="mjx-mrow"><span id="MJXc-Node-7178" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7179" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7180" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7181" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7182" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.418em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.418em; top: -1.542em;"><span id="MJXc-Node-7183" class="mjx-mrow"><span id="MJXc-Node-7184" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7185" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7186" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7187" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-7188" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7189" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 4.418em; bottom: -1.042em;"><span id="MJXc-Node-7190" class="mjx-mrow"><span id="MJXc-Node-7191" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7192" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7193" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7194" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.418em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-7195" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7196" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.864em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.864em; top: -1.542em;"><span id="MJXc-Node-7197" class="mjx-mrow"><span id="MJXc-Node-7198" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7199" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7200" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7201" class="mjx-texatom"><span id="MJXc-Node-7202" class="mjx-mrow"><span id="MJXc-Node-7203" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7204" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7205" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7206" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7207" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7208" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7209" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 5.864em; bottom: -1.042em;"><span id="MJXc-Node-7210" class="mjx-mrow"><span id="MJXc-Node-7211" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7212" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7213" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7214" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 5.864em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-7215" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7216" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-7217" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">30</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.741em;"><span id="MJXc-Node-7218" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">49</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.109em; vertical-align: -0.741em;"></span></span><span id="MJXc-Node-7219" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mn>30</mn><mn>49</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-559">P(A|D)=\frac{P(A\cap D)}{P(D)}=\frac{P(D|A)P(A)}{P(D)}=\frac{30}{49}.</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>32</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - QCM<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1245"> [Signaler une erreur]</a><a id="exo32" href="javascript:ajouteexo(1245,32,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1245&#39;,&#39;imageenonce1245&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1245"></a><div class="inner" id="innerenonce1245">Un questionnaire à choix multiples propose <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-560-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7220" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7221" class="mjx-mrow"><span id="MJXc-Node-7222" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">m</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script type="math/tex" id="MathJax-Element-560">m</script> réponses pour chaque question. Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-561-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7223" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7224" class="mjx-mrow"><span id="MJXc-Node-7225" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-561">p</script> la probabilité qu'un étudiant connaisse la bonne réponse à une question donnée. S'il ignore la réponse, il choisit au hasard l'une des réponses proposées. Quelle est pour le correcteur la probabilité qu'un étudiant connaisse vraiment la bonne réponse lorsqu'il l'a donnée?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1245&#39;,&#39;imageindication1245&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1245"></a><div class="inner" id="innerindication1245" style="display:none;">Utiliser la formule de Bayes.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1245&#39;,&#39;imagecorrige1245&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1245"></a><div class="inner" id="innercorrige1245" style="display:none;">On note :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-562-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;L&#39;&amp;#xE9;tudiant donne la bonne r&amp;#xE9;ponse&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7226" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7227" class="mjx-mrow"><span id="MJXc-Node-7228" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7229" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7230" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-7231" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">{</span></span><span id="MJXc-Node-7232" class="mjx-texatom"><span id="MJXc-Node-7233" class="mjx-mrow"><span id="MJXc-Node-7234" class="mjx-mtext"><span class="mjx-char" style="padding-top: 0.528em; padding-bottom: 0.236em;"><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">L'</span><span class="mjx-charbox MJXc-TeX-unknown-R" style="padding-bottom: 0.273em; width: 0.493em;">é</span><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">tudiant donne la bonne r</span><span class="mjx-charbox MJXc-TeX-unknown-R" style="padding-bottom: 0.273em; width: 0.493em;">é</span><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">ponse</span></span></span></span></span><span id="MJXc-Node-7235" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">}</span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>B</mi><mo>=</mo><mrow><mo>{</mo><mrow class="MJX-TeXAtom-ORD"><mtext>L'étudiant donne la bonne réponse</mtext></mrow><mo>}</mo></mrow></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-562">B=\left\{\textrm{L'étudiant donne la bonne réponse}\right\}</script>
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-563-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;L&#39;&amp;#xE9;tudiant connait la bonne r&amp;#xE9;ponse&lt;/mtext&gt;&lt;/mrow&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7236" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7237" class="mjx-mrow"><span id="MJXc-Node-7238" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7239" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7240" class="mjx-mrow MJXc-space3"><span id="MJXc-Node-7241" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">{</span></span><span id="MJXc-Node-7242" class="mjx-texatom"><span id="MJXc-Node-7243" class="mjx-mrow"><span id="MJXc-Node-7244" class="mjx-mtext"><span class="mjx-char" style="padding-top: 0.528em; padding-bottom: 0.236em;"><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">L'</span><span class="mjx-charbox MJXc-TeX-unknown-R" style="padding-bottom: 0.273em; width: 0.493em;">é</span><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">tudiant connait la bonne r</span><span class="mjx-charbox MJXc-TeX-unknown-R" style="padding-bottom: 0.273em; width: 0.493em;">é</span><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em;">ponse</span></span></span></span></span><span id="MJXc-Node-7245" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.645em; padding-bottom: 0.645em;">}</span></span></span><span id="MJXc-Node-7246" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>C</mi><mo>=</mo><mrow><mo>{</mo><mrow class="MJX-TeXAtom-ORD"><mtext>L'étudiant connait la bonne réponse</mtext></mrow><mo>}</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-563">C=\left\{\textrm{L'étudiant connait la bonne réponse}\right\}.</script>
On cherche <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-564-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7247" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7248" class="mjx-mrow"><span id="MJXc-Node-7249" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7250" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7251" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span id="MJXc-Node-7252" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7253" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7254" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7255" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7256" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7257" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7258" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7259" class="mjx-texatom"><span id="MJXc-Node-7260" class="mjx-mrow"><span id="MJXc-Node-7261" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7262" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7263" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>B</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-564">P_B(C)=P(C|B)</script>, et l'énoncé donne :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-565-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7264" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7265" class="mjx-mrow"><span id="MJXc-Node-7266" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7267" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7268" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7269" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7270" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7271" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-7272" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7273" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-7274" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7275" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7276" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7277" class="mjx-texatom"><span id="MJXc-Node-7278" class="mjx-mrow"><span id="MJXc-Node-7279" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7280" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7281" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7282" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7283" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7284" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7285" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-7286" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7287" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7288" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7289" class="mjx-texatom"><span id="MJXc-Node-7290" class="mjx-mrow"><span id="MJXc-Node-7291" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7292" class="mjx-texatom"><span id="MJXc-Node-7293" class="mjx-mrow"><span id="MJXc-Node-7294" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.268em;"><span id="MJXc-Node-7296" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7295" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span></span></span></span><span id="MJXc-Node-7297" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7298" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7299" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.078em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.078em; top: -1.368em;"><span id="MJXc-Node-7300" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 1.078em; bottom: -0.722em;"><span id="MJXc-Node-7301" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.078em;"></span></span><span class="mjx-vsize" style="height: 2.089em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-7302" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo>,</mo><mtext>&nbsp;</mtext><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>,</mo><mtext>&nbsp;</mtext><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>C</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>m</mi></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-565">P(C)=p,\ P(B|C)=1,\ P(B|\bar C)=\frac{1}{m}.</script>
