From 0eb717651e8d021a09eca906da6c348b53a602c7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?F=C3=A9lix=20MARQUET?= Date: Fri, 20 Dec 2024 08:26:12 +0100 Subject: [PATCH] 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. --- Ma feuille d'exercice - Bibm@th.net.html | 1021 ++++++++++ .../Icone_etoile.png | Bin 0 -> 935 bytes .../MathJax.js.téléchargement | 19 + .../ajouteexo.js.téléchargement | 47 + .../bas(1).png | Bin 0 -> 1225 bytes .../bas.png | Bin 0 -> 1225 bytes .../bibliotheques.png | Bin 0 -> 841 bytes .../bibmathnet.js.téléchargement | 4 + .../bibstyle.css | 1766 +++++++++++++++++ .../deletemobile.png | Bin 0 -> 1010 bytes .../droite.png | Bin 0 -> 1240 bytes .../etudiants1.png | Bin 0 -> 9634 bytes .../etudiants2.png | Bin 0 -> 10768 bytes .../fb.png | Bin 0 -> 5268 bytes .../fbmobile.png | Bin 0 -> 1263 bytes .../flechemenumobileferme.png | Bin 0 -> 562 bytes .../flechemenumobileouvert.png | Bin 0 -> 583 bytes .../fonctions.js.téléchargement | 134 ++ .../forum.png | Bin 0 -> 1126 bytes .../forumgrand.png | Bin 0 -> 2532 bytes Ma feuille d'exercice - 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Bibm@th
Mon compte > Accéder à ma feuille d'exercices >

Exercice 1 - Triangles [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
On trace dans un plan n≥3 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?
Indication
Corrigé
Un triangle est déterminé par 3 droites (ses côtés). Il y a autant de triangles que de possibilités de choisir 3 droites parmi n, c'est-à-dire (n3).
+
Exercice 2 - Podium! [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Une course oppose 20 concurrents, dont Émile. +
    +
  1. Combien y-a-t-il de podiums possibles? +
  2. Combien y-a-t-il de podiums possibles où Émile est premier? +
  3. Combien y-a-t-il de podiums possibles dont Émile fait partie? +
  4. On souhaite récompenser les 3 premiers en leur offrant un prix identique à chacun. Combien y-a-t-il de distributions de récompenses possibles? +
Indication
Corrigé
+
Exercice 3 - Les boulangeries [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Dans une ville, il y a quatre boulangeries qui ferment un jour par semaine. +
    +
  1. Déterminer le nombre de façons d'attribuer un jour de fermeture hebdomadaire à chaque boulangerie. +
  2. Déterminer le nombre de façons d'attribuer un jour de fermeture hebdomadaire à chaque boulangerie si plusieurs boulangeries ne peuvent fermer le même jour. +
  3. 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. +
Indication
Corrigé
+
Exercice 4 - Autour d'une table [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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?
Indication
Corrigé
+
Exercice 5 - Addition, multiplication et combinaison [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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?
Indication
Corrigé
+
Exercice 6 - Le cadenas [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Un cadenas possède un code à 3 chiffres, chacun des chiffres pouvant être un chiffre de 1 à 9. +
    +
  1. +
      +
    1. Combien y-a-t-il de codes possibles? +
    2. Combien y-a-t-il de codes se terminant par un chiffre pair? +
    3. Combien y-a-t-il de codes contenant au moins un chiffre 4? +
    4. Combien y-a-t-il de codes contenant exactement un chiffre 4? +
    +
  2. Dans les questions suivantes, on souhaite que le code comporte obligatoirement trois chiffres distincts. +
      +
    1. Combien y-a-t-il de codes possibles? +
    2. Combien y-a-t-il de codes se terminant par un chiffre impair? +
    3. Combien y-a-t-il de codes comprenant le chiffre 6? +
    +
Indication
Corrigé
+
Exercice 7 - Les hélicoptères [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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 ?
Indication
Corrigé
+
Exercice 8 - Comité de joueurs [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Fred et Émile font partie d'une équipe de 8 joueurs (6 garçons et 2 filles). On décide de fabriquer un comité de 3 joueurs. +
    +
  1. Combien y-a-t-il de comités possibles? +
  2. Combien y-a-t-il de comités contenant exactement 2 garçons et 1 fille? +
  3. Combien y-a-t-il de comités contenant au moins deux garçons? +
  4. On veut que Fred et Émile soient ensemble dans le comité. Combien y-a-t-il de comités possibles? +
  5. On ne veut pas que Fred et Émile soient ensemble dans le comité. Combien y-a-t-il de comités possibles? +
Indication
Corrigé
+
Exercice 9 - Les trois mousquetaires [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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. +
    +
  1. Combien de possibilités s'offrent à lui? +
  2. Combien de choix a-t-il tels que les deux bottes forment une paire (une droite et une gauche quelconques)? +
  3. Combien de choix a-t-il tels que les deux bottes appartiennent à deux personnes différentes? +
Indication
Corrigé
+
Exercice 10 - Chaussures dans mon armoire [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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. +
    +
  1. Combien y-a-t-il de choix possibles ? +
  2. Combien y-a-t-il de choix où j'obtiens deux chaussures de même couleur ? +
  3. Combien de choix amènent un pied gauche et un pied droit ? +
  4. Combien de choix amènent une chaussure droite et une chaussure gauche de même couleur ? +
  5. Combien de choix amènent à deux chaussures qui ne sont pas de la même paire? +
Indication
Corrigé
+
Exercice 11 - Le tournoi de tennis [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Un tournoi de tennis comporte 2n joueurs. De combien de façons peut-on organiser le premier tour dans le cas où : +
    +
  1. on s’intéresse à la fois aux joueurs qui sont opposés et à l’ordre des matches ; +
  2. on ne s’intéresse qu’à la connaissance des joueurs opposés. +
Indication
Corrigé
+
Exercice 12 - Dénombrement de tirages successifs et simultanés, avec ou sans remise [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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. +
    +
  1. On suppose que les jetons sont tirés successivement en remettant à chaque fois le jeton tiré. +
      +
    1. Donner le nombre de résultats possibles. +
    2. Combien de ces résultats amènent +
        +
      1. exactement 1 jeton noir ? +
      2. au moins 1 jeton noir ? +
      3. au plus un jeton noir ? +
      4. 2 fois plus de jetons noirs que de jetons blancs ? +
      +
    +
  2. Mêmes questions en supposant que les jetons sont tirés successivement sans remise. +
  3. Mêmes questions en supposant que les jetons sont tirés simultanément. +
Indication
Corrigé
+
Exercice 13 - Bridge [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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. +
    +
  1. 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. +
  2. 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,...)? +
  3. Combien y-a-t-il de façons que deux joueurs quelconques aient chacun une seule couleur? +
  4. 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. +
Indication
Corrigé
+
Exercice 14 - Le poker [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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 : +
    +
  1. 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). +
  2. carré : main contenant 4 cartes de la même valeur (4 as par exemple). +
  3. 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). +
  4. quinte : main formée de 5 cartes consécutives et qui ne sont pas toutes de la même couleur. +
  5. 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). +
Indication
Corrigé
+
Exercice 15 - Tirages dans un jeu de cartes [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
On tire simultanément 5 cartes d'un jeu de 32 cartes. Combien de tirages différents peut-on obtenir : +
    +
  1. sans imposer de contraintes sur les cartes. +
  2. contenant 5 carreaux ou 5 piques. +
  3. contenant 2 carreaux et 3 piques. +
  4. contenant au moins un roi. +
  5. contenant au plus un roi. +
  6. contenant exactement 2 rois et exactement 3 piques. +
Indication
Corrigé
+
Exercice 16 - Ranger des livres [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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 : +
    +
  1. si les livres doivent être groupés par matières. +
  2. si on exige seulement que les livres mathématiques soient groupés (les livres des autres disciplines pouvant l'être ou non). +
Indication
Corrigé
+
Exercice 17 - Autour d'une table! [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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?
Indication
Corrigé
+
Exercice 18 - Groupe d'étudiants [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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.
+Indiquer la répartition des étudiants par discipline, ainsi que le nombre total d'étudiants dans le groupe.
Indication
Corrigé
+
Exercice 19 - Dans une entreprise... [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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?
Indication
Corrigé
+
Exercice 20 - Indépendance et contexte [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
    +
  1. Une urne contient 12 boules numérotées de 1 à 12. On en tire une hasard, et on considère les événements +A="tirage d'un nombre pair'', +B="tirage d'un multiple de 3''. +Les événements A et B sont-ils indépendants? +
  2. Reprendre la question avec une urne contenant 13 boules. +
Indication
Corrigé
+
Exercice 21 - Indépendance deux à deux et indépendance mutuelle [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Votre voisine a deux enfants dont vous ignorez le sexe. On considère les trois événement suivants : +
    +
  • A="les deux enfants sont de sexes différents" +
  • B="l'ainé est une fille" +
  • C="le cadet est un garçon". +
+Montrer que A, B et C 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 1/2.
Indication
Corrigé
+
Exercice 22 - Probabilité d'une réunion et indépendance [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Soient A1,…,An n événements d’un espace probabilisé (Ω,P). On les suppose mutuellement indépendants et de probabilités respectives pi=P(Ai). + Donner une expression simple de P(A1∪⋯∪An) en fonction de p1,…,pn.
+Application : on suppose qu'une personne est soumise à n expériences indépendantes les unes des autres et qu'à chaque expérience, elle ait une probabilité p d'avoir un accident. Quelle est la probabilité qu'elle ait au moins un accident?
Indication
Corrigé
+
Exercice 23 - Circuit électrique [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
    +
  1. Soient A, B, C trois événements. Montrer que : +P(A∪B∪C)=P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C). +
  2. On dispose de 3 composants électriques C1, C2 et C3 dont la probabilité de fonctionnement est pi, et de fonctionnement totalement indépendant les uns des autres. Donner la probabilité de fonctionnement du circuit +
      +
    1. si les composants sont disposés en série. +
    2. si les composants sont disposés en parallèle. +
    3. si le circuit est mixte : C1 est disposé en série avec le sous-circuit constitué de C2 et C3 en parallèle. +
    +
Indication
Corrigé
+
Exercice 24 - Tirages successifs dans une urne [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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?