D'après la formule des probabilités totales, on a :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-566-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7303" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7304" class="mjx-mrow"><span id="MJXc-Node-7305" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7306" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7307" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7308" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7309" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7310" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7311" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7312" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7313" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7314" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7315" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7316" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7317" class="mjx-texatom"><span id="MJXc-Node-7318" class="mjx-mrow"><span id="MJXc-Node-7319" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7320" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7321" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7322" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7323" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7324" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7325" class="mjx-texatom"><span id="MJXc-Node-7326" class="mjx-mrow"><span id="MJXc-Node-7327" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.268em;"><span id="MJXc-Node-7329" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7328" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span></span></span></span><span id="MJXc-Node-7330" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7331" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7332" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7333" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7334" class="mjx-texatom"><span id="MJXc-Node-7335" class="mjx-mrow"><span id="MJXc-Node-7336" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7337" class="mjx-texatom"><span id="MJXc-Node-7338" class="mjx-mrow"><span id="MJXc-Node-7339" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.268em;"><span id="MJXc-Node-7341" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7340" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span></span></span></span><span id="MJXc-Node-7342" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7343" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7344" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.804em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.804em; top: -1.542em;"><span id="MJXc-Node-7345" class="mjx-mrow"><span id="MJXc-Node-7346" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7347" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-7348" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7349" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7350" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7351" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span><span id="MJXc-Node-7352" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7353" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-denominator" style="width: 5.804em; bottom: -0.722em;"><span id="MJXc-Node-7354" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 5.804em;"></span></span><span class="mjx-vsize" style="height: 2.264em; vertical-align: -0.722em;"></span></span><span id="MJXc-Node-7355" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>C</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>C</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow><mi>m</mi></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-566">P(B)=P(C)P(B|C)+P(\bar C)P(B|\bar C)=\frac{(m-1)p+1}{m}.</script>
D'après la formule de Bayes :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-567-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7356" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7357" class="mjx-mrow"><span id="MJXc-Node-7358" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7359" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7360" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7361" class="mjx-texatom"><span id="MJXc-Node-7362" class="mjx-mrow"><span id="MJXc-Node-7363" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7364" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7365" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7366" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7367" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.815em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.815em; top: -1.542em;"><span id="MJXc-Node-7368" class="mjx-mrow"><span id="MJXc-Node-7369" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7370" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7371" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7372" class="mjx-texatom"><span id="MJXc-Node-7373" class="mjx-mrow"><span id="MJXc-Node-7374" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7375" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7376" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7377" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7378" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7379" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7380" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 5.815em; bottom: -1.042em;"><span id="MJXc-Node-7381" class="mjx-mrow"><span id="MJXc-Node-7382" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7383" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7384" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7385" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 5.815em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-7386" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7387" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.804em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.804em; top: -1.178em;"><span id="MJXc-Node-7388" class="mjx-mrow"><span id="MJXc-Node-7389" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-7390" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="mjx-denominator" style="width: 5.804em; bottom: -1.042em;"><span id="MJXc-Node-7391" class="mjx-mrow"><span id="MJXc-Node-7392" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7393" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7394" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7395" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">m</span></span><span id="MJXc-Node-7396" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7397" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7398" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7399" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 5.804em;"></span></span><span class="mjx-vsize" style="height: 2.221em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-7400" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>C</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>m</mi><mi>p</mi></mrow><mrow><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>p</mi></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-567">P(C|B)=\frac{P(B|C)P(C)}{P(B)}=\frac{mp}{1+(m-1)p}.</script></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>33</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Dés pipés<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1246"> [Signaler une erreur]</a><a id="exo33" href="javascript:ajouteexo(1246,33,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1246&#39;,&#39;imageenonce1246&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1246"></a><div class="inner" id="innerenonce1246">On dispose de 100 dés dont 25 sont pipés. Pour chaque dé pipé, la probabilité d'obtenir le chiffre 6 lors d'un lancer vaut <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-568-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7401" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7402" class="mjx-mrow"><span id="MJXc-Node-7403" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">1</span></span><span id="MJXc-Node-7404" class="mjx-texatom"><span id="MJXc-Node-7405" class="mjx-mrow"><span id="MJXc-Node-7406" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.458em; padding-bottom: 0.587em;">/</span></span></span></span><span id="MJXc-Node-7407" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.328em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-568">1/2</script>.
<ol class="enumeratechiffre">
<li> On tire un dé au hasard parmi les 100 dés. On lance ce dé et on obtient 6. Quelle est la probabilité que ce dé soit pipé?
</li><li> Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-569-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;msup&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x2217;&lt;/mo&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7408" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7409" class="mjx-mrow"><span id="MJXc-Node-7410" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span><span id="MJXc-Node-7411" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.263em; padding-bottom: 0.393em;">∈</span></span><span id="MJXc-Node-7412" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-7413" class="mjx-texatom"><span id="MJXc-Node-7414" class="mjx-mrow"><span id="MJXc-Node-7415" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.458em; padding-bottom: 0.328em;">N</span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.615em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7416" class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.198em; padding-bottom: 0.328em;">∗</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">N</mi></mrow><mo>∗</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-569">n\in\mathbb N^*</script>. On tire un dé au hasard parmi 100 dés. On lance ce dé <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-570-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7417" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7418" class="mjx-mrow"><span id="MJXc-Node-7419" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-570">n</script> fois et on obtient <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-571-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7420" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7421" class="mjx-mrow"><span id="MJXc-Node-7422" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-571">n</script> fois le chiffre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-572-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7423" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7424" class="mjx-mrow"><span id="MJXc-Node-7425" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.393em; padding-bottom: 0.393em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-572">6</script>.
Quelle est la probabilité <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-573-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 107%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7426" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7427" class="mjx-mrow"><span id="MJXc-Node-7428" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7429" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.458em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7430" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.198em; padding-bottom: 0.263em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>n</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-573">p_n</script> pour que ce dé soit pipé. Interpréter le résultat.
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1246&#39;,&#39;imageindication1246&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1246"></a><div class="inner" id="innerindication1246" style="display:none;"><ol class="enumeratechiffre">
<li> Formule de Bayes!
</li><li>
</li></ol>
</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1246&#39;,&#39;imagecorrige1246&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1246"></a><div class="inner" id="innercorrige1246" style="display:none;"><ol class="enumeratechiffre">
<li> Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-574-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7431" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7432" class="mjx-mrow"><span id="MJXc-Node-7433" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-574">T</script> l'événement : ``le dé est pipé'' et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-575-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7434" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7435" class="mjx-mrow"><span id="MJXc-Node-7436" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math></span></span><script type="math/tex" id="MathJax-Element-575">C</script> l'événement ``le lancer amène un 6''. On cherche <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-576-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7437" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7438" class="mjx-mrow"><span id="MJXc-Node-7439" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7440" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-7441" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span id="MJXc-Node-7442" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7443" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7444" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-576">P_C(T)</script>.
On va calculer cette probabilité en utilisant la formule de Bayes. En effet, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-577-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7445" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7446" class="mjx-mrow"><span id="MJXc-Node-7447" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7448" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7449" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7450" class="mjx-texatom MJXc-space1"><span id="MJXc-Node-7451" class="mjx-mrow"><span id="MJXc-Node-7452" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.288em;"><span id="MJXc-Node-7454" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7453" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span></span></span></span><span id="MJXc-Node-7455" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>T</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-577">(T,\bar T)</script> est un système complet
d'événements de probabilités non nulles, avec <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-578-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;25&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7456" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7457" class="mjx-mrow"><span id="MJXc-Node-7458" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7459" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7460" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7461" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7462" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7463" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25</span></span><span id="MJXc-Node-7464" class="mjx-texatom"><span id="MJXc-Node-7465" class="mjx-mrow"><span id="MJXc-Node-7466" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-7467" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">100</span></span><span id="MJXc-Node-7468" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7469" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7470" class="mjx-texatom"><span id="MJXc-Node-7471" class="mjx-mrow"><span id="MJXc-Node-7472" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-7473" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mn>25</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>100</mn><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-578">P(T)=25/100=1/4</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-579-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7474" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7475" class="mjx-mrow"><span id="MJXc-Node-7476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7477" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7478" class="mjx-texatom"><span id="MJXc-Node-7479" class="mjx-mrow"><span id="MJXc-Node-7480" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.288em;"><span id="MJXc-Node-7482" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7481" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span></span></span></span><span id="MJXc-Node-7483" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7484" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7485" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7486" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7487" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7488" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7489" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7490" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7491" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7492" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-7493" class="mjx-texatom"><span id="MJXc-Node-7494" class="mjx-mrow"><span id="MJXc-Node-7495" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-7496" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>T</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>4</mn></math></span></span><script type="math/tex" id="MathJax-Element-579">P(\bar T)=1-P(T)=3/4</script>. L'énoncé nous dit aussi que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-580-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7497" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7498" class="mjx-mrow"><span id="MJXc-Node-7499" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7500" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7501" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span><span id="MJXc-Node-7502" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7503" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7504" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7505" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7506" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7507" class="mjx-texatom"><span id="MJXc-Node-7508" class="mjx-mrow"><span id="MJXc-Node-7509" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-7510" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-580">P_T(C)=1/2</script> et bien sûr <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-581-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7511" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7512" class="mjx-mrow"><span id="MJXc-Node-7513" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7514" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.358em; padding-right: 0.071em;"><span id="MJXc-Node-7515" class="mjx-texatom" style=""><span id="MJXc-Node-7516" class="mjx-mrow"><span id="MJXc-Node-7517" class="mjx-texatom"><span id="MJXc-Node-7518" class="mjx-mrow"><span id="MJXc-Node-7519" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.288em;"><span id="MJXc-Node-7521" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7520" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-7522" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7523" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7524" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7525" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7526" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7527" class="mjx-texatom"><span id="MJXc-Node-7528" class="mjx-mrow"><span id="MJXc-Node-7529" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-7530" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mi>T</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn></math></span></span><script type="math/tex" id="MathJax-Element-581">P_{\bar T}(C)=1/6</script>. La formule
de Bayes donne alors
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-582-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right left right left right left right left right left right left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7531" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7532" class="mjx-mrow"><span id="MJXc-Node-7533" class="mjx-mtable" style="vertical-align: -4.049em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-7534" class="mjx-mtr" style="height: 2.818em;"><span id="MJXc-Node-7535" class="mjx-mtd" style="padding: 0px; text-align: right; width: 2.697em;"><span id="MJXc-Node-7536" class="mjx-mrow" style="margin-top: 0.542em;"><span id="MJXc-Node-7537" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7538" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-7539" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span id="MJXc-Node-7540" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7541" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7542" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-7543" class="mjx-mtd" style="padding: 0px; text-align: left; width: 13.032em;"><span id="MJXc-Node-7544" class="mjx-mrow"><span id="MJXc-Node-7545" class="mjx-mi"></span><span id="MJXc-Node-7546" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7547" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 11.459em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 11.459em; top: -1.542em;"><span id="MJXc-Node-7548" class="mjx-mrow"><span id="MJXc-Node-7549" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7550" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7551" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7552" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7553" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7554" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7555" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span><span id="MJXc-Node-7556" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7557" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7558" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 11.459em; bottom: -1.125em;"><span id="MJXc-Node-7559" class="mjx-mrow"><span id="MJXc-Node-7560" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7561" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7562" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span><span id="MJXc-Node-7563" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7564" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7565" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7566" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7567" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7568" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7569" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7570" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7571" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7572" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.358em; padding-right: 0.071em;"><span id="MJXc-Node-7573" class="mjx-texatom" style=""><span id="MJXc-Node-7574" class="mjx-mrow"><span id="MJXc-Node-7575" class="mjx-texatom"><span id="MJXc-Node-7576" class="mjx-mrow"><span id="MJXc-Node-7577" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.288em;"><span id="MJXc-Node-7579" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7578" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-7580" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7581" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-7582" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7583" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7584" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7585" class="mjx-texatom"><span id="MJXc-Node-7586" class="mjx-mrow"><span id="MJXc-Node-7587" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.288em;"><span id="MJXc-Node-7589" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7588" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span></span></span></span><span id="MJXc-Node-7590" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 11.459em;"></span></span><span class="mjx-vsize" style="height: 2.668em; vertical-align: -1.125em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-7591" class="mjx-mtr" style="height: 3.552em;"><span id="MJXc-Node-7592" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-7593" class="mjx-mrow" style="margin-top: 0.865em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-7594" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: left;"><span id="MJXc-Node-7595" class="mjx-mrow"><span id="MJXc-Node-7596" class="mjx-mi"></span><span id="MJXc-Node-7597" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7598" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 6.807em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 6.807em; top: -1.865em;"><span id="MJXc-Node-7599" class="mjx-mrow"><span id="MJXc-Node-7600" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7601" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.622em;"><span id="MJXc-Node-7602" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.372em; vertical-align: -0.44em;"></span></span><span id="MJXc-Node-7603" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-7604" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7605" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-7606" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span></span></span><span class="mjx-denominator" style="width: 6.807em; bottom: -1.387em;"><span id="MJXc-Node-7607" class="mjx-mrow"><span id="MJXc-Node-7608" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7609" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-7610" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span id="MJXc-Node-7611" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-7612" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7613" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.622em;"><span id="MJXc-Node-7614" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.372em; vertical-align: -0.44em;"></span></span><span id="MJXc-Node-7615" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7616" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7617" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.633em;"><span id="MJXc-Node-7618" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.38em; vertical-align: -0.448em;"></span></span><span id="MJXc-Node-7619" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-7620" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.339em;"><span id="MJXc-Node-7621" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.622em;"><span id="MJXc-Node-7622" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.387em; vertical-align: -0.44em;"></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 6.807em;"></span></span><span class="mjx-vsize" style="height: 3.252em; vertical-align: -1.387em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-7623" class="mjx-mtr" style="height: 2.228em;"><span id="MJXc-Node-7624" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-7625" class="mjx-mrow" style="margin-top: 0.368em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-7626" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: left;"><span id="MJXc-Node-7627" class="mjx-mrow"><span id="MJXc-Node-7628" class="mjx-mi"></span><span id="MJXc-Node-7629" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7630" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.7em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 0.7em; top: -1.368em;"><span id="MJXc-Node-7631" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 0.7em; bottom: -0.711em;"><span id="MJXc-Node-7632" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.7em;"></span></span><span class="mjx-vsize" style="height: 2.078em; vertical-align: -0.711em;"></span></span><span id="MJXc-Node-7633" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msub><mi>P</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><msub><mi>P</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>P</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mi>T</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>T</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>×</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-582">\begin{align*}
P_C(T)&=\frac{P(T)P_T(C)}{P_T(C)P(T)+P_{\bar T}(C)P(\bar T)}\\
&=\frac{\frac 14\times\frac 12}{\frac 12\times\frac 14+\frac 16\times \frac 34}\\
&=\frac 12.
\end{align*}</script>
</li><li> Introduisons les événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-583-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7634" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7635" class="mjx-mrow"><span id="MJXc-Node-7636" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-7637" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-7638" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-583">C_k</script> définis par ``le <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-584-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7639" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7640" class="mjx-mrow"><span id="MJXc-Node-7641" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-584">k</script>-ième lancer amène un 6'' et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-585-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x22C2;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7642" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7643" class="mjx-mrow"><span id="MJXc-Node-7644" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7645" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7646" class="mjx-munderover MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-7647" class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.528em; padding-bottom: 0.528em;">⋂</span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.392em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7653" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7648" class="mjx-texatom" style=""><span id="MJXc-Node-7649" class="mjx-mrow"><span id="MJXc-Node-7650" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span><span id="MJXc-Node-7651" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7652" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span><span id="MJXc-Node-7654" class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-7655" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-7656" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><munderover><mo>⋂</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>C</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-585">D=\bigcap_{k=1}^n C_k</script>.