Indication
Corrigé
+
Exercice 25 - Deux ateliers [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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 3% pour l'atelier 1 et 4% pour l'atelier 2. On prélève une pièce au hasard dans l'ensemble de la production d'une journée. +Déterminer +
    +
  1. la probabilité que cette pièce provienne de l'atelier 1; +
  2. la probabilité que cette pièce provienne de l'atelier 1 et est défectueuse; +
  3. la probabilité que cette pièce provienne de l'atelier 1 sachant qu'elle est défectueuse. +
Indication
Corrigé
+
Exercice 26 - A partir de dénombrement [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Une urne contient 8 boules blanches et 2 boules noires, indiscernables au toucher. On tire sans remise et successivement 3 boules de cette urne. +
    +
  1. Quelle est la probabilité qu'au moins une boule noire figure dans le tirage? +
  2. Sachant qu'au moins une boule noire figure dans le tirage, quelle est la probabilité que la première boule tirée soit noire? +
Indication
Corrigé
+
Exercice 27 - [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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 n-ième jour, alors la probabilité +pour qu'elle ne fume pas le jour suivant est 0,3; +mais si elle a fumé le n-ième jour, alors la probabilité +pour qu'elle ne fume pas le jour suivant est 0,9. +Pour n≥0, on note Fn l'événement ``la personne fume le n-ième jour'' et pn la probabilité de Fn. +En particulier on a p0=1. +
    +
  1. Démontrer que pn+1=−0,6pn+0,7. +
  2. La personne va-t-elle s'arrêter de fumer? +
Indication
Corrigé
+
Exercice 28 - Déplacement d'un pion sur un triangle [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
On considère trois points distincts du plan nommés A, B et C. +Nous allons étudier le déplacement aléatoire d’un pion se déplaçant sur ces trois points. A l’étape n=0, on suppose que le pion se trouve sur le point A. +Ensuite, le mouvement aléatoire du pion respecte les deux règles suivantes : +
    +
  • le mouvement du pion de l’étape n à l’étape n+1 ne dépend que de la position du pion à l’étape n; +
  • pour passer de l'étape n à l'étape n+1, 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. +
+Pour tout n∈N, on note An l’évènement "le pion se trouve en A à l’étape n", Bn l’évènement "le pion se trouve en B à l’étape n" +et Cn l’évènement "le pion se trouve en C à l’étape n". On note également, pour tout n∈N, +an=P(An), bn=P(Bn), cn=P(Cn) et Vn=(anbncn). +
    +
  1. Calculer les nombres an, bn et cn pour n=0,1. +
  2. Pour n∈N, exprimer an+1 en fonction de an, bn et cn. Faire de même pour bn+1 et cn+1. +
  3. Donner une matrice M telle que, pour tout n∈N, on a Vn+1=MVn. +
  4. On admet que, pour tout n∈N, on a + Mn=13⋅4n(4n+24n−14n−14n−14n+24n−14n−14n−14n+2). +En déduire une expression de an, bn et cn pour tout n∈N. +
  5. Déterminer les limites respectives des suites (an), (bn) et (cn). Interpréter le résultat. +
Indication
Corrigé
+
Exercice 29 - Sauts de puce [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Une particule se trouve à l'instant 0 au point d'abscisse a (a entier), sur un segment gradué de 0 à N (on suppose donc 0≤a≤N). +A chaque instant, elle fait un bond de +1 avec la probabilité p (0<p<1/2), ou un bond de −1 avec la probabilité q=1−p. +Autrement dit, si xn est l'abscisse de la particule à l'instant n, on a : +xn+1={xn+1avec probabilité pxn−1avec probabilité 1−p. +Le processus se termine lorsque la particule atteint une des extrémités du segment (i.e. s'il existe xn avec xn=0 ou xn=N). +
    +
  1. Écrire une fonction python puce(a,N,p) qui simule cette marche aléatoire et qui retourne l'endroit où la puce sort du segment (0 ou N) et le nombre de pas nécessaires pour que le processus s'arrête. +
  2. On note ua la probabilité pour que la particule partant de a, le processus s'arrête en 0. +
      +
    1. Que vaut u0? uN? +
    2. Montrer que si 0<a<N, alors ua=pua+1+qua−1. +
    3. En déduire l'expression exacte de ua. +
    +
  3. On note va la probabilité pour que la particule partant de a, le processus s'arrête en N. Reprendre les questions précédentes avec va au lieu de ua. +
  4. Calculer ua+va. Qu'en déduisez-vous? +
Indication
Corrigé
+
Exercice 30 - La puce [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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 n elle se trouve sur un sommet donné, elle saute à l'instant n+1 vers l'un des deux sommets voisins avec probabilité 1/2. On note Xn la position de la puce à l'instant n. +
    +
  1. On suppose dans cette question seulement que X0 suit une loi uniforme sur l'ensemble {1,2,3}. Décrire la loi de X1 puis la loi de Xn pour n≥1.
    +On s'intéresse maintenant au cas général. On note Un=(P[Xn=1]P[Xn=2]P[Xn=3]). +
  2. Montrer qu'il existe une matrice A que l'on déterminera telle que pour tout n≥0, Un+1=AUn. +
  3. Montrer qu'il existe une matrice orthogonale P, qu'on ne cherchera pas à calculer, telle que +A=P(1000−1/2000−1/2)Pt . +
  4. En déduire que la suite (Un)n≥0 converge vers un vecteur U∞ vérifiant U∞=AU∞. +
  5. Que vaut U∞ ? +
Indication
Corrigé
+
Exercice 31 - Clés USB [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Le gérant d'un magasin d'informatique a reçu un lot de clés USB. 5% des boites sont abîmées. Le gérant estime que : +
    +
  • 60% des boites abîmées contiennent au moins une clé défectueuse. +
  • 98% des boites non abîmées ne contiennent aucune clé défectueuse. +
+Un client achète une boite du lot. On désigne par A l'événement : ``la boite est abîmée'' et par D l'événement ``la boite achetée contient au moins une clé défectueuse''. +
    +
  1. Donner les probabilités de P(A), P(A¯), P(D|A), P(D|A¯), P(D¯|A) et P(D¯|A¯). En déduire la probabilité de D. +
  2. Le client constate qu'une des clés achetées est défectueuse. Quelle est +la probabilité pour qu'il ait acheté une boite abîmée? +
Indication
Corrigé
+
Exercice 32 - QCM [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Un questionnaire à choix multiples propose m réponses pour chaque question. Soit p 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?
Indication
Corrigé
+
Exercice 33 - Dés pipés [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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 1/2. +
    +
  1. 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? +
  2. Soit n∈N∗. On tire un dé au hasard parmi 100 dés. On lance ce dé n fois et on obtient n fois le chiffre 6. + Quelle est la probabilité pn pour que ce dé soit pipé. Interpréter le résultat. +
Indication
Corrigé
+
Exercice 34 - Pièces défectueuses [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
Une usine fabrique des pièces, avec une proportion de 0,05 de pièces défectueuses. Le contrôle des +fabrications est tel que : +
    +
  • si la pièce est bonne, elle est acceptée avec la probabilité 0,96. +
  • si la pièce est mauvaise, elle est refusée avec la probabilité 0,98. +
+On choisit une pièce au hasard et on la contrôle. Quelle est la probabilité +
    +
  1. qu'il y ait une erreur de contrôle? +
  2. qu'une pièce acceptée soit mauvaise? +
Indication
Corrigé
+
Exercice 35 - La forêt [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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?
Indication
Corrigé
+
Exercice 36 - Menteur! [Signaler une erreur] [Ajouter à ma feuille d'exos]
Enoncé
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?
Indication
Corrigé
+
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setXandrConfig(){const{siteIabCategories:siteIabCategories,pageIabCategories:pageIabCategories,pageTopics:pageTopics,brandSafe:brandSafe}=r89Data.config.contextual,hasContextual="done"===r89.contextual.status,xandrKeys=r89.helpers.removeNulls({iab_content_taxonomy:siteIabCategories.length>0?siteIabCategories:null,page_iab_taxonomy:hasContextual&&pageIabCategories.length>0?pageIabCategories:null,page_topics:hasContextual&&pageTopics.length>0?pageTopics:null,page_bs:hasContextual&&null!==brandSafe?brandSafe:null,btng_aprvd:r89Data.config.cmp.betting&&r89.cmp.checkCustomPurpose(50)?"1":null});r89Data.config.xandrKeys=xandrKeys,Object.keys(xandrKeys).length>0&&r89_pbjs.setConfig({appnexusAuctionKeywords:xandrKeys})}()}))}function initAPS(seller){apstag.init({pubID:"d02f0482-a50f-427c-ac01-9856371f1f6b",adServer:"googletag",schain:{complete:1,ver:"1.0",nodes:[{asi:"refinery89.com",sid:seller.id,hp:1,name:seller.name}]}})}var commonjsGlobal="undefined"!=typeof globalThis?globalThis:"undefined"!=typeof window?window:"undefined"!=typeof global?global:"undefined"!=typeof self?self:{};function getDefaultExportFromCjs(x){return x&&x.__esModule&&Object.prototype.hasOwnProperty.call(x,"default")?x.default:x}function getAugmentedNamespace(n){if(n.__esModule)return n;var f=n.default;if("function"==typeof f){var a=function a2(){return this instanceof a2?Reflect.construct(f,arguments,this.constructor):f.apply(this,arguments)};a.prototype=f.prototype}else a={};return Object.defineProperty(a,"__esModule",{value:!0}),Object.keys(n).forEach((function(k){var d=Object.getOwnPropertyDescriptor(n,k);Object.defineProperty(a,k,d.get?d:{enumerable:!0,get:function(){return n[k]}})})),a}var sha256={exports:{}};var core={exports:{}};const require$$0=getAugmentedNamespace(Object.freeze(Object.defineProperty({__proto__:null,default:{}},Symbol.toStringTag,{value:"Module"})));var hasRequiredCore,CryptoJS;function requireCore(){return 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Latin1.parse(unescape(encodeURIComponent(utf8Str)))}},BufferedBlockAlgorithm=C_lib.BufferedBlockAlgorithm=Base.extend({reset:function(){this._data=new WordArray.init,this._nDataBytes=0},_append:function(data){"string"==typeof data&&(data=Utf8.parse(data)),this._data.concat(data),this._nDataBytes+=data.sigBytes},_process:function(doFlush){var processedWords,data=this._data,dataWords=data.words,dataSigBytes=data.sigBytes,blockSize=this.blockSize,nBlocksReady=dataSigBytes/(4*blockSize),nWordsReady=(nBlocksReady=doFlush?Math2.ceil(nBlocksReady):Math2.max((0|nBlocksReady)-this._minBufferSize,0))*blockSize,nBytesReady=Math2.min(4*nWordsReady,dataSigBytes);if(nWordsReady){for(var offset=0;offset>>7)^(gamma0x<<14|gamma0x>>>18)^gamma0x>>>3,gamma1x=W[i2-2],gamma1=(gamma1x<<15|gamma1x>>>17)^(gamma1x<<13|gamma1x>>>19)^gamma1x>>>10;W[i2]=gamma0+W[i2-7]+gamma1+W[i2-16]}var maj=a&b^a&c^b&c,sigma0=(a<<30|a>>>2)^(a<<19|a>>>13)^(a<<10|a>>>22),t1=h+((e<<26|e>>>6)^(e<<21|e>>>11)^(e<<7|e>>>25))+(e&f^~e&g)+K[i2]+W[i2];h=g,g=f,f=e,e=d+t1|0,d=c,c=b,b=a,a=t1+(sigma0+maj)|0}H2[0]=H2[0]+a|0,H2[1]=H2[1]+b|0,H2[2]=H2[2]+c|0,H2[3]=H2[3]+d|0,H2[4]=H2[4]+e|0,H2[5]=H2[5]+f|0,H2[6]=H2[6]+g|0,H2[7]=H2[7]+h|0},_doFinalize:function(){var data=this._data,dataWords=data.words,nBitsTotal=8*this._nDataBytes,nBitsLeft=8*data.sigBytes;return dataWords[nBitsLeft>>>5]|=128<<24-nBitsLeft%32,dataWords[14+(nBitsLeft+64>>>9<<4)]=Math2.floor(nBitsTotal/4294967296),dataWords[15+(nBitsLeft+64>>>9<<4)]=nBitsTotal,data.sigBytes=4*dataWords.length,this._process(),this._hash},clone:function(){var clone=Hasher.clone.call(this);return clone._hash=this._hash.clone(),clone}});C.SHA256=Hasher._createHelper(SHA2562),C.HmacSHA256=Hasher._createHmacHelper(SHA2562)}(Math),CryptoJS.SHA256));var encHex$1={exports:{}};encHex$1.exports=function(CryptoJS){return CryptoJS.enc.Hex}(requireCore());const encHex=getDefaultExportFromCjs(encHex$1.exports);window.r89=window.r89||{},window.googletag=window.googletag||{},googletag.cmd=googletag.cmd||[],window.r89_pbjs=window.r89_pbjs||{},r89_pbjs.que=r89_pbjs.que||[],r89Data.config.viewport={width:Math.max(document.documentElement.clientWidth||0,window.innerWidth||0),height:Math.max(document.documentElement.clientHeight||0,window.innerHeight||0)},r89Data.config.preview=window.location.search.includes("preview=1")?1:0,Object.assign(window.r89,{logStorage:[],log:function log(str){r89.logStorage.push(str),r89Data.config.preview&&console.log("R89 New | "+str)},helpers:function helpers(){return{filterUndefined:function(el){return void 0!