Nous cherchons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-586-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7657" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7658" class="mjx-mrow"><span id="MJXc-Node-7659" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7660" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7661" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span></span><span id="MJXc-Node-7662" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7663" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7664" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>D</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-586">P_D(T)</script> que l'on calcule toujours par la formule de Bayes. Il faut juste remarquer que maintenant,
par indépendance des événements <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-587-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7665" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7666" class="mjx-mrow"><span id="MJXc-Node-7667" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span id="MJXc-Node-7668" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-7669" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">k</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-587">C_k</script>, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-588-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7670" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7671" class="mjx-mrow"><span id="MJXc-Node-7672" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7673" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7674" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span><span id="MJXc-Node-7675" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7676" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7677" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7678" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7679" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7680" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7681" class="mjx-texatom"><span id="MJXc-Node-7682" class="mjx-mrow"><span id="MJXc-Node-7683" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-7684" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-7685" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7686" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7687" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><msup><mo stretchy="false">)</mo><mi>n</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-588">P_T(D)=(1/2)^n</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-589-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7688" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7689" class="mjx-mrow"><span id="MJXc-Node-7690" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7691" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.358em; padding-right: 0.071em;"><span id="MJXc-Node-7692" class="mjx-texatom" style=""><span id="MJXc-Node-7693" class="mjx-mrow"><span id="MJXc-Node-7694" class="mjx-texatom"><span id="MJXc-Node-7695" class="mjx-mrow"><span id="MJXc-Node-7696" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.288em;"><span id="MJXc-Node-7698" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7697" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-7699" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7700" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7701" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7702" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7703" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7704" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7705" class="mjx-texatom"><span id="MJXc-Node-7706" class="mjx-mrow"><span id="MJXc-Node-7707" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-7708" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span><span id="MJXc-Node-7709" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7710" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7711" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mi>T</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>6</mn><msup><mo stretchy="false">)</mo><mi>n</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-589">P_{\bar T}(D)=(1/6)^n</script>. On trouve donc :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-590-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right left right left right left right left right left right left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mover&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7712" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7713" class="mjx-mrow"><span id="MJXc-Node-7714" class="mjx-mtable" style="vertical-align: -4.719em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-7715" class="mjx-mtr" style="height: 2.818em;"><span id="MJXc-Node-7716" class="mjx-mtd" style="padding: 0px; text-align: right; width: 2.745em;"><span id="MJXc-Node-7717" class="mjx-mrow" style="margin-top: 0.542em;"><span id="MJXc-Node-7718" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7719" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7720" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span></span></span><span id="MJXc-Node-7721" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7722" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7723" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-7724" class="mjx-mtd" style="padding: 0px; text-align: left; width: 13.168em;"><span id="MJXc-Node-7725" class="mjx-mrow"><span id="MJXc-Node-7726" class="mjx-mi"></span><span id="MJXc-Node-7727" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7728" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 11.595em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 11.595em; top: -1.542em;"><span id="MJXc-Node-7729" class="mjx-mrow"><span id="MJXc-Node-7730" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7731" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7732" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7733" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7734" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7735" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7736" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span><span id="MJXc-Node-7737" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7738" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7739" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 11.595em; bottom: -1.125em;"><span id="MJXc-Node-7740" class="mjx-mrow"><span id="MJXc-Node-7741" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7742" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7743" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span><span id="MJXc-Node-7744" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7745" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7746" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7747" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7748" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7749" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-7750" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7751" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7752" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-7753" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.358em; padding-right: 0.071em;"><span id="MJXc-Node-7754" class="mjx-texatom" style=""><span id="MJXc-Node-7755" class="mjx-mrow"><span id="MJXc-Node-7756" class="mjx-texatom"><span id="MJXc-Node-7757" class="mjx-mrow"><span id="MJXc-Node-7758" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.288em;"><span id="MJXc-Node-7760" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7759" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-7761" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7762" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">D</span></span><span id="MJXc-Node-7763" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7764" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7765" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7766" class="mjx-texatom"><span id="MJXc-Node-7767" class="mjx-mrow"><span id="MJXc-Node-7768" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.288em;"><span id="MJXc-Node-7770" class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.295em;">¯</span></span></span><span class="mjx-op"><span id="MJXc-Node-7769" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span></span></span></span><span id="MJXc-Node-7771" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 11.595em;"></span></span><span class="mjx-vsize" style="height: 2.668em; vertical-align: -1.125em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-7772" class="mjx-mtr" style="height: 4.597em;"><span id="MJXc-Node-7773" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-7774" class="mjx-mrow" style="margin-top: 1.399em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-7775" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: left;"><span id="MJXc-Node-7776" class="mjx-mrow"><span id="MJXc-Node-7777" class="mjx-mi"></span><span id="MJXc-Node-7778" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7779" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 10.114em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 10.114em; top: -2.399em;"><span id="MJXc-Node-7780" class="mjx-mrow"><span id="MJXc-Node-7781" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7782" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.622em;"><span id="MJXc-Node-7783" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.372em; vertical-align: -0.44em;"></span></span><span id="MJXc-Node-7784" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-7785" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-7786" class="mjx-mrow"><span id="MJXc-Node-7787" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">(</span></span><span id="MJXc-Node-7788" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7789" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-7790" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span id="MJXc-Node-7791" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.276em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7792" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span></span></span><span class="mjx-denominator" style="width: 10.114em; bottom: -1.899em;"><span id="MJXc-Node-7793" class="mjx-mrow"><span id="MJXc-Node-7794" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7795" class="mjx-mrow"><span id="MJXc-Node-7796" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">(</span></span><span id="MJXc-Node-7797" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7798" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.611em;"><span id="MJXc-Node-7799" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.364em; vertical-align: -0.432em;"></span></span><span id="MJXc-Node-7800" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.276em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7801" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-7802" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-7803" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7804" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.622em;"><span id="MJXc-Node-7805" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.372em; vertical-align: -0.44em;"></span></span><span id="MJXc-Node-7806" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7807" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-7808" class="mjx-mrow"><span id="MJXc-Node-7809" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">(</span></span><span id="MJXc-Node-7810" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.318em;"><span id="MJXc-Node-7811" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.633em;"><span id="MJXc-Node-7812" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">6</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.38em; vertical-align: -0.448em;"></span></span><span id="MJXc-Node-7813" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.937em; padding-bottom: 0.937em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.276em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7814" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-7815" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-7816" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.339em;"><span id="MJXc-Node-7817" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.622em;"><span id="MJXc-Node-7818" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">4</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 0.495em;"></span></span><span class="mjx-vsize" style="height: 1.387em; vertical-align: -0.44em;"></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 10.114em;"></span></span><span class="mjx-vsize" style="height: 4.297em; vertical-align: -1.899em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-7819" class="mjx-mtr" style="height: 2.522em;"><span id="MJXc-Node-7820" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-7821" class="mjx-mrow" style="margin-top: 0.368em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-7822" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: left;"><span id="MJXc-Node-7823" class="mjx-mrow"><span id="MJXc-Node-7824" class="mjx-mi"></span><span id="MJXc-Node-7825" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7826" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.336em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.336em; top: -1.368em;"><span id="MJXc-Node-7827" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span><span class="mjx-denominator" style="width: 4.336em; bottom: -1.005em;"><span id="MJXc-Node-7828" class="mjx-mrow"><span id="MJXc-Node-7829" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7830" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7831" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-7832" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.59em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-7833" class="mjx-texatom" style=""><span id="MJXc-Node-7834" class="mjx-mrow"><span id="MJXc-Node-7835" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7836" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span><span id="MJXc-Node-7837" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7838" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 4.336em;"></span></span><span class="mjx-vsize" style="height: 2.372em; vertical-align: -1.005em;"></span></span><span id="MJXc-Node-7839" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msub><mi>P</mi><mi>D</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><msub><mi>P</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>P</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mover><mi>T</mi><mo stretchy="false">¯</mo></mover></mrow></mrow></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>T</mi><mo stretchy="false">¯</mo></mover></mrow><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>×</mo><msup><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow><mi>n</mi></msup></mrow><mrow><msup><mrow><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow><mi>n</mi></msup><mo>×</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>)</mo></mrow><mi>n</mi></msup><mo>×</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mn>3</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-590">\begin{align*}
P_D(T)&=\frac{P(T)P_T(D)}{P_T(D)P(T)+P_{\bar T}(D)P(\bar T)}\\
&=\frac{\frac 14\times\left(\frac 12\right)^n}{\left(\frac 12\right)^n\times\frac 14+\left(\frac 16\right)^n\times \frac 34}\\
&=\frac {1}{1+3^{-n+1}}.
\end{align*}</script>
En particulier, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-591-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7840" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7841" class="mjx-mrow"><span id="MJXc-Node-7842" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7843" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-7844" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.47em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-7845" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span id="MJXc-Node-7846" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>p</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-591">(p_n)</script> tend vers 1 lorsque <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-592-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7847" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7848" class="mjx-mrow"><span id="MJXc-Node-7849" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-592">n</script> tend vers l'infini, ce qui est conforme à l'intuition : plus <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-593-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7850" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7851" class="mjx-mrow"><span id="MJXc-Node-7852" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-593">n</script> est grand, plus le dé est très certainement pipé.
</li></ol>
</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>34</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Pièces défectueuses<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1247"> [Signaler une erreur]</a><a id="exo34" href="javascript:ajouteexo(1247,34,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1247&#39;,&#39;imageenonce1247&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1247"></a><div class="inner" id="innerenonce1247">Une usine fabrique des pièces, avec une proportion de 0,05 de pièces défectueuses. Le contrôle des
fabrications est tel que :
<ul class="rien">
<li> si la pièce est bonne, elle est acceptée avec la probabilité 0,96.
</li><li> si la pièce est mauvaise, elle est refusée avec la probabilité 0,98.
</li></ul>
On choisit une pièce au hasard et on la contrôle. Quelle est la probabilité
<ol class="enumeratechiffre">
<li> qu'il y ait une erreur de contrôle?
</li><li> qu'une pièce acceptée soit mauvaise?
</li></ol></div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1247&#39;,&#39;imageindication1247&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1247"></a><div class="inner" id="innerindication1247" style="display:none;"><ol class="enumeratechiffre">
<li> Décomposer l'événement en "bonne et refusée" et "mauvaise et acceptée", ces deux événements étant incompatibles.
</li><li> C'est la formule de Bayes!
</li></ol></div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1247&#39;,&#39;imagecorrige1247&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1247"></a><div class="inner" id="innercorrige1247" style="display:none;"><ol class="enumeratechiffre">
<li> On note <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-594-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7853" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7854" class="mjx-mrow"><span id="MJXc-Node-7855" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-594">A</script> l'événement "la pièce est acceptée par le contrôle", et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-595-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7856" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7857" class="mjx-mrow"><span id="MJXc-Node-7858" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-595">B</script> l'événement "la pièce est bonne".
L'événement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-596-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7859" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7860" class="mjx-mrow"><span id="MJXc-Node-7861" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math></span></span><script type="math/tex" id="MathJax-Element-596">E</script> "Il y a une erreur au contrôle" se décompose en <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-597-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7862" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7863" class="mjx-mrow"><span id="MJXc-Node-7864" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7865" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-7866" class="mjx-munderover MJXc-space2"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7868" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7867" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∩</mo><mover><mi>B</mi><mo accent="false">¯</mo></mover></math></span></span><script type="math/tex" id="MathJax-Element-597">A\cap\overline{B}</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-598-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7869" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7870" class="mjx-mrow"><span id="MJXc-Node-7871" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7873" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.051em 0px -0.19em; letter-spacing: -0.281em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7872" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span><span id="MJXc-Node-7874" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-7875" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>A</mi><mo accent="false">¯</mo></mover><mo>∩</mo><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-598">\overline{A}\cap B</script>. Ces deux derniers événements sont incompatibles, on a donc :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-599-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-7876" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7877" class="mjx-mrow"><span id="MJXc-Node-7878" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7879" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7880" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-7881" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7882" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7883" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7884" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7885" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7886" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-7887" class="mjx-munderover MJXc-space2"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7889" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7888" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-7890" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7891" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-7892" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7893" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7894" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7896" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.051em 0px -0.19em; letter-spacing: -0.281em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7895" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span><span id="MJXc-Node-7897" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-7898" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7899" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7900" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo accent="false">¯</mo></mover><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-599">P(E)=P(A\cap\overline{B})+P(\overline{A}\cap B).</script>
Maintenant, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-600-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7901" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7902" class="mjx-mrow"><span id="MJXc-Node-7903" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7904" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7905" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7906" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-7907" class="mjx-munderover MJXc-space2"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7909" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7908" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-7910" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7911" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7912" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7913" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7914" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7915" class="mjx-texatom"><span id="MJXc-Node-7916" class="mjx-mrow"><span id="MJXc-Node-7917" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7918" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7920" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7919" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-7921" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7922" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7923" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7924" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7926" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7925" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-7927" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-600">P(A\cap\overline{B})=P(A|\overline{B})P(\overline{B})</script>. Or, <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-601-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;05&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7928" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7929" class="mjx-mrow"><span id="MJXc-Node-7930" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7931" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7932" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7934" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7933" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-7935" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7936" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7937" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-7938" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7939" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">05</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>05</mn></math></span></span><script type="math/tex" id="MathJax-Element-601">P(\overline{B})=0,05</script>, et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-602-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;02&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7940" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7941" class="mjx-mrow"><span id="MJXc-Node-7942" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7943" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7944" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-7945" class="mjx-texatom"><span id="MJXc-Node-7946" class="mjx-mrow"><span id="MJXc-Node-7947" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7948" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7950" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7949" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-7951" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7952" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7953" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-7954" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-7955" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7956" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7957" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7959" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.051em 0px -0.19em; letter-spacing: -0.281em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7958" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span><span id="MJXc-Node-7960" class="mjx-texatom"><span id="MJXc-Node-7961" class="mjx-mrow"><span id="MJXc-Node-7962" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7963" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7965" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7964" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-7966" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7967" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7968" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-7969" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-7970" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">02</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo accent="false">¯</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>02</mn></math></span></span><script type="math/tex" id="MathJax-Element-602">P(A|\overline{B})=1-P(\overline{A}|\overline{B})=0,02</script>.
De même, on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-603-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;&amp;#x2229;&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7971" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7972" class="mjx-mrow"><span id="MJXc-Node-7973" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7974" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7975" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7977" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.051em 0px -0.19em; letter-spacing: -0.281em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7976" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span><span id="MJXc-Node-7978" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.353em;">∩</span></span><span id="MJXc-Node-7979" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7980" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7981" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-7982" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7983" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7984" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-7986" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.051em 0px -0.19em; letter-spacing: -0.281em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-7985" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span><span id="MJXc-Node-7987" class="mjx-texatom"><span id="MJXc-Node-7988" class="mjx-mrow"><span id="MJXc-Node-7989" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-7990" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7991" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-7992" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7993" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-7994" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-7995" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo accent="false">¯</mo></mover><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo accent="false">¯</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-603">P(\overline{A}\cap B)=P(\overline{A}|B)P(B)</script> et on a <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-604-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;04&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-7996" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-7997" class="mjx-mrow"><span id="MJXc-Node-7998" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-7999" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8000" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-8002" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.051em 0px -0.19em; letter-spacing: -0.281em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-8001" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span></span></span></span><span id="MJXc-Node-8003" class="mjx-texatom"><span id="MJXc-Node-8004" class="mjx-mrow"><span id="MJXc-Node-8005" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8006" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-8007" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8008" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8009" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8010" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8011" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8012" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8013" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-8014" class="mjx-texatom"><span id="MJXc-Node-8015" class="mjx-mrow"><span id="MJXc-Node-8016" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8017" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span><span id="MJXc-Node-8018" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8019" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8020" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8021" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8022" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">04</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mover><mi>A</mi><mo accent="false">¯</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>04</mn></math></span></span><script type="math/tex" id="MathJax-Element-604">P(\overline{A}|B)=1-P(A|B)=0,04</script>.