==el},randomInt:function(min,max){return Math.floor(Math.random()*(max-min+1)+min)},uuid:function(){return"10000000-1000-4000-8000-100000000000".replace(/[018]/g,(c=>(c^crypto.getRandomValues(new Uint8Array(1))[0]&15>>c/4).toString(16)))},createStickyCloseButton:function(slotParent,topOrBottom){const 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loadScripts({scripts:scripts}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},main:function(){r89.log("LoadScripts: Main"),"uncalled"==this.status&&(r89.events.trackFunction("loadScriptsMain"),r89.loadScripts.setStatus("loading"),r89.loadScripts.gpt())},done:function(){r89.log("LoadScripts: Done"),r89.events.trackFunction("loadScriptsDone"),setTimeout((()=>{r89.log("Timeout: "+r89Data.config.scriptTimeout.timeout),r89.loadScripts.setStatus("done"),r89.init.main()}),r89Data.config.scriptTimeout.timeout)},gpt:function(){r89.log("LoadScripts: GPT"),r89.events.trackFunction("loadScriptsGPT");const script=document.createElement("script");script.src="https://securepubads.g.doubleclick.net/tag/js/gpt.js",script.type="text/javascript",script.async=!0,document.head.appendChild(script),googletag.cmd.push((function(){r89.events.trackFunction("loadScriptsGPTPushed"),r89.loadScripts.pbjs()}))},pbjs:function(){if(r89.events.trackFunction("loadScriptsPBJS"),r89Data.config.integrations.pbjs){r89.log("LoadScripts: PBJS");const script=document.createElement("script");script.src=scripts.pbjs,script.type="text/javascript",script.async=!0,document.head.appendChild(script),r89_pbjs.que.push((function(){r89.loadScripts.aps()}))}else r89.log("LoadScripts: No PBJS"),r89.loadScripts.aps()},aps:function(){r89.events.trackFunction("loadScriptsAPS"),r89Data.config.integrations.aps?(r89.log("LoadScripts: APS"),function(a9,a,p,s,t,A,g){function q(c,r){a[a9]._Q.push([c,r])}a[a9]||(a[a9]={init:function(){q("i",arguments)},fetchBids:function(){q("f",arguments)},setDisplayBids:function(){},targetingKeys:function(){return[]},_Q:[]},(A=p.createElement(s)).async=!0,A.src="//c.amazon-adsystem.com/aax2/apstag.js",(g=p.getElementsByTagName(s)[0]).parentNode.insertBefore(A,g))}("apstag",window,document,"script"),r89.loadScripts.done()):(r89.log("LoadScripts: No APS"),r89.loadScripts.done())}}}(r89Data),init:function init({seller:seller,cmp:cmp2,exchange_rate:exchange_rate}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},main:function(){if(r89.log("Init: Main"),r89.log("Init: LoadScripts Status - "+r89.loadScripts.getStatus()),r89.log("Init: CMP Status - "+r89.cmp.getStatus()),"done"===r89.loadScripts.getStatus()&&"done"===r89.cmp.getStatus()){r89Data.config.prioritize_custom_ad_units&&r89.customAdunits.insert(),r89.events.trackFunction("initMain"),r89.log("Init: Loading"),r89.init.setStatus("loading");const integrations={gpt:r89Data.config.integrations.gpt?initGPT():r89.log("Init: No GPT"),pbjs:r89Data.config.integrations.pbjs?initPBJS(seller,cmp2,exchange_rate):r89.log("Init: No PBJS"),aps:r89Data.config.integrations.aps?initAPS(seller):r89.log("Init: No APS")};Object.keys(integrations).forEach((key=>{r89.log(`Init: ${key}`)})),r89.init.done()}},done:function(){r89.log("Init: Done"),this.setStatus("done"),r89.adunits.placeAds()}}}(r89Data),demandTier:function demandTier(){return{main:function(){try{var country;fetch("https://d294j4en0095q1.cloudfront.net/demandTiersFloors.json",{mode:"cors",method:"GET"}).then((response=>{if(response.ok)return response;throw new Error("Network response was not ok")})).then((function(data){return 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el&&el.has_amazon&&bidders.push("aps"),r89Data.config.integrations.pbjs&&"bidPBJS"in el&&bidders.push("pbjs");const requestManager={adserverRequestSent:!1};function headerBidderBack(bidder){!0!==requestManager.adserverRequestSent&&("aps"===bidder?requestManager.aps=!0:"pbjs"===bidder&&(requestManager.pbjs=!0),function allBiddersBack(){return bidders.map((function(bidder){return requestManager[bidder]})).filter(Boolean).length===bidders.length}()&&sendAdserverRequest())}function sendAdserverRequest(){!0!==requestManager.adserverRequestSent&&(requestManager.adserverRequestSent=!0,googletag.cmd.push((function(){googletag.pubads().refresh([el.slot])})))}bidders.forEach((function(bidder){requestManager[bidder]=!1})),bidders.includes("aps")&&apstag.fetchBids({slots:[el.bidAPS],timeout:timeout},(function(){googletag.cmd.push((function(){apstag.setDisplayBids(),headerBidderBack("aps")}))})),bidders.includes("pbjs")&&(el.bidPBJS.forEach((slot=>{if(void 0!==slot.adSlotId&&void 0===slot.floors){let floors=r89.helpers.getPrebidFloor(slot.adSlotId);floors&&(slot.floors=floors),delete slot.adSlotId}})),r89_pbjs.que.push((function(){r89_pbjs.requestBids({timeout:timeout,adUnits:el.bidPBJS,bidsBackHandler:function(){googletag.cmd.push((function(){r89_pbjs.setTargetingForGPTAsync([el.id]),headerBidderBack("pbjs")}));try{const amazonVideoPlayer=document.querySelector('[id^="apsVideoDiv"]'),childAmazonIframe=amazonVideoPlayer.getElementsByTagName("iframe")[0];childAmazonIframe.id.includes("apstag")&&(amazonVideoPlayer.style.maxWidth="100%",childAmazonIframe.style.maxWidth="100%")}catch(err){r89.log("AMAZON_DID_NOT_WON")}}})}))),0===bidders.length?sendAdserverRequest():window.setTimeout((function(){sendAdserverRequest()}),1.5*timeout)},infiniteScroll:function(){r89.log("Bids: Request Infinite Scroll");const timeout=r89Data.config.timeouts.default,refreshItems=r89.infiniteScroll.placedItems.map((el=>el.slot)),slotIds=r89.infiniteScroll.placedItems.map((el=>el.id)),slotsAPS=r89.infiniteScroll.placedItems.map((el=>el.bidAPS)).filter(r89.helpers.filterUndefined),slotsPBJS=r89.infiniteScroll.placedItems.map((el=>el.bidPBJS)).filter(r89.helpers.filterUndefined),bidders=[];r89Data.config.integrations.aps&&slotsAPS.length>0&&bidders.push("aps"),r89Data.config.integrations.pbjs&&slotsPBJS.length>0&&bidders.push("pbjs");const requestManager={adserverRequestSent:!1};function headerBidderBack(bidder){!0!==requestManager.adserverRequestSent&&("aps"===bidder?requestManager.aps=!0:"pbjs"===bidder&&(requestManager.pbjs=!0),function allBiddersBack(){return bidders.map((function(bidder){return requestManager[bidder]})).filter(Boolean).length===bidders.length}()&&sendAdserverRequest())}function sendAdserverRequest(){!0!==requestManager.adserverRequestSent&&(requestManager.adserverRequestSent=!0,googletag.cmd.push((function(){googletag.pubads().refresh(refreshItems)})))}bidders.forEach((function(bidder){requestManager[bidder]=!1})),bidders.includes("aps")&&slotsAPS.length>0&&apstag.fetchBids({slots:slotsAPS,timeout:timeout},(function(){googletag.cmd.push((function(){apstag.setDisplayBids(),headerBidderBack("aps")}))})),bidders.includes("pbjs")&&slotsPBJS.length>0&&r89_pbjs.que.push((function(){r89_pbjs.requestBids({timeout:timeout,adUnits:slotsPBJS,bidsBackHandler:function(){googletag.cmd.push((function(){r89_pbjs.setTargetingForGPTAsync(slotIds),headerBidderBack("pbjs")}))}})})),0===bidders.length?sendAdserverRequest():window.setTimeout((function(){sendAdserverRequest()}),1.5*timeout)}}}(),load:function load({preload:preload,cmp:cmpConfig,preconnect:preconnect}){return{scrollLoaded:!1,scroll:function(){r89.load.scrollLoaded||(r89.load.scrollLoaded=!0,window.removeEventListener("scroll",r89.load.scroll),r89.load.contentLoaded())},contentLoadedState:!1,contentLoaded:function(){r89.load.preload(),r89.log("Load: ContentLoaded - "+document.readyState),"complete"===document.readyState||"interactive"===document.readyState?(r89.log("Load: ContentLoaded - Complete"),r89.load.contentLoadedState=!0,r89.load.main()):(r89.log("Load: ContentLoaded - Event Listener"),document.addEventListener("readystatechange",(event=>{r89.log("Load: Onreadystatechange - "+document.readyState),"interactive"!==document.readyState&&"complete"!==document.readyState||r89.load.contentLoadedState||(r89.load.contentLoadedState=!0,r89.load.main())})))},main:function(){if(r89.log("Load: Main"),r89.events.trackFunction("loadMain"),r89.load.setEnvironment(),r89.load.setCurrentUrl(),!r89.load.checkUrlFilters())return r89.log("Load: Canceled with URL filter"),!1;r89Data.config.is_mobile&&r89Data.config.scriptTimeout.active&&(r89Data.config.scriptTimeout.timeout=r89Data.config.scriptTimeout.options[r89.helpers.randomInt(0,r89Data.config.scriptTimeout.options.length-1)]),r89.adunits.main()},setEnvironment:function(){r89Data.config.viewport.width=Math.max(document.documentElement.clientWidth||0,window.innerWidth||0),r89Data.config.viewport.height=Math.max(document.documentElement.clientHeight||0,window.innerHeight||0),r89Data.config.viewport.width>=r89Data.config.desktop_width?(r89Data.config.is_mobile=0,r89Data.config.is_desktop=1):(r89Data.config.is_mobile=1,r89Data.config.is_desktop=0);let ranges=[[0,320],[320,360],[360,375],[375,414],[414,768],[768,992],[992,1024],[1024,1152],[1152,1280],[1280,1440],[1440,1680],[1680,1920]];r89Data.config.viewport.width>1920?r89Data.config.screenWidthKey="1920-plus":ranges.every((v=>!(r89Data.config.viewport.width2&&-1==r89Data.config.currentUrl.titleTags.indexOf(kv_tags[i2])&&r89Data.config.currentUrl.titleTags.push(kv_tags[i2])},checkUrlFilters:function(){if(void 0!==r89.pageConfig&&void 0!==r89.pageConfig.noAds&&1==r89.pageConfig.noAds)return!1;if(-1!==r89Data.config.urlFilters.exact.indexOf(r89Data.config.currentUrl.path))return!1;for(var i2=0;i2el.wrapper)).forEach((function(wrapper2){wrapper2.remove()}))}r89Data.config.outstream&&r89.outstream.adunit.wrapper&&r89.outstream.adunit.wrapper.remove(),this.resetKeys()},resetKeys:function(){r89.adunits.batches=[],r89.adunits.currentBatch=0,r89.adunits.placedItems={},clearInterval(r89.bids.trackViewableTimeInterval),clearInterval(r89.bids.trackRefreshInterval),this.resetSession()},resetSession:function(){r89.session.newPageview(),this.done()},done:function(){r89.load.main()},customTargetingKeys:function(){googletag.cmd.push((function(){if(void 0!