On obtient finalement :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-605-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;95&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;04&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;05&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;02.&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-8023" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8024" class="mjx-mrow"><span id="MJXc-Node-8025" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8026" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8027" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.026em;">E</span></span><span id="MJXc-Node-8028" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8029" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8030" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8031" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8032" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">95</span></span><span id="MJXc-Node-8033" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-8034" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8035" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8036" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">04</span></span><span id="MJXc-Node-8037" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8038" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8039" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8040" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">05</span></span><span id="MJXc-Node-8041" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-8042" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8043" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8044" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">02.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mn>95</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>04</mn><mo>+</mo><mn>0</mn><mo>,</mo><mn>05</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>02.</mn></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-605">P(E)=0,95\times 0,04+0,05\times 0,02.</script>
</li><li> Dans cette question, on cherche <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-606-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8045" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8046" class="mjx-mrow"><span id="MJXc-Node-8047" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8048" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8049" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-8051" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-8050" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-8052" class="mjx-texatom"><span id="MJXc-Node-8053" class="mjx-mrow"><span id="MJXc-Node-8054" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8055" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-8056" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mover><mi>B</mi><mo accent="false">¯</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-606">P(\overline{B}|A)</script> alors que l'on connait les probabilités conditionnelles sachant <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-607-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8057" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8058" class="mjx-mrow"><span id="MJXc-Node-8059" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-607">B</script>. Ceci nous invite à utiliser la formule de Bayes.
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-608-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mover&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo accent=&quot;false&quot;&gt;&amp;#x00AF;&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;05&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;02&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;95&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;96&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;05&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;02&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;913&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;&amp;#x2243;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;001.&lt;/mn&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-8060" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8061" class="mjx-mrow"><span id="MJXc-Node-8062" class="mjx-mtable" style="vertical-align: -2.647em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-8063" class="mjx-mtr" style="height: 3.336em;"><span id="MJXc-Node-8064" class="mjx-mtd" style="padding: 0px; text-align: right; width: 3.316em;"><span id="MJXc-Node-8065" class="mjx-mrow" style="margin-top: 0.767em;"><span id="MJXc-Node-8066" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8067" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8068" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-8070" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-8069" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-8071" class="mjx-texatom"><span id="MJXc-Node-8072" class="mjx-mrow"><span id="MJXc-Node-8073" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8074" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-8075" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8076" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-8077" class="mjx-mrow" style="margin-top: 0.843em;"><span id="MJXc-Node-8078" class="mjx-mi"></span><span id="MJXc-Node-8079" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8080" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 19.216em;"><span id="MJXc-Node-8081" class="mjx-mrow"><span id="MJXc-Node-8082" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 12.63em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 12.63em; top: -1.843em;"><span id="MJXc-Node-8083" class="mjx-mrow"><span id="MJXc-Node-8084" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8085" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8086" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-8088" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-8087" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-8089" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8090" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8091" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8092" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-8093" class="mjx-texatom"><span id="MJXc-Node-8094" class="mjx-mrow"><span id="MJXc-Node-8095" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8096" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-8098" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-8097" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-8099" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 12.63em; 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padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8120" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8121" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8122" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">A</span></span><span id="MJXc-Node-8123" class="mjx-texatom"><span id="MJXc-Node-8124" class="mjx-mrow"><span id="MJXc-Node-8125" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8126" class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span id="MJXc-Node-8128" class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px 0.048em 0px -0.187em; letter-spacing: -0.274em;">¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span id="MJXc-Node-8127" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em;">B</span></span></span></span></span><span id="MJXc-Node-8129" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 12.63em;"></span></span><span class="mjx-vsize" style="height: 3.186em; vertical-align: -1.343em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-8130" class="mjx-mtr" style="height: 2.457em;"><span id="MJXc-Node-8131" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-8132" class="mjx-mrow" style="margin-top: 0.402em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8133" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-8134" class="mjx-mrow" style="margin-top: 0.402em;"><span id="MJXc-Node-8135" class="mjx-mi"></span><span id="MJXc-Node-8136" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8137" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-8138" class="mjx-mrow"><span id="MJXc-Node-8139" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 11.646em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 11.646em; top: -1.402em;"><span id="MJXc-Node-8140" class="mjx-mrow"><span id="MJXc-Node-8141" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; 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padding-bottom: 0.353em;">913</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.7em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-8168" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.295em;">≃</span></span><span id="MJXc-Node-8169" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8170" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8171" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">001.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><mover><mi>B</mi><mo accent="false">¯</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mi>B</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><mn>0</mn><mo>,</mo><mn>05</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>02</mn></mrow><mrow><mn>0</mn><mo>,</mo><mn>95</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>96</mn><mo>+</mo><mn>0</mn><mo>,</mo><mn>05</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>02</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>913</mn></mfrac><mo>≃</mo><mn>0</mn><mo>,</mo><mn>001.</mn></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-608">\begin{eqnarray*}
P(\overline{B}|A)&=&\frac{P(\overline{B})P(A|\overline{B})}{P(B)P(A|B)+P(\overline{B})P(A|\overline{B})}\\
&=&\frac{0,05\times 0,02}{0,95\times 0,96+0,05\times 0,02}=\frac{1}{913}\simeq 0,001.