==r89.pageConfig&&"object"==typeof r89.pageConfig.targetingKeys)for(var targetingKey in r89.pageConfig.targetingKeys)googletag.pubads().setTargeting(targetingKey,r89.pageConfig.targetingKeys[targetingKey])}))}}}(),adunits:function adunits({adunits:adunitsConfig,batches:batches,inimage_adunits:inImageAdsConfig,inimage_bidders:inImageBiddersConfig,video_overlay:videoOverlayConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},items:adunitsConfig,performance:{},batchNames:batches,batches:[],currentBatch:0,allBatchesPlaced:0,placedItems:{},lazyLoadDistances:[0,250,500,500,500,750,1e3,1250,1500,1750,2e3],main:function(){r89.log("Adnits: Main"),r89.events.trackFunction("adunitsMain"),r89Data.config.stickySidebar&&r89.stickySidebar.main(),this.prepareBatches(),this.placeWrappers()},pickLazyLoadDistance:function(){return this.lazyLoadDistances[Math.floor(Math.random()*this.lazyLoadDistances.length)]},prepareBatches:function(){this.batchNames.forEach((function(item){"video_overlay"!==item&&(this.placedItems[item]=[],this.batches.push(item))}),this)},placeWrappers:function(){r89.log("Adunits: PlaceWrappers"),this.setStatus("placeWrappers"),this.items.forEach(this.placeWrapper,this),this.attachObserversOnWrappers(),this.placeWrappersDone()},attachObserversOnWrappers:function(){const observer=new IntersectionObserver((function(entries,observer2){entries.forEach((entry=>{let targetId=entry.target.id.replace("-wrapper",""),el=r89.adunits.searchBatches(targetId);el&&(el.isIntersectingAt50=entry.isIntersecting)}))}),{root:null,threshold:.5});document.querySelectorAll('[id^=r89-][id$="-wrapper"]').forEach((element=>{try{"undefined"!==element.id&&observer.observe(document.getElementById(element.id))}catch(err){r89.log("FAILED",element)}}))},placeWrapper:function(el,index){if(!r89.helpers.checkScreenSize(r89Data,el))return!1;r89.log("Adunits: Attempt to place "+el.div_id);let selector_divs=[];if(el.selector_all)selector_divs=document.querySelectorAll(el.selector);else{let selector_div=null;const selector_split=el.selector.split(",");for(let i2=0;i20?selector_divs.forEach((function(selector_div){const id=r89Data.config.prefix+el.div_id+"-"+count;++count;const adunit=document.createElement("div");adunit.id=id;const wrapper2=document.createElement("div");if(wrapper2.id=id+"-wrapper",el.wrapper_style_parsed&&Object.keys(el.wrapper_style_parsed).forEach((key=>{wrapper2.style[key]=el.wrapper_style_parsed[key]})),el.show_label){const ad_unit_label=document.createElement("div");ad_unit_label.className="r89-ad-label",ad_unit_label.innerHTML=r89Data.config.ad_label_text,r89Data.config.labelStyle&&Object.keys(r89Data.config.labelStyle).forEach((key=>{ad_unit_label.style[key]=r89Data.config.labelStyle[key]})),ad_unit_label.style.display="none";const ad_unit_label_wrapper=document.createElement("div");el.wrapper_sticky&&(ad_unit_label_wrapper.style.position="sticky",ad_unit_label_wrapper.style.top=r89Data.config.adunit_wrapper_sticky_top),ad_unit_label_wrapper.appendChild(ad_unit_label),wrapper2.appendChild(ad_unit_label_wrapper)}if(el.wrapper_sticky){const stickyWrapper=document.createElement("div");stickyWrapper.style.position="sticky",stickyWrapper.style.top=r89Data.config.adunit_wrapper_sticky_top,wrapper2.appendChild(stickyWrapper),stickyWrapper.appendChild(adunit)}else wrapper2.appendChild(adunit);if(r89Data.config.positionNames.includes(el.position.name)){if("insertBefore"===el.position.name?selector_div.parentNode.insertBefore(wrapper2,selector_div):"insertAfter"===el.position.name?null===selector_div.nextElementSibling?selector_div.parentNode.appendChild(wrapper2):selector_div.parentNode.insertBefore(wrapper2,selector_div.nextElementSibling):"appendChild"===el.position.name&&selector_div.appendChild(wrapper2),el.id=id,el.wrapper=wrapper2,el.rendered=!1,el.viewable=!1,el.viewableTime=0,el.visible=!1,el.doRefresh=!1,el.refreshed=!1,el.isEmpty=!0,el.isIntersectingAt50=!1,el.failedCount=0,el.has_amazon&&(el.bidAPS={slotID:id,sizes:el.aps_sizes,slotName:el.slot_name}),el.bidder&&el.bidder in r89Data.bidders){const mediaTypes={};el.pbjs_sizes.length>0&&(mediaTypes.banner={sizes:el.pbjs_sizes}),el.has_native&&(mediaTypes.native={type:"image"}),el.bidPBJS={code:id,mediaTypes:mediaTypes,bids:r89Data.bidders[el.bidder],adSlotId:el.ad_slot_id}}el.lazy_load_ab_testing&&(el.lazy_load_distance=this.pickLazyLoadDistance()),this.placedItems[el.batch].push({...el})}}),this):r89.log("Adunits: Selector not found "+el.div_id)},placeWrappersDone:function(){this.setStatus("placeWrappersDone"),"uncalled"!==r89.reset.getStatus()?(r89.log("Adunits: PlaceWrappersDone - Reset"),r89.adunits.placeAds()):(r89.log("Adunits: PlaceWrappersDone"),r89.cmp.load(),0===Object.values(r89.adunits.placedItems).flat().length&&r89.events.push("pageview",{type:"noWrappers"}))},placeAds:function(){r89.log("Adunits: PlaceAds"),this.setStatus("placeAds"),r89.events.trackFunction("adunitsPlaceAds"),this.batches=this.batches.filter((function(batch){return"outstream"===batch&&r89Data.config.outstream?1:this.placedItems[batch].length}),this),1===inImageAdsConfig.active&&inImageBiddersConfig["InImage-R89"].length>0&&r89.inImageR89.main(),videoOverlayConfig.active&&r89.video_overlay.main(),this.placeBatch()},placeBatch:function(){if(r89.log("Adunits: PlaceBatch"),this.setStatus("placeBatch"),this.batches.length>this.currentBatch){const batch=this.batches[this.currentBatch];r89.log("Adunits: PlaceBatch "+batch),this.placedItems[batch].forEach(this.placeAd,this),"lazy"===batch?this.lazyLoadObservers(batch):"outstream"===batch?r89.outstream.main():r89.bids.request(batch)}else this.placeBatchesDone()},placeBatchesDone:function(){r89.log("Adunits: PlaceBatchesDone"),this.setStatus("placeBatchesDone"),this.allBatchesPlaced?r89.log("Adunits: PlaceBatchesDone command ran twice. Stopping further script."):(this.allBatchesPlaced=1,r89Data.config.refresh.active&&(r89.bids.trackViewableTime(),r89.bids.trackRefresh()),r89.inImage.main())},placeAd:function(el){googletag.cmd.push((function(){r89.log("Adunits: PlaceAd "+el.id);const slot=googletag.defineSlot(el.slot_name,el.gpt_sizes,el.id);slot.setTargeting("ad_slot",el.ad_slot.name);let floor=r89.helpers.getGAMFloor(el.ad_slot_id);floor&&slot.setTargeting("flr",floor),"object"==typeof r89.adunits.performance[el.gam_ad_unit]&&(slot.setTargeting("au_vb",r89.adunits.performance[el.gam_ad_unit].au_vb),slot.setTargeting("au_cb",r89.adunits.performance[el.gam_ad_unit].au_cb)),el.lazy_load&&slot.setTargeting("lazy_load_dist",el.lazy_load_distance.toString()),slot.addService(googletag.pubads()),googletag.display(el.id),el.slot=slot}))},updateCurrentBatch:function(){++this.currentBatch},searchBatches:function(id){for(let key in r89.adunits.placedItems){let items=r89.adunits.placedItems[key].filter((function(item){return item.id===id}));if(1===items.length)return items[0]}return!1},lazyLoadObservers:function(batch){r89.log("Adunits: LazyLoadObservers"),this.placedItems[batch].forEach(this.lazyLoadObserver,this),this.updateCurrentBatch(),this.placeBatch()},lazyLoadObserver:function(el){el.lazy_load_ab_testing&&r89.events.trackEvent("lazyLoadDistance",el.gam_ad_unit+";"+el.lazy_load_distance);const options={rootMargin:"0px 0px "+el.lazy_load_distance+"px 0px",threshold:0};new IntersectionObserver((function(entries,observer2){entries.forEach((entry=>{entry.isIntersecting&&(r89.log("Adunit: Observed "+el.id),"requestIdleCallback"in window?requestIdleCallback((function(){r89.bids.lazy(el)}),{timeout:250}):r89.bids.lazy(el),observer2.unobserve(entry.target))}))}),options).observe(el.wrapper)}}}(r89Data),initialize:function initialize(){r89.log("Init: PublisherCall Initialize"),"done"===r89.init.getStatus()?r89.reset.main():r89.load.contentLoaded()},stickySidebar:function stickySidebar({sticky_sidebar:stickySidebarConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},sidebar:stickySidebarConfig,main:function(){if(this.sidebar.active&&r89Data.config.viewport.width>=this.sidebar.screen_width&&(wrapper=document.body.querySelector(this.sidebar.selector),null!==wrapper)){let wrapper_height=wrapper.clientHeight-this.sidebar.decrease_height;this.sidebar.minimum_elements>1&&wrapper_height0){let widgets_inserted=!1;if(widgets.forEach((function(widget){let selector_div=null;const selector_split=widget.selector.split(",");for(let i2=0;i2{entry.isIntersecting&&(r89.log("Outbrain: Lazy load insert."),"requestIdleCallback"in window?requestIdleCallback((function(){document.head.appendChild(scriptElm)}),{timeout:250}):document.head.appendChild(scriptElm),observer2.unobserve(entry.target))}))}),options).observe(r89.outbrain.wrapper)}else document.head.appendChild(scriptElm)}}}this.done()},done:function(){r89.customAdunits.main()}}}(r89Data),outstream:function outstream({outstream:outstreamConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},adunit:outstreamConfig.adunit,deal_id_exception:outstreamConfig.deal_id_exception,line_item_exception:outstreamConfig.line_item_exception,ad_slot_id:outstreamConfig.ad_slot_id,main:function(){r89.log("Outstream: Main"),r89Data.config.outstream?this.adunit.is_desktop&&r89Data.config.is_desktop?(this.adunit.bidder=this.adunit.desktop_bidder,this.adunit.sizes=this.adunit.desktop_sizes,this.adunit.gpt_sizes=this.adunit.desktop_gpt_sizes,this.adunit.aps_sizes=this.adunit.desktop_aps_sizes,this.adunit.pbjs_sizes=this.adunit.desktop_pbjs_sizes,this.placeWrapper(this.adunit)):this.adunit.is_mobile&&r89Data.config.is_mobile?(this.adunit.selector=this.adunit.selector_mobile,this.adunit.bidder=this.adunit.mobile_bidder,this.adunit.sizes=this.adunit.mobile_sizes,this.adunit.gpt_sizes=this.adunit.mobile_gpt_sizes,this.adunit.aps_sizes=this.adunit.mobile_aps_sizes,this.adunit.pbjs_sizes=this.adunit.mobile_pbjs_sizes,this.placeWrapper(this.adunit)):this.done():this.done()},placeWrapper:function(el){let selector_div=null,selector_split=el.selector.split(",");for(let i2=0;i20&&el.bidPBJS.push({code:id,mediaTypes:{banner:{sizes:el.pbjs_sizes}},bids:banner_bidders,adSlotId:r89.outstream.ad_slot_id}),el.bidPBJS.push({code:id,mediaTypes:{video:{context:"outstream",playerSize:[640,360],mimes:["video/mp4"],protocols:[1,2,3,4,5,6,7,8],playbackmethod:[2],skip:0,minduration:3,maxduration:120,linearity:1,api:[1,2],plcmt:4,renderer:{url:"https://tags.refinery89.com/video/js/renderV2.js",render:function(bidz){bidz.renderer.push((()=>{bidz.adLabel=1==r89.outstream.adunit.show_label?r89Data.config.ad_label_text:"",bidz.show_close_button=r89.outstream.adunit.show_close_button,bidz.scroll_position_mobile=r89.outstream.adunit.mobile_scrolling_position,bidz.scroll_position_desktop=r89.outstream.adunit.desktop_scrolling_position,bidz.is_scrollable=r89.outstream.adunit.corner_scroll,bidz.is_refresh=!1,bidz.deal_id_exceptions=outstreamConfig.deal_id_exception,bidz.line_item_exceptions=outstreamConfig.line_item_exception,r89VideoJS.renderAds(bidz)}))}}}},bids:video_bidders,adSlotId:r89.outstream.ad_slot_id})}googletag.cmd.push((function(){var slot=googletag.defineSlot(el.slot_name,el.gpt_sizes,el.id);slot.setTargeting("ad_slot","Outstream-Video");let floor=r89.helpers.getGAMFloor(r89.outstream.ad_slot_id);floor&&slot.setTargeting("flr",floor),slot.addService(googletag.pubads()),googletag.display(el.id),el.slot=slot})),r89.bids.outstream(el)}}else this.done()},done:function(){r89.adunits.updateCurrentBatch(),r89.adunits.placeBatch()}}}(r89Data),inImage:function inImage(){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},checkDevice:function(){return 1==r89Data.config.in_image.is_mobile&&1==r89Data.config.is_mobile||1==r89Data.config.in_image.is_desktop&&1==r89Data.config.is_desktop},checkSelector:function(){if(r89Data.config.in_image.selector){return null!==window.document.querySelector(r89Data.config.in_image.selector)}return!0},main:function(){if(r89.log("InImage: Main"),"uncalled"==r89.reset.getStatus()&&r89Data.config.in_image.active&&this.checkDevice()&&this.checkSelector()){r89.log("InImage: Insert");const script=document.createElement("script");script.src=r89Data.config.in_image.script,script.type="text/javascript",script.async=!0,window.document.head.appendChild(script),this.setStatus("loaded")}this.done()},done:function(){r89.log("InImage: Done"),this.setStatus("done"),r89.elasticNative.main()}}}(),elasticNative:function elasticNative({elastic_native:elasticNativeConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},adunit:elasticNativeConfig,viewTime:0,inView:!1,measureCalled:!1,intervalTime:2500,measuredTime:0,label:{text:"▼ Gesponsord",style:{color:"#999999",textAlign:"center",fontSize:"12px",lineHeight:"16px",margin:"0px 0px 15px 0px",padding:"4px 0 6px 5px",fontFamily:"Arial",borderBottom:"1px solid #ccc"}},main:function(){if(this.setStatus="called",this.adunit){r89.log("Elastic Native: Insert");let selector_div=null;const selector_split=this.adunit.selector.split(",");for(let i2=0;i2{entry.isIntersecting?r89.elasticNative.inView=!0:r89.elasticNative.inView=!1}))}),{rootMargin:"0px 0px 0px 0px",threshold:0}).observe(r89.elasticNative.adunit.wrapper);var inViewInterval=setInterval((function(){r89.elasticNative.inView&&(r89.elasticNative.measuredTime+=r89.elasticNative.intervalTime),!1===r89.elasticNative.measureCalled&&r89.elasticNative.measuredTime>=r89.elasticNative.viewTime&&(r89.elasticNative.callMeasure(),clearInterval(inViewInterval))}),r89.elasticNative.intervalTime)},callMeasure:function(){r89.log("Elastic: Refresh Tracking Slot"),googletag.cmd.push((function(){googletag.pubads().refresh([r89.elasticNative.gamMeasureSlot])})),r89.elasticNative.measureCalled=!0}}}(r89Data),customAdunits:function customAdunits({custom_adunits:customAdunitsConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},adunits:customAdunitsConfig,main:function(){r89.log("CustomAdunits: Main"),r89.events.trackFunction("customAdunitsCalled"),"uncalled"==r89.reset.getStatus()&&"uncalled"==this.getStatus()&&this.adunits.length>0&&r89.customAdunits.insert(),this.done()},insert:function(){if(r89.customAdunits.adunits.length>0){const adunits2=r89.customAdunits.adunits.filter((function(adunit){return!(!adunit.is_desktop||!r89Data.config.is_desktop)||!(!adunit.is_mobile||!r89Data.config.is_mobile)}));if(adunits2.length>0){adunits2.map((el=>el.code)).forEach((function(code){const script=document.createElement("script");script.innerHTML=code,document.head.appendChild(script)})),r89.events.trackFunction("customAdunitsLoaded")}}r89.customAdunits.setStatus("done")},done:function(){r89.log("CustomAdunits: Done"),this.setStatus("done"),r89.fallbackAdunits.main()}}}(r89Data),fallbackAdunits:function fallbackAdunits({fallbacks:fallbackAdUnitsConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},items:fallbackAdUnitsConfig,insertCount:0,main:function(){if(r89.log("Fallback: Main"),this.items.length>0){const configVars=[["minScreenWidth","min_screen_width"],["maxScreenWidth","max_screen_width"],["minScreenHeight","min_screen_height"],["maxScreenHeight","max_screen_height"],["lazyLoad","lazy_load_distance"],["sizes","sizes"]];r89.pushAd=function(divId,adunit,config){r89.log("Fallback: Insert "+adunit+" in #"+divId);const source=r89.fallbackAdunits.items.find((element=>element.div_id==adunit));if(!source)return void r89.log("Fallback: Ad unit not found.");r89.log("Fallback: Ad unit found.");const el=Object.assign({},source);el.selector=divId.replace("#",""),"object"==typeof config&&configVars.forEach((configVar=>{configVar[0]in config&&(el[configVar[1]]=config[configVar[0]],"lazy_load_distance"==configVar[1]&&config[configVar[0]]&&(el.lazy_load=1),"sizes"==configVar[1]&&(el.gpt_sizes=el.sizes,el.aps_sizes=el.sizes,el.pbjs_sizes=el.sizes))})),r89.fallbackAdunits.placeWrapper(el)},"object"==typeof r89.callAds&&r89.callAds.forEach((item=>{r89.pushAd(item[0],item[1],item[2])}))}this.done()},placeWrapper:function(el){let place=!1;if(null===el.min_screen_width&&null!==el.max_screen_width&&el.max_screen_width>=r89Data.config.viewport.width&&(place=!0),null===el.max_screen_width&&null!==el.min_screen_width&&el.min_screen_width<=r89Data.config.viewport.width&&(place=!0),null!==el.min_screen_width&&null!==el.max_screen_width&&el.min_screen_width<=r89Data.config.viewport.width&&el.max_screen_width>=r89Data.config.viewport.width&&(place=!0),!place)return r89.log("Fallback: Screen resolution requirements not met."),!1;r89.log("Fallback: Attempt to place "+el.div_id);const selector_div=document.getElementById(el.selector);if(!selector_div)return r89.log("Fallback: Selector div not found."),!1;r89.fallbackAdunits.insertCount++;const id=r89Data.config.prefix+el.div_id+"-"+r89.fallbackAdunits.insertCount,adunit=document.createElement("div");adunit.id=id;const wrapper2=document.createElement("div");if(wrapper2.id=id+"-wrapper","wrapper_style_parsed"in el&&Object.keys(el.wrapper_style_parsed).forEach((key=>{wrapper2.style[key]=el.wrapper_style_parsed[key]})),wrapper2.appendChild(adunit),selector_div.appendChild(wrapper2),el.id=id,el.wrapper=wrapper2,el.rendered=!1,el.viewable=!1,el.viewableTime=0,el.visible=!1,el.doRefresh=!1,el.refreshed=!1,el.isEmpty=!0,el.failedCount=0,el.has_amazon&&(el.bidAPS={slotID:id,sizes:el.aps_sizes,slotName:el.slot_name}),el.bidder&&el.bidder in r89Data.bidders){const mediaTypes={};el.pbjs_sizes.length>0&&(mediaTypes.banner={sizes:el.pbjs_sizes}),el.has_native&&(mediaTypes.native={type:"image"}),el.bidPBJS={code:id,mediaTypes:mediaTypes,bids:r89Data.bidders[el.bidder]}}googletag.cmd.push((function(){r89.log("Fallback: Place ad "+el.id);const slot=googletag.defineSlot(el.slot_name,el.gpt_sizes,el.id);slot.setTargeting("ad_slot",el.ad_slot.name),slot.addService(googletag.pubads()),googletag.display(el.id),el.slot=slot,r89.adunits.placedItems.fallback.push({...el}),el.lazy_load?r89.fallbackAdunits.lazyLoad(el):r89.fallbackAdunits.request(el)}))},request:function(el){"requestIdleCallback"in window?requestIdleCallback((function(){r89.bids.fallback(el)}),{timeout:250}):r89.bids.fallback(el)},lazyLoad:function(el){const options={rootMargin:"0px 0px "+el.lazy_load_distance+"px 0px",threshold:0};new IntersectionObserver((function(entries,observer2){entries.forEach((entry=>{entry.isIntersecting&&(r89.log("Fallback: Observed "+el.id),r89.fallbackAdunits.request(el),observer2.unobserve(entry.target))}))}),options).observe(el.wrapper)},done:function(){r89.log("Fallback: Done"),this.setStatus("done")}}}(r89Data),infiniteScroll:function infiniteScroll({adunits_infinite_scroll:infiniteScrollConfig,inimage_adunits:inImageConfig,inimage_bidders:inImageBiddersConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},selector:null,items:infiniteScrollConfig,placedItems:[],count:0,main:function(){r89.log("Infinite Scroll: Main"),this.load()},load:function(selector){if(r89.log("Infinite Scroll: Load"),++this.count,void 0===selector&&r89Data.config.infinite_scroll)this.selector=document.querySelector(r89Data.config.infinite_scroll.selector);else if("string"==typeof selector)this.selector=document.querySelector(selector);else{if("object"!=typeof selector)return void r89.log("Infinite Scroll: Selector not defined");this.selector=selector}null!=this.selector?(this.placedItems=[],this.items.forEach(this.placeWrapper,this),this.placedItems.forEach(this.placeAd,this),1===inImageConfig.active&&inImageBiddersConfig["InImage-R89"].length>0&&r89.inImageR89.main(!0),r89.bids.infiniteScroll()):r89.log("Infinite Scroll: Selector not found")},placeWrapper:function(el,index){if(!r89.helpers.checkScreenSize(r89Data,el))return!1;r89.log("Adunits: Attempt to place "+el.div_id);let selector_divs=[];if(el.selector_all)selector_divs=this.selector.querySelectorAll(el.selector);else{let selector_div=null;const selector_split=el.selector.split(",");for(let i2=0;i20?selector_divs.forEach((function(selector_div){const id=r89Data.config.prefix+el.div_id+"-infinite-"+this.count+"-"+count;++count;const adunit=document.createElement("div");adunit.id=id;const wrapper2=document.createElement("div");if(wrapper2.id=id+"-wrapper",el.wrapper_style_parsed&&Object.keys(el.wrapper_style_parsed).forEach((key=>{wrapper2.style[key]=el.wrapper_style_parsed[key]})),el.show_label){const ad_unit_label=document.createElement("div");ad_unit_label.className="r89-ad-label",ad_unit_label.innerHTML=r89Data.config.ad_label_text,r89Data.config.labelStyle&&Object.keys(r89Data.config.labelStyle).forEach((key=>{ad_unit_label.style[key]=r89Data.config.labelStyle[key]}));const ad_unit_label_wrapper=document.createElement("div");el.wrapper_sticky&&(ad_unit_label_wrapper.style.position="sticky",ad_unit_label_wrapper.style.top=r89Data.config.adunit_wrapper_sticky_top),ad_unit_label_wrapper.appendChild(ad_unit_label),wrapper2.appendChild(ad_unit_label_wrapper)}if(el.wrapper_sticky){const stickyWrapper=document.createElement("div");stickyWrapper.style.position="sticky",stickyWrapper.style.top=r89Data.config.adunit_wrapper_sticky_top,wrapper2.appendChild(stickyWrapper),stickyWrapper.appendChild(adunit)}else wrapper2.appendChild(adunit);if(r89Data.config.positionNames.includes(el.position.name)){if("insertBefore"==el.position.name?selector_div.parentNode.insertBefore(wrapper2,selector_div):"insertAfter"==el.position.name?null===selector_div.nextElementSibling?selector_div.parentNode.appendChild(wrapper2):selector_div.parentNode.insertBefore(wrapper2,selector_div.nextElementSibling):"appendChild"==el.position.name&&selector_div.appendChild(wrapper2),el.id=id,el.wrapper=wrapper2,el.rendered=!1,el.viewable=!1,el.viewableTime=0,el.visible=!1,el.doRefresh=!1,el.isIntersectingAt50=!1,el.failedCount=0,el.refreshed=!1,el.isEmpty=!0,el.has_amazon&&(el.bidAPS={slotID:id,sizes:el.aps_sizes,slotName:el.slot_name}),el.bidder&&el.bidder in r89Data.bidders){const mediaTypes={};el.pbjs_sizes.length>0&&(mediaTypes.banner={sizes:el.pbjs_sizes}),el.has_native&&(mediaTypes.native={type:"image"}),el.bidPBJS={code:id,mediaTypes:mediaTypes,bids:r89Data.bidders[el.bidder]}}this.placedItems.push({...el})}}),this):r89.log("Infinite Scroll: Selector not found "+el.div_id)},placeAd:function(el){googletag.cmd.push((function(){r89.log("Adunits: PlaceAd "+el.id);const slot=googletag.defineSlot(el.slot_name,el.gpt_sizes,el.id);slot.setTargeting("ad_slot",el.ad_slot.name),"object"==typeof r89.adunits.performance[el.gam_ad_unit]&&(slot.setTargeting("au_vb",r89.adunits.performance[el.gam_ad_unit].au_vb),slot.setTargeting("au_cb",r89.adunits.performance[el.gam_ad_unit].au_cb)),slot.addService(googletag.pubads()),googletag.display(el.id),el.slot=slot}))}}}(r89Data),inImageR89:function inImageR89({inimage_adunits:inImageAdsConfig,inimage_bidders:inImageBiddersConfig,seller:seller,gam_ad_unit:gamAdunit,gam_network_id:gamNetworkId}){return{main:function(infiniteScroll2=!1){try{if(r89.inimageR89={},!1===infiniteScroll2){r89.inImageR89.ads=[];let css_scripts=["https://tags.refinery89.com/static/css/inimage.css"];for(let i2=0;i20){let imageOrientation2=function(width,height){let category="horizontal";return category=width/height>1?["horizontal",horizontal_image_class]:["vertical",vertical_image_class],category};var adImages=[],inimageDivs=[];r89.inimageR89.FAILSAFE_TIMEOUT=2e3,r89.inimageR89.inimageAdunits=[],r89.inimageR89.targetingDivs="",r89.inimageR89.inimageAdunits=[],infiniteScroll2?r89.inimageR89.counter++:r89.inimageR89.counter=images.length;var horizontal_image_class=inImageAdsConfig.ad_units[0].horizontal_class_name,vertical_image_class=inImageAdsConfig.ad_units[0].vertical_class_name;vertical_image_class||(vertical_image_class="middle"),horizontal_image_class||(horizontal_image_class="bottom-right");for(let i2=0;i2=468?horizontalSizes:verticalSizes;let adSizes=[];for(let i2=0;i2\n\t\t\t\t\t\t \n\t\t\t\t\t\t\t \n\t\t\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t ';return inImageAdsConfig.ad_units[0].show_label||(htmlLabel='\n\t\t\t\t\t\t