\end{eqnarray*}</script>
</li></ol></div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>35</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - La forêt<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=2019"> [Signaler une erreur]</a><a id="exo35" href="javascript:ajouteexo(2019,35,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce2019&#39;,&#39;imageenonce2019&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce2019"></a><div class="inner" id="innerenonce2019">Une forêt se compose de trois types d'arbres : 30% sont des chênes, 50% des peupliers, et 20% des hêtres. Suite à une tempête, une maladie se déclare et touche 10% des chênes, 4% des peupliers, et 25% des hêtres. Sachant qu'un arbre est malade, quelle est la probabilité que ce soit un chêne? un peuplier? un hêtre?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication2019&#39;,&#39;imageindication2019&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication2019"></a><div class="inner" id="innerindication2019" style="display:none;">Utiliser la formule de Bayes.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige2019&#39;,&#39;imagecorrige2019&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige2019"></a><div class="inner" id="innercorrige2019" style="display:none;">Notons <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-609-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8172" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8173" class="mjx-mrow"><span id="MJXc-Node-8174" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></span></span><script type="math/tex" id="MathJax-Element-609">M</script> l'événement "l'arbre étudié est malade", et respectivement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-610-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8175" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8176" class="mjx-mrow"><span id="MJXc-Node-8177" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-8178" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8179" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-8180" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8181" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">Q</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>Q</mi></math></span></span><script type="math/tex" id="MathJax-Element-610">C,H,Q</script> l'événement "l'arbre étudié est un chêne (resp. un hêtre, un peuplier)". On cherche à connaitre <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-611-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8182" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8183" class="mjx-mrow"><span id="MJXc-Node-8184" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-8185" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8186" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span id="MJXc-Node-8187" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8188" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-8189" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-611">P_M(C)</script>. Par la formule de Bayes,
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-612-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mtable columnalign=&quot;right center left&quot; rowspacing=&quot;3pt&quot; columnspacing=&quot;0 thickmathspace&quot; displaystyle=&quot;true&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;04&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;25&lt;/mn&gt;&lt;mo&gt;&amp;#x00D7;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd /&gt;&lt;mtd&gt;&lt;mi&gt;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-8190" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8191" class="mjx-mrow"><span id="MJXc-Node-8192" class="mjx-mtable" style="vertical-align: -3.571em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-8193" class="mjx-mtr" style="height: 2.776em;"><span id="MJXc-Node-8194" class="mjx-mtd" style="padding: 0px; text-align: right; width: 2.959em;"><span id="MJXc-Node-8195" class="mjx-mrow" style="margin-top: 0.542em;"><span id="MJXc-Node-8196" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-8197" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8198" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span id="MJXc-Node-8199" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8200" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-8201" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8202" class="mjx-mtd" style="padding: 0px 0.139em 0px 0px; width: 1.056em;"><span id="MJXc-Node-8203" class="mjx-mrow" style="margin-top: 0.542em;"><span id="MJXc-Node-8204" class="mjx-mi"></span><span id="MJXc-Node-8205" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8206" class="mjx-mtd" style="padding: 0px 0px 0px 0.139em; text-align: left; width: 19.155em;"><span id="MJXc-Node-8207" class="mjx-mrow"><span id="MJXc-Node-8208" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 18.915em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 18.915em; top: -1.542em;"><span id="MJXc-Node-8209" class="mjx-mrow"><span id="MJXc-Node-8210" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-8211" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-8212" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span id="MJXc-Node-8213" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8214" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-8215" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8216" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8217" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8218" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-8219" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 18.915em; bottom: -1.084em;"><span id="MJXc-Node-8220" class="mjx-mrow"><span id="MJXc-Node-8221" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-8222" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-8223" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span></span></span><span id="MJXc-Node-8224" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8225" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-8226" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8227" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8228" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8229" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.045em;">C</span></span><span id="MJXc-Node-8230" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8231" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8232" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-8233" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8234" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span></span></span><span id="MJXc-Node-8235" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8236" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-8237" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8238" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8239" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8240" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-8241" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8242" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8243" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-8244" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.229em; padding-right: 0.071em;"><span id="MJXc-Node-8245" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">Q</span></span></span></span><span id="MJXc-Node-8246" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8247" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span><span id="MJXc-Node-8248" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8249" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8250" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8251" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">Q</span></span><span id="MJXc-Node-8252" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 18.915em;"></span></span><span class="mjx-vsize" style="height: 2.626em; vertical-align: -1.084em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-8253" class="mjx-mtr" style="height: 2.616em;"><span id="MJXc-Node-8254" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-8255" class="mjx-mrow" style="margin-top: 0.402em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8256" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-8257" class="mjx-mrow" style="margin-top: 0.402em;"><span id="MJXc-Node-8258" class="mjx-mi"></span><span id="MJXc-Node-8259" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8260" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-8261" class="mjx-mrow"><span id="MJXc-Node-8262" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 15.98em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 15.98em; top: -1.402em;"><span id="MJXc-Node-8263" class="mjx-mrow"><span id="MJXc-Node-8264" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8265" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8266" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8267" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-8268" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8269" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8270" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span></span><span class="mjx-denominator" style="width: 15.98em; bottom: -0.913em;"><span id="MJXc-Node-8271" class="mjx-mrow"><span id="MJXc-Node-8272" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8273" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8274" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8275" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-8276" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8277" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8278" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span><span id="MJXc-Node-8279" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8280" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8281" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8282" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.411em; padding-bottom: 0.353em;">04</span></span><span id="MJXc-Node-8283" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-8284" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8285" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8286" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span><span id="MJXc-Node-8287" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8288" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8289" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8290" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">25</span></span><span id="MJXc-Node-8291" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.178em; padding-bottom: 0.353em;">×</span></span><span id="MJXc-Node-8292" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">0</span></span><span id="MJXc-Node-8293" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8294" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 15.98em;"></span></span><span class="mjx-vsize" style="height: 2.316em; vertical-align: -0.913em;"></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-8295" class="mjx-mtr" style="height: 2.249em;"><span id="MJXc-Node-8296" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-8297" class="mjx-mrow" style="margin-top: 0.367em;"><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8298" class="mjx-mtd" style="padding: 0.15em 0.139em 0px 0px;"><span id="MJXc-Node-8299" class="mjx-mrow" style="margin-top: 0.367em;"><span id="MJXc-Node-8300" class="mjx-mi"></span><span id="MJXc-Node-8301" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-8302" class="mjx-mtd" style="padding: 0.15em 0px 0px 0.139em; text-align: left;"><span id="MJXc-Node-8303" class="mjx-mrow"><span id="MJXc-Node-8304" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.367em;"><span id="MJXc-Node-8305" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">3</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-8306" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.099em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-8307" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" displaystyle="true"><mtr><mtd><msub><mi>P</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><msub><mi>P</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>P</mi><mi>C</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>P</mi><mi>H</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>P</mi><mi>Q</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>3</mn></mrow><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>+</mo><mn>0</mn><mo>,</mo><mn>04</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>5</mn><mo>+</mo><mn>0</mn><mo>,</mo><mn>25</mn><mo>×</mo><mn>0</mn><mo>,</mo><mn>2</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo></mtd><mtd><mfrac><mn>3</mn><mn>10</mn></mfrac><mo>.</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-612">\begin{eqnarray*}
P_M(C)&=&\frac{P_C(M)P(C)}{P_C(M)P(C)+P_H(M)P(H)+P_Q(M)P(Q)}\\
&=&\frac{0,1\times 0,3}{0,1\times 0,3+0,04\times 0,5+0,25\times 0,2}\\
&=&\frac{3}{10}.