\n\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t \n\t\t\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t
'),htmlLabel},attachCloseButtonListeners:function(){document.querySelectorAll("[id^=remove-inimage-close-overlay-]").forEach((function(el){el.addEventListener("click",(function(){let identifier=el.id.slice(-1);document.getElementById("overlay-inimage-ad-"+identifier).remove(),el.closest(".r89-ad-label").remove()}))}))},destroyAdSlots:function(inimageSlot){googletag.destroySlots(inimageSlot)},attachLazyLoadObserver:function(el,infiniteScroll2){let options={};options=infiniteScroll2?{rootMargin:"0px 0px 20px 0px",threshold:0}:{rootMargin:"0px 0px 100px 0px",threshold:0};new IntersectionObserver((function(entries,observer2){entries.forEach((entry=>{if(entry.isIntersecting){const img=entry.target.getElementsByTagName("img");if(img.length){const min_img_height=inImageAdsConfig.ad_units[0].min_image_height,min_img_width=inImageAdsConfig.ad_units[0].min_image_width,imageSelected=img[0],identifier=entry.target.id.split("-")[2],imageHeight=imageSelected.offsetHeight,imageWidth=imageSelected.offsetWidth;if(imageWidth>min_img_width||imageHeight>min_img_height){let adSizes=r89.inImageR89.generateAdSizes(imageWidth,imageHeight);if(adSizes.length){if(0==inImageAdsConfig.ad_units[0].max_ads||r89.inImageR89.adsPlaced{try{let number=entry.target.id.split("-")[2];entry.isIntersecting?r89.inImageR89.ads["overlay-inimage-ad-"+number].intersecting=!0:r89.inImageR89.ads["overlay-inimage-ad-"+number].intersecting=!1,r89.log("PREPARED_SLOTS",r89.inImageR89.ads)}catch(err){r89.log(err)}}))}),{rootMargin:"0px",threshold:.5}).observe(document.querySelector("#r89-inimage-"+i2))},refreshInImageR89:function(){setInterval((function(){let keys=Object.keys(r89.inImageR89.ads);for(let i2=0;i2{event.slot.getSlotElementId().includes(slotId)&&event.isEmpty&&(r89.inImageR89.ads[event.slot.getSlotElementId()].viewable=!0)})),googletag.pubads().addEventListener("impressionViewable",(function(event){event.slot.getSlotElementId().includes(slotId)&&(r89.inImageR89.ads[event.slot.getSlotElementId()].viewable=!0)}))}}}(r89Data),video_overlay:function videoOverlay({video_overlay:videoOverlayConfig,gam_ad_unit:gamAdunit,gam_network_id:gamNetworkId}){return{main:function(){let player2,youtube_element=document.querySelectorAll('iframe[src*="youtube"][src*="/embed/"]');if(youtube_element.length>0){let bidForInstreamAndDisplayInstream2=function(auto_play,ad_type,ev=!1,i3){r89_pbjs.que.push((function(){r89_pbjs.addAdUnits(instreamAdUnitBids[0]),r89_pbjs.setConfig({userSync:{syncEnabled:!1},deviceAccess:!0,debug:!1,cache:{url:"https://prebid.adnxs.com/pbc/v1/cache"},instreamTracking:{enabled:!0},disableAjaxTimeout:!0,bidderTimeout:3e4,consentManagement:{gdpr:{cmpApi:"iab",timeout:1e4,allowAuctionWithoutConsent:!0}}}),r89_pbjs.requestBids({bidsBackHandler:function(){r89_pbjs.getHighestCpmBids(gamAdunit+"-Video-Overlay-Preroll-"+i3);const randNum=Math.floor(9999999*Math.random()).toString(),currentUrl=window.location.href,video_param={iu:"/"+gamNetworkId+"/"+gamAdunit+"/"+gamAdunit+"-Video-Overlay-Preroll",output:"vast",correlator:randNum,vpos:"preroll",wta:1,vpmute:0,vpa:"click",description_url:currentUrl,gdfp_req:1,env:"vp",sz:"640x480|640x360",cust_params:{tier:r89Data.config.demandTier,ad_slot:"Overlay-Video",website_id:r89Data.config.website_id.toString()}},videoUrl=r89_pbjs.adServers.dfp.buildVideoUrl({adUnit:instreamAdUnitBids[0],params:video_param});""!==videoUrl&&("mid-roll"===ad_type&&ev.target.pauseVideo(),initVideo(i3,videoUrl,auto_play,ad_type))}})}))},checkScript2=function(){if(void 0!==this.window.loaded||"undefined"==typeof videojs)setTimeout((function(){checkScript2()}),100);else for(let i3=0;i3{try{entry.isIntersecting?(r89_overlay_skipped_finished.includes(gamAdunit+"-Video-Overlay-Preroll-"+i3)&&(r89.log("STOP_OBSERVING",gamAdunit+"-Video-Overlay-Preroll-"+i3),observer2.unobserve(document.querySelector("#"+gamAdunit+"-Video-Overlay-Preroll-"+i3))),r89_overlay_played.includes(gamAdunit+"-Video-Overlay-Preroll-"+i3)?r89.log("ALREADY_ADDED",r89_overlay_played):(r89_overlay_played.push(gamAdunit+"-Video-Overlay-Preroll-"+i3),bidForInstreamAndDisplayInstream2(!1,"pre-roll",!1,i3))):r89.log("NOT_IN_VIEW")}catch(err){r89.log(err)}}))}),{rootMargin:"0px",threshold:1}).observe(document.querySelector("#"+gamAdunit+"-Video-Overlay-Preroll-"+i3))},assignExistingClasses2=function(element,classList){return element.className+=classList},setVideoAdUnit2=function(i3,ad_unit_name){if(instreamAdUnitCode.push(ad_unit_name+"-"+i3),r89.log("YT_ELEMENT",youtube_element[i3]),youtube_element[i3].previousElementSibling||youtube_element[i3].nextElementSibling){const grand_pa=youtube_element[i3].parentNode;var youtube_parent=document.createElement("div");youtube_element[i3].nextElementSibling?(grand_pa.insertBefore(youtube_parent,youtube_element[i3].nextElementSibling),youtube_parent.append(youtube_element[i3])):youtube_element[i3].previousElementSibling&&(grand_pa.insertBefore(youtube_parent,youtube_element[i3]),youtube_parent.append(youtube_element[i3]))}else youtube_parent=youtube_element[i3].parentNode;if(youtube_parent.id=instreamAdUnitCode[i3],0==youtube_parent.classList.length){const iframeWidth=youtube_element[i3].offsetWidth,iframeHeight=youtube_element[i3].offsetHeight;youtube_parent.style.width=iframeWidth+"px",youtube_parent.style.height=iframeHeight+"px"}youtube_parent.className+=" containerz-instream ";const video_div=document.createElement("div");video_div.id="video-instream"+i3,youtube_parent.appendChild(video_div);const match=youtube_element[i3].src.match(/.*(?:youtu.be\/|v\/|u\/\w\/|embed\/|watch\?v=)([^#\&\?]*).*/);match&&11==match[1].length&&match[1],prepareVideoBidders2(instreamAdUnitCode[i3])},prepareVideoBidders2=function(code){instreamAdUnitBids.push({code:code,divId:code,mediaTypes:{video:{context:"instream",playerSize:[640,360],mimes:["video/mp4","application/javascript"],protocols:[1,2,3,4,5,6,7],playbackmethod:[1,2,3],skip:1,linearity:1,api:[1,2,7,8,9],maxduration:30,minduration:0,startdelay:0,plcmt:1,placement:1,w:640,h:480}},bids:videoOverlayConfig.bidders})};const js_scripts=["https://imasdk.googleapis.com/js/sdkloader/ima3.js","https://tags.refinery89.com/video/js/video1.min.js","https://tags.refinery89.com/video/js/video2.min.js","https://tags.refinery89.com/video/js/video3.js","https://www.youtube.com/iframe_api"],css_scripts=["https://tags.refinery89.com/video/css/video2.min.css","https://tags.refinery89.com/video/css/video3.css"];for(var i2=0;i2\n \n \n `;document.getElementById(instreamAdUnitCode[i3]).insertAdjacentHTML("afterbegin",playerHTML);const real_height=document.getElementById("removeable-instream-div-"+i3).offsetHeight,real_width=youtube_element[i3].offsetWidth;real_width>0&&(document.getElementById("removeable-instream-div-"+i3).style.width=real_width+"px"),document.getElementById("removeable-instream-div-"+i3).style.height=real_height>0?real_height+"px":"100%";try{let onPlayerStateChange2=function(event2){if(Math.floor(player2.getDuration())>30){if(videoOverlayConfig["mid-roll"].active&&1===event2.r89Data){const total_length=Math.floor(player2.getDuration()),mid_point=Math.floor(total_length/2);var interval=setInterval((function(){Math.floor(player2.getCurrentTime())===mid_point?(clearInterval(interval),bidForInstreamAndDisplayInstream2(!0,"mid-roll",event2)):r89.log("WAITING")}),1e3)}videoOverlayConfig["post-roll"].active&&0===event2.r89Data&&bidForInstreamAndDisplayInstream2(!0,"post-roll")}else r89.log("VIDEO_NOT_LONG_ENOUGH_FOR_MID_ROLL_AND_POST_ROLL")},onPlayerReady2=function(event2){null===event2.r89Data&&0==videoOverlayConfig.autoplay&&(r89.log("PLAYING"),event2.target.playVideo())};const player=videojs("content_instream_video_"+i3);player.ima({id:"content_instream_video_"+i3,adTagUrl:vastXML,debug:!1}),player.playsinline(!0);let startEvent="click";(navigator.userAgent.match(/iPhone/i)||navigator.userAgent.match(/iPad/i)||navigator.userAgent.match(/Android/i))&&(startEvent="touchend"),document.getElementsByClassName("placeholder-instream-"+i3)[0].addEventListener("click",(function(){const wrapperDiv=document.getElementById("content_instream_video_"+i3);wrapperDiv.addEventListener(startEvent,initAdDisplayContainer(wrapperDiv))}));var initAdDisplayContainer=function(wrapperDiv=!1){player.ima.initializeAdDisplayContainer(),wrapperDiv.removeEventListener(startEvent,initAdDisplayContainer)};player.src("https://tags.refinery89.com/video/video/ads.mp4"),(auto_play||videoOverlayConfig.autoplay&&"pre-roll"==ad_type)&&player.on("adsready",(function(){"pre-roll"===ad_type&&player.muted(!0);const promise=player.play();void 0!==promise&&promise.then((_=>{})).catch((error=>{r89.log("AUTO_PLAY_ERROR",error)}))}));const contentPlayer=document.getElementById("content_instream_video_"+i3+"_html5_api");(navigator.userAgent.match(/iPad/i)||navigator.userAgent.match(/iPhone/i)||navigator.userAgent.match(/Android/i))&&contentPlayer.hasAttribute("controls")&&contentPlayer.removeAttribute("controls"),player.on("adserror",(function(response){r89_overlay_skipped_finished.push(gamAdunit+"-Video-Overlay-Preroll-"+i3),document.getElementById("removeable-instream-div-"+i3).remove(),"pre-roll"===ad_type&&(player2=new YT.Player("video-instream"+i3,{height:youtube_element[0].height,width:youtube_element[0].width,videoId:video_id,playerVars:{autoplay:0,origin:window.location.origin},events:{onStateChange:onPlayerStateChange2}})),youtube_element[i3].className.length>0&&assignExistingClasses2(document.getElementById("video-instream"+i3),youtube_element[i3].className),youtube_element[i3].remove(),!1===player.isDisposed()&&player.dispose()})),player.on("ads-manager",(function(response){const adsManager=response.adsManager;adsManager.addEventListener(google.ima.AdEvent.Type.COMPLETE,(function(){adsManager.destroy(),r89_overlay_skipped_finished.push(gamAdunit+"-Video-Overlay-Preroll-"+i3),document.getElementById("removeable-instream-div-"+i3).remove(),"pre-roll"===ad_type&&(player2=new YT.Player("video-instream"+i3,{height:youtube_element[0].height,width:youtube_element[0].width,videoId:video_id,playerVars:{autoplay:0,origin:window.location.origin},events:{onStateChange:onPlayerStateChange2,onReady:onPlayerReady2}})),youtube_element[i3].className.length>0&&assignExistingClasses2(document.getElementById("video-instream"+i3),youtube_element[i3].className),youtube_element[i3].remove(),0==player.isDisposed()&&player.dispose()})),adsManager.addEventListener(google.ima.AdEvent.Type.SKIPPED,(function(){adsManager.destroy(),r89_overlay_skipped_finished.push(gamAdunit+"-Video-Overlay-Preroll-"+i3),document.getElementById("removeable-instream-div-"+i3).remove(),"pre-roll"===ad_type&&(player2=new 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    Bibm@th
    Mon compte > Accéder à ma feuille d'exercices >

    Exercice 1 - Loi d'un dé truqué [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On considère un dé cubique truqué dont les faces sont numérotés de 1 à 6 et on note X la variable aléatoire donnée par le numéro de la face du dessus. On suppose que le dé est truqué de sorte que la probabilité d'obtenir une face est proportionnelle au numéro inscrit sur cette face. +
      +
    1. Déterminer la loi de X, calculer son espérance. +
    2. On pose Y=1/X. Déterminer la loi de Y, et son espérance. +
    Indication
    Corrigé
      +
    1. X prend ses valeurs dans {1,…,6}. Par hypothèse, il existe un réel a tel que P(X=k)=ka. Maintenant, +puisque PX est une loi de probabilité, on a : +∑k=16P(X=k)=1⟺a6×72=1⟹a=1/21. +On a donc : +k123456P(X=k)121221321421521621 +On vérifie aisément en appliquant la formule E(X)=∑k=16kP(X=k) que E(X)=133. +
    2. On a X=k⟺Y=1/k. Y prend donc ses valeurs dans {1,1/2,1/3,1/4,1/5,1/6}, et la loi est donnée par : +1/k1/11/21/31/41/51/6P(Y=1/k)121221321421521621 +Le calcul de l'espérance n'est pas plus difficile, et donne : +E(Y)=27. +Attention à l'erreur suivante : ce n'est pas parce que Y=1/X que E(Y)=1/E(X)!!!. +
    +
    Exercice 2 - En plein dans le mille! [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Un joueur tire sur une cible de 10cm de rayon, constituée de couronnes concentriques, +délimitées par des cercles de rayons 1,2, ..., 10 cm, et numérotées respectivement de 10 à 1. La +probabilité d’atteindre la couronne k est proportionnelle à l’aire de cette couronne, et on suppose que +le joueur atteint sa cible à chaque lancer. Soit X la variable aléatoire qui à chaque lancer associe le numéro de la cible. +
      +
    1. Quelle est la loi de probabilité de X ? +
    2. Le joueur gagne k euros s’il atteint la couronne numérotée k pour k compris entre 6 et 10, tandis qu’il perd 2 euros s’il atteint l’une des couronnes périphériques numérotées de 1 à 5. Le jeu est-il +favorable au joueur ? +
    Indication
    Corrigé
    +
    Exercice 3 - Plus grand nombre tiré [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On lance deux dés parfaitement équilibrés. On note X le plus grand des numéros obtenus. Déterminer la loi de la variable aléatoire X.
    Indication
    Corrigé
    +
    Exercice 4 - Garagiste [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Un garagiste dispose de deux voitures de location. Chacune est utilisable en moyenne 4 jours sur 5. Il loue les voitures avec une marge brute de 300 euros par jour et par voiture. +On considère X la variable aléatoire égale au nombre de clients se présentant chaque jour pour louer une voiture. On suppose que X(Ω)={0,1,2,3} avec +P(X=0)=0,1  P(X=1)=0,3  P(X=2)=0,4  P(X=3)=0,2. +
      +
    1. On note Z le nombre de voitures disponibles par jour. Déterminer la loi de Z. On pourra considérer dans la suite que X et Z sont indépendantes. +
    2. On note Y la variable aléatoire : " nombre de clients satisfaits par jour". Déterminer la loi de Y. +
    3. Calculer la marge brute moyenne par jour. +
    Indication
    Corrigé
    +
    Exercice 5 - Analyse de sang [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On cherche à dépister une maladie détectable à l'aide d'un examen sanguin. On suppose que dans notre population, il y a une proportion p de personnes qui n'ont pas cette maladie. +
      +
    1. On analyse le sang de r personnes de la population, avec r entier au moins égal à 2. On suppose que l'effectif de la population est suffisamment grand pour que le choix de ces r personnes s'apparente à un tirage avec remise. Quelle est la probabilité qu'aucune de ces personnes ne soit atteinte de la maladie? +
    2. On regroupe le sang de ces r personnes, puis on procède à l'analyse de sang. Si l'analyse est négative, aucune de ces personnes n'est malade et on arrête. Si l'analyse est positive, on fait toutes les analyses individuelles (on avait pris soin de conserver une partie du sang recueilli avant l'analyse groupée). On note Y la variable aléatoire qui donne le nombre d'analyses de sang effectuées. Donner la loi de probabilité de Y et calculer son espérance en fonction de r et de p. +
    3. On s'intéresse à une population de n personnes, et on effectue des analyses collectives après avoir mélangé les prélèvements par groupe de r personnes, où r est un diviseur de n. Montrer que le nombre d'analyses que l'on peut espérer économiser, par rapport à la démarche consistant à tester immédiatement toutes les personnes, est égal à npr−nr. +
    4. Dans cette question, on suppose que p=0,9 et on admet qu'il existe un réel a>1 de sorte que la fonction x↦px−1x est croissante sur [1,a] et décroissante sur [a,+∞[. Écrire un algorithme permettant de déterminer +pour quelle valeur de l'entier r le nombre pr−1r est maximal. +
    +
    Indication
    Corrigé
    +
    Exercice 6 - Trouver le paramètre d'une loi uniforme connaissant son espérance [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soit X une variable aléatoire suivant une loi uniforme sur {0,1,…,a}, où a∈N. On suppose que E(X)=6. Déterminer a.
    Indication
    Corrigé
    +
    Exercice 7 - Avion [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    A et B sont deux avions ayant respectivement 4 et 2 moteurs. Les moteurs sont supposés indépendants les uns des autres, et ils ont une probabilité p de tomber en panne. +Chaque avion arrive à destination si strictement moins de la moitié de ses moteurs tombe en panne. Quel avion choisissez-vous? (on discutera en fonction de p).
    Indication
    Corrigé
    +
    Exercice 8 - Lancer de pièces [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On lance n fois une pièce parfaitement équilibrée. Quelle est la probabilité d'obtenir strictement plus de piles que de faces.
    Indication
    Corrigé
    +
    Exercice 9 - Tirage avec remises [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Une urne contient N boules numérotées de 1 à N. On en tire n en effectuant des tirages avec remise. On note X et Y le plus petit et le plus grand des nombres obtenus. Déterminer la loi de X et la loi de Y.
    Indication
    Corrigé
    +
    Exercice 10 - Le restaurateur [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Un restaurateur accueille chaque soir 70 clients. Il sait qu'en moyenne, deux clients sur cinq prennent une crème brûlée. Il pense que s'il prépare 30 crèmes brûlées, dans plus de 70\% des cas, la demande sera satisfaite. +
      +
    1. A-t-il raison? +
    2. Combien de crèmes brûlées doit-il fabriquer au minimum pour que la demande soit satisfaite dans au moins 90\% des cas. +
    Indication
    Corrigé
    +
    Exercice 11 - Code de la route! [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    L'examen du code de la route se compose de 40 questions. Pour chaque question, on a le choix entre 4 réponses possibles. Une seule de ces réponses est correcte. Un candidat se présente à l'examen. Il arrive qu’il connaisse la réponse +à certaines questions. Il répond alors à coup sûr. S’il ignore la réponse, il +choisit au hasard entre les 4 réponses proposées. On suppose toutes les questions +indépendantes et que pour chacune de ces questions, la probabilité que +le candidat connaisse la vraie réponse est p. On note, pour 1≤i≤40, Ai l'événement : "le candidat donne la bonne réponse à la i-ème +question". On note S la variable aléatoire égale au nombre total de bonnes réponses. +
      +
    1. Calculer P(Ai). +
    2. Quelle est la loi de S (justifier!)? +
    3. A quelle condition sur p le candidat donnera en moyenne au moins 36 bonnes réponses? +
    Indication
    Corrigé
    +
    Exercice 12 - Minimum et maximum de deux dés [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On lance deux dés équilibrés, on note U1 et U2 les variables aléatoires correspondant aux résultats obtenus. +On appelle X=min(U1,U2) et Y=max(U1,U2). +
      +
    1. Donner la loi de X. En déduire E(X). +
    2. Exprimer X+Y en fonction de U1 et U2. En déduire E(Y). +
    3. Exprimer XY en fonction de U1 et U2. En déduire Cov(X,Y). X et Y sont-elles indépendantes? +
    Indication
    Corrigé
    +
    Exercice 13 - Simuler une variable aléatoire discrète [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    La plupart des langages de programmation dispose d'une fonction ALEA() qui renvoie un nombre aléatoire suivant la loi uniforme sur [0,1]. Écrire un algorithme dont les données sont un entier N≥1 et un tableau p[1],…,p[N] de réels positifs vérifiant ∑i=1np[i]=1 et qui retourne un nombre aléatoire X tel que P(X=i)=p[i].
    Indication
    Corrigé
    +
    Exercice 14 - Deux fois pile [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On joue à pile ou face avec une pièce non équilibrée. A chaque lancer, la probabilité d'obtenir pile est 2/3, et donc celle d'obtenir face est 1/3. +Les lancers sont supposés indépendants, et on note X la variable aléatoire réelle égale au nombre de lancers nécessaires pour obtenir, pour la première fois, deux piles consécutifs. +Pour n≥1, on note pn la probabilité P(X=n). +
      +
    1. Expliciter les événements (X=2), (X=3), (X=4), et déterminer la valeur de p2, p3, p4. +
    2. Montrer que l'on a pn=29pn−2+13pn−1, n≥4. +
    3. En déduire l'expression de pn pour tout n. +
    4. Rappeler, pour q∈]−1,1[, l'expression de ∑n=0+∞nqn, et calculer alors E(X). Interpréter. +
    Indication
    Corrigé
    +
    Exercice 15 - Maximum d'une loi de Poisson [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soit X une variable aléatoire réelle suivant une loi de Poisson P(λ). +Pour quelle(s) valeur(s) de k∈N la probabilité P(X=k) est maximale?
    Indication
    Corrigé
    +
    Exercice 16 - Retrouver une loi connaissant son conditionnement [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soit X et Y deux variables aléatoires. On suppose que X suit une loi de Poisson P(λ) et que la loi de Y conditionnée par (X=n) est la loi binomiale B(n,p), pour tout n∈N. Quelle est la loi de Y?