\end{eqnarray*}</script>
De la même façon, on trouve que
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-613-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mtext&gt;&amp;#xA0;&lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-8308" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8309" class="mjx-mrow"><span id="MJXc-Node-8310" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-8311" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8312" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span id="MJXc-Node-8313" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8314" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-8315" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8316" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8317" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-8318" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">5</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-8319" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-8320" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8321" class="mjx-mtext MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.289em; padding-bottom: 0.353em;">&nbsp;</span></span><span id="MJXc-Node-8322" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.109em;"><span id="MJXc-Node-8323" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-8324" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.081em;">M</span></span></span></span><span id="MJXc-Node-8325" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8326" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.47em;">Q</span></span><span id="MJXc-Node-8327" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8328" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8329" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.368em;"><span id="MJXc-Node-8330" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.733em;"><span id="MJXc-Node-8331" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">10</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 1.2em;"></span></span><span class="mjx-vsize" style="height: 2.1em; vertical-align: -0.733em;"></span></span><span id="MJXc-Node-8332" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>P</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>5</mn><mn>10</mn></mfrac><mo>,</mo><mtext>&nbsp;</mtext><msub><mi>P</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>2</mn><mn>10</mn></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-613">P_M(H)=\frac{5}{10},\ P_M(Q)=\frac2{10}.</script>
On vérifie bien que la somme des probabilités vaut 1.</div></div></div>
<div class="exo"><div class="titreexo"><span>Exercice </span><span>36</span> <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"><img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/Icone_etoile.png" align="bottom"> - Menteur!<a href="http://www.bibmath.net/ressources/signalerreur.php?numero=1252"> [Signaler une erreur]</a><a id="exo36" href="javascript:ajouteexo(1252,36,1)"> [Ajouter à ma feuille d'exos]</a></div><div class="enonce"><a href="javascript:showAll(&#39;innerenonce1252&#39;,&#39;imageenonce1252&#39;);" class="lienexo">Enoncé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/bas.png" id="imageenonce1252"></a><div class="inner" id="innerenonce1252">Vous jouez à pile ou face avec un autre joueur. Il parie sur pile, lance la pièce, et obtient pile. Quelle est la probabilité pour qu'il soit un tricheur?</div></div><div class="indication"><a href="javascript:showAll(&#39;innerindication1252&#39;,&#39;imageindication1252&#39;);" class="lienexo">Indication <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imageindication1252"></a><div class="inner" id="innerindication1252" style="display:none;">Introduire la proportion <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-614-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8333" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8334" class="mjx-mrow"><span id="MJXc-Node-8335" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-614">x</script> de tricheurs dans la population, et donner, à l'aide du théorème de Bayes, une estimation de la probabilité qu'il soit un tricheur en fonction de <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-615-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8336" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8337" class="mjx-mrow"><span id="MJXc-Node-8338" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-615">x</script>. Il faudra convenir de certaines probabilités de bon sens.</div></div><div class="corrige"><a href="javascript:showAll(&#39;innercorrige1252&#39;,&#39;imagecorrige1252&#39;);" class="lienexo">Corrigé <img src="./Ma feuille d&#39;exercice - Bibm@th.net_files/droite.png" id="imagecorrige1252"></a><div class="inner" id="innercorrige1252" style="display:none;">Soit <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-616-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8339" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8340" class="mjx-mrow"><span id="MJXc-Node-8341" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-616">x</script> la proportion de tricheurs dans la population. On note respectivement <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-617-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8342" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8343" class="mjx-mrow"><span id="MJXc-Node-8344" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8345" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8346" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span><span id="MJXc-Node-8347" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8348" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-8349" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.528em;">,</span></span><span id="MJXc-Node-8350" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>,</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-617">P,F,H,T</script> les événements "le joueur obtient pile'', "le joueur obtient face'', "Le joueur est honnête'', "le joueur est un tricheur''. Il semble raisonnable de convenir que <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-618-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8351" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8352" class="mjx-mrow"><span id="MJXc-Node-8353" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8354" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8355" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8356" class="mjx-texatom"><span id="MJXc-Node-8357" class="mjx-mrow"><span id="MJXc-Node-8358" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8359" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-8360" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8361" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8362" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8363" class="mjx-texatom"><span id="MJXc-Node-8364" class="mjx-mrow"><span id="MJXc-Node-8365" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-8366" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-618">P(P|H)=1/2</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-619-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8367" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8368" class="mjx-mrow"><span id="MJXc-Node-8369" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8370" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8371" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.106em;">F</span></span><span id="MJXc-Node-8372" class="mjx-texatom"><span id="MJXc-Node-8373" class="mjx-mrow"><span id="MJXc-Node-8374" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8375" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-8376" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8377" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8378" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8379" class="mjx-texatom"><span id="MJXc-Node-8380" class="mjx-mrow"><span id="MJXc-Node-8381" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-8382" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-619">P(F|H)=1/2</script> et <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-620-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8383" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8384" class="mjx-mrow"><span id="MJXc-Node-8385" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8386" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8387" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8388" class="mjx-texatom"><span id="MJXc-Node-8389" class="mjx-mrow"><span id="MJXc-Node-8390" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8391" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-8392" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8393" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8394" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-620">P(P|T)=1</script> (un tricheur fait vraiment ce qu'il veut!). On cherche donc <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-621-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8395" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8396" class="mjx-mrow"><span id="MJXc-Node-8397" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8398" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8399" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-8400" class="mjx-texatom"><span id="MJXc-Node-8401" class="mjx-mrow"><span id="MJXc-Node-8402" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8403" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8404" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>P</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-621">P(T|P)</script>. De la formule de Bayes, on déduit :
<span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-622-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; text-align: center; position: relative;"><span id="MJXc-Node-8405" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8406" class="mjx-mrow"><span id="MJXc-Node-8407" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8408" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8409" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-8410" class="mjx-texatom"><span id="MJXc-Node-8411" class="mjx-mrow"><span id="MJXc-Node-8412" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8413" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8414" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8415" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8416" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 12.78em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 12.78em; top: -1.542em;"><span id="MJXc-Node-8417" class="mjx-mrow"><span id="MJXc-Node-8418" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8419" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8420" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8421" class="mjx-texatom"><span id="MJXc-Node-8422" class="mjx-mrow"><span id="MJXc-Node-8423" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8424" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-8425" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8426" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8427" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8428" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-8429" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-denominator" style="width: 12.78em; bottom: -1.042em;"><span id="MJXc-Node-8430" class="mjx-mrow"><span id="MJXc-Node-8431" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8432" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8433" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8434" class="mjx-texatom"><span id="MJXc-Node-8435" class="mjx-mrow"><span id="MJXc-Node-8436" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8437" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-8438" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8439" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8440" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8441" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.12em;">T</span></span><span id="MJXc-Node-8442" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8443" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8444" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8445" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8446" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8447" class="mjx-texatom"><span id="MJXc-Node-8448" class="mjx-mrow"><span id="MJXc-Node-8449" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">|</span></span></span></span><span id="MJXc-Node-8450" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-8451" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span><span id="MJXc-Node-8452" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.109em;">P</span></span><span id="MJXc-Node-8453" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8454" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.47em; padding-bottom: 0.295em; padding-right: 0.057em;">H</span></span><span id="MJXc-Node-8455" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 12.78em;"></span></span><span class="mjx-vsize" style="height: 2.585em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-8456" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8457" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 6.567em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 6.567em; top: -1.144em;"><span id="MJXc-Node-8458" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span><span class="mjx-denominator" style="width: 6.567em; bottom: -1.042em;"><span id="MJXc-Node-8459" class="mjx-mrow"><span id="MJXc-Node-8460" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8461" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8462" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8463" class="mjx-texatom"><span id="MJXc-Node-8464" class="mjx-mrow"><span id="MJXc-Node-8465" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">/</span></span></span></span><span id="MJXc-Node-8466" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8467" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">(</span></span><span id="MJXc-Node-8468" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span><span id="MJXc-Node-8469" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">−</span></span><span id="MJXc-Node-8470" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8471" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.47em; padding-bottom: 0.586em;">)</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 6.567em;"></span></span><span class="mjx-vsize" style="height: 2.186em; vertical-align: -1.042em;"></span></span><span id="MJXc-Node-8472" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.061em; padding-bottom: 0.295em;">=</span></span><span id="MJXc-Node-8473" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.494em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.494em; top: -1.368em;"><span id="MJXc-Node-8474" class="mjx-mrow"><span id="MJXc-Node-8475" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">2</span></span><span id="MJXc-Node-8476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="mjx-denominator" style="width: 2.494em; bottom: -0.793em;"><span id="MJXc-Node-8477" class="mjx-mrow"><span id="MJXc-Node-8478" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span><span id="MJXc-Node-8479" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.295em; padding-bottom: 0.411em;">+</span></span><span id="MJXc-Node-8480" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.353em; padding-bottom: 0.353em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.286em; width: 2.494em;"></span></span><span class="mjx-vsize" style="height: 2.16em; vertical-align: -0.793em;"></span></span><span id="MJXc-Node-8481" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.172em; padding-bottom: 0.353em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>T</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>T</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>H</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-622">P(T|P)=\frac{P(P|T)P(T)}{P(P|T)P(T)+P(P|H)P(H)}=\frac{x}{x+1/2(1-x)}=\frac{2x}{x+1}.</script>
Le résultat est plus ou moins réconfortant suivant la proportion de tricheurs <span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-623-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;" role="presentation" style="font-size: 119%; position: relative;"><span id="MJXc-Node-8482" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8483" class="mjx-mrow"><span id="MJXc-Node-8484" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.236em; padding-bottom: 0.295em;">x</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-623">x</script> dans la population!</div></div></div>
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