    Indication
    Corrigé
    +
    Exercice 17 - Un problème chinois! [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On suppose qu'à la naissance, la probabilité qu'un nouveau-né soit un garçon est égale à 1/2. On suppose que +tous les couples ont des enfants jusqu'à obtenir un garçon. On souhaite évaluer la proportion de garçons dans une génération +de cette population. On note X le nombre d'enfants d'un couple pris au hasard dans la population. +
      +
    1. Donner la loi de la variable aléatoire X. +
    2. On suppose qu'une génération en âge de procréer est constituée de N couples, et on note X1,⋯,XN le nombre d'enfants respectif de chaque couple. On note enfin P la proportion de garçons issus de cette génération. +Exprimer P en fonction de X1,…,XN. +
    3. Quelle est la limite de P lorsque N tend vers l'infini. Qu'en pensez-vous? +
    Indication
    Corrigé
    +
    Exercice 18 - Le concierge [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Un concierge rentre d'une soirée. Il dispose de n clefs dont une seule ouvre la porte de son domicile, +mais il ne sait plus laquelle. +
      +
    1. Il essaie les clefs les unes après les autres en éliminant après chaque essai la clef qui n'a pas convenu. +Trouver le nombre moyen d'essais nécessaires pour trouver la bonne clef. +
    2. En réalité, la soirée était bien arrosée, et après chaque essai, le concierge remet la clef essayée dans le trousseau. +Trouver le nombre moyen d'essais nécessaires pour trouver la bonne clef. +
    Indication
    Corrigé
    +
    Exercice 19 - Densité ou non? [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Parmi les fonctions suivantes définies sur R, déterminer lesquelles sont la densité d'une variable aléatoire à densité. Calculer le cas échéant leur fonction de répartition et préciser si elles admettent une espérance. +1. f1(x)={cos⁡x si x∈[0,π/2]0 sinon.2. f2(x)=11+x2, x∈R3. f3(x)=ex(ex+1)2, x∈R4. f4(x)={1+x si x∈[−1,0]1−x si x∈[0,1]0 sinon5. f5(x)={1|x|3 si |x|>10 sinon6. f6(x)=sin⁡x+1, x∈R. +
    Indication
    Corrigé
    +
    Exercice 20 - Loi de Cauchy [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soit f la fonction définie sur R par f(x)=a1+x2. Déterminer a pour que f soit la densité de probabilité d'une variable aléatoire X. Déterminer la fonction de répartition de X. X admet-elle une espérance?
    Indication
    Corrigé
    +
    Exercice 21 - Une densité [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soit la fonction f définie sur R par f(x)=axx si x≥1 et f(x)=0 sinon. +
      +
    1. Déterminer le réel a pour que f soit une densité de probabilité d'une certaine variable aléatoire X. +
    2. Déterminer la fonction de répartition associée à X. +
    3. X admet-elle une espérance? Si oui, la déterminer. +
    Indication
    Corrigé
    +
    Exercice 22 - La station-service [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Dans une station-service, la demande hebdomadaire en essence, en milliers de litres, +est une variable aléatoire X de densité f(x)=c(1−x)41[0,1]. +
      +
    1. Déterminer c. +
    2. La station est réapprovisionnée chaque lundi à 20h. Quelle doit être la capacité du réservoir d'essence +pour que la probabilité d'épuiser ce réservoir soit inférieure à 10−5? +
    +
    Indication
    Corrigé
    +
    Exercice 23 - Loi de Laplace [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On considère une variable aléatoire X dont la densité est donnée par +f(x)=ce−|x|. +
      +
    1. Calculer c. +
    2. Démontrer que X admet des moments de tout ordre. Les calculer. +
    Indication
    Corrigé
    +
    Exercice 24 - Exponentiel des deux côtés! [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soit f la fonction définie sur R par : +f(x)={a.3−xsi x>0a.3xsi x<0. +
      +
    1. Déterminer a pour que f soit une densité de probabilité. +
    2. Soit X une variable aléatoire admettant f pour densité. Déterminer la fonction de répartition de X. Montrer que X admet une espérance E(X) et la calculer. +
    3. On pose Y=3X. Déterminer la fonction de répartition de Y. Y admet-elle une espérance? +
    Indication
    Corrigé
    +
    Exercice 25 - Carré de la loi uniforme [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soit X une variable aléatoire suivant une loi uniforme sur [a,b], avec 0<a<b. +Donner la fonction de répartition, la densité, l'espérance et la variance de Y=X2.
    Indication
    Corrigé
    +
    Exercice 26 - Loi uniforme, moyenne et écart-type [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soit X une variable aléatoire suivant une loi uniforme sur [a,b]. On note m sa moyenne et σ son écart-type. Calculer la probabilité P(X∈[m−σ,m+σ]).
    Indication
    Corrigé
    +
    Exercice 27 - Temps d'attente [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    A partir de 7heures du matin, les bus passent toutes les quinze minutes à un arrêt précis. Un usager se présente à cet arrêt entre 7h et 7h30. On fait l'hypothèse que l'heure exacte de son arrivée, représentée par le nombre de minutes après 7h, est une variable aléatoire uniformément répartie sur l'intervalle [0,30]. Quelle est la probabilité que l'usager attende moins de cinq minutes le prochain bus? Qu'il l'attende plus de dix minutes?
    Indication
    Corrigé
    +
    Exercice 28 - Désintégration radioactive. [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    La durée de vie des atomes de radon suit une loi exponentielle. La probabilité qu'un atome de radon ne soit pas désintégré en 40s +sachant qu'il ne l'est pas en 12s vaut 22. Quelle est la probabilité qu'il ne soit pas désintégré avant 76s sachant qu'il ne l'est pas en 20s?
    Indication
    Corrigé
    +
    Exercice 29 - Chaîne de fabrication [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Une usine fabrique des cadres de vélo. Pour qu'une pièce soit terminée, il faut +qu'elle passe par la chaine A puis par la chaine B. +Le temps de passage exprimé en minutes pour un objet sur la chaine +A est une variable aléatoire M suivant une loi exponentielle de +paramètre 2. +Le temps de passage exprimé en minutes pour un objet sur la chaine +B est une variable aléatoire N suivant une loi uniforme sur [0,1] +Les variables M et N sont indépendantes. +
      +
    1. Rappeler l'expression d'une densité de probabilité v de M +et d'une densité w de N. +
    2. On note S la variable aléatoire représentant le temps total +de fabrication d'une pièce. +Exprimer S en fonction de M et de N et déterminer le temps moyen +de fabrication d'une pièce. +
    Indication
    Corrigé
    +
    Exercice 30 - Tailles [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    La taille d'un homme âgé de 25 ans suit une loi normale de moyenne 175cm et d'écart-type 6cm. +
      +
    1. Quel est le pourcentage d'hommes ayant une taille supérieure à 1m85? +
    2. Parmi les hommes mesurant plus de 1m80, quelle proportion mesure plus de 1m92? +
    Indication
    Corrigé
    +
    Exercice 31 - Javelot! [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On suppose que la distance en mètres parcourues par un javelot lancé par un athlète A suit une loi normale. Au cours d'un entraînement, on constate que +
      +
    • exactement 10% des javelots atteignent plus de 75 mètres. +
    • exactement 25% des javelots atteignent moint de 50 mètres. +
    +Calculer la longueur moyenne parcourue par un javelot ainsi que l'écart-type de cette longueur.
    Indication
    Corrigé
    +
    Exercice 32 - C'est le pied! [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    On a observé que la longueur d'un pied adulte, en cm, suivait une loi normale N(26,36). Une entreprise décide de fabriquer des chaussettes, en proposant trois tailles. +
      +
    1. Déterminer un intervalle centré en 26 qui concentre au moins 95% des tailles. +
    2. Diviser l'intervalle obtenu en trois intervalles égaux, qui détermineront les trois tailles. +
    3. Déterminer quelle part de production on doit réserver à chacune des tailles. +
    Indication
    Corrigé
    +
    Exercice 33 - Somme [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Un grossiste fournit en viande hachée trois cantines. Il reçoit chaque matin leurs commandes. Ce sont des variables aléatoires indépendantes +suivant des lois normales d'espérance respective 55 kg, 65 kg et 30kg, et d'écart-type respectif 4 kg, 10 kg et 3 kg. Calculer la quantité de viande dont +le grossiste dois disposer pour que le risque de ne pouvoir satisfaire la demande soit inférieur à 5\%.
    Indication
    Corrigé
    +
    Exercice 34 - L'ascenseur [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    La capacité des ascenseurs est déterminée par le fait que la +masse d’une personne suit une loi normale de moyenne 75 kg et d’écart-type +5 kg. Dans un ascenseur du type WH1 le nombre maximum de personnes est +de 9. Un voyant lumineux affiche qu’il y a surpoids pour une masse supérieure +à 700 kg, dans ce cas l’ascenseur ne démarre pas. +
      +
    1. Calculer la probabilité qu’il y ait surpoids, quand un groupe de 9 personnes +monte dans l’ascenseur. +
    2. Une enquête récente a montré qu’aux USA, le poids moyen est de 76 kg +et l’écart-type de 6 kg. Le building de la World-Company, à New York, +est équipé d’un ascenseur du type WH1. Calculer la probabilité qu’il y +ait surpoids, quand un groupe de 9 personnes monte dans cet ascenseur. +
    Indication
    Corrigé
    +
    Exercice 35 - Loi jointe uniforme [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soit (X,Y) un couple de variables aléatoires suivant une loi uniforme sur {0,…,n}2. +
      +
    1. Déterminer la loi de X, la loi de Y, la loi de X+Y. +
    2. X et Y sont-elles indépendantes? +
    Indication
    Corrigé
    +
    Exercice 36 - Couple géométrique [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Soient X et Y deux variables aléatoires à valeurs dans N∗, telles que : +P((X=i)∩(Y=j))=a2i+j, +pour tous i,j de N∗. +
      +
    1. Calculer a. +
    2. Déterminer les lois marginales de X et Y. +
    3. X et Y sont-elles indépendantes? +
    +
    Indication
    Corrigé
    +
    Exercice 37 - Guichet de poste [Signaler une erreur] [Ajouter à ma feuille d'exos]
    Enoncé
    Dans un bureau de poste, il y a deux guichets. Chacune des personnes arrivant à la poste choisit le premier guichet avec une probabilité p, +ou le deuxième guichet avec une probabilité q=1−p. Les personnes effectuent leur choix de façon indépendante. En une heure, le nombre X de personnes arrivés à la poste suit une loi de Poisson P(m). +On désigne par Y le nombre de personnes ayant choisi le premier guichet. +
      +
    1. Exprimer la probabilité conditionnelle de Y=k sachant que X=n. +
    2. En déduire la loi conjointe du couple (X,Y). +
    3. Déterminer la loi de Y. On trouvera que Y suit une loi de Poisson de paramètre mp. +
    Indication
    Corrigé
    +
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setXandrConfig(){const{siteIabCategories:siteIabCategories,pageIabCategories:pageIabCategories,pageTopics:pageTopics,brandSafe:brandSafe}=r89Data.config.contextual,hasContextual="done"===r89.contextual.status,xandrKeys=r89.helpers.removeNulls({iab_content_taxonomy:siteIabCategories.length>0?siteIabCategories:null,page_iab_taxonomy:hasContextual&&pageIabCategories.length>0?pageIabCategories:null,page_topics:hasContextual&&pageTopics.length>0?pageTopics:null,page_bs:hasContextual&&null!==brandSafe?brandSafe:null,btng_aprvd:r89Data.config.cmp.betting&&r89.cmp.checkCustomPurpose(50)?"1":null});r89Data.config.xandrKeys=xandrKeys,Object.keys(xandrKeys).length>0&&r89_pbjs.setConfig({appnexusAuctionKeywords:xandrKeys})}()}))}function initAPS(seller){apstag.init({pubID:"d02f0482-a50f-427c-ac01-9856371f1f6b",adServer:"googletag",schain:{complete:1,ver:"1.0",nodes:[{asi:"refinery89.com",sid:seller.id,hp:1,name:seller.name}]}})}var commonjsGlobal="undefined"!=typeof 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el&&el.has_amazon&&bidders.push("aps"),r89Data.config.integrations.pbjs&&"bidPBJS"in el&&bidders.push("pbjs");const requestManager={adserverRequestSent:!1};function headerBidderBack(bidder){!0!==requestManager.adserverRequestSent&&("aps"===bidder?requestManager.aps=!0:"pbjs"===bidder&&(requestManager.pbjs=!0),function allBiddersBack(){return bidders.map((function(bidder){return requestManager[bidder]})).filter(Boolean).length===bidders.length}()&&sendAdserverRequest())}function sendAdserverRequest(){!0!==requestManager.adserverRequestSent&&(requestManager.adserverRequestSent=!0,googletag.cmd.push((function(){googletag.pubads().refresh([el.slot])})))}bidders.forEach((function(bidder){requestManager[bidder]=!1})),bidders.includes("aps")&&apstag.fetchBids({slots:[el.bidAPS],timeout:timeout},(function(){googletag.cmd.push((function(){apstag.setDisplayBids(),headerBidderBack("aps")}))})),bidders.includes("pbjs")&&(el.bidPBJS.forEach((slot=>{if(void 0!==slot.adSlotId&&void 0===slot.floors){let floors=r89.helpers.getPrebidFloor(slot.adSlotId);floors&&(slot.floors=floors),delete slot.adSlotId}})),r89_pbjs.que.push((function(){r89_pbjs.requestBids({timeout:timeout,adUnits:el.bidPBJS,bidsBackHandler:function(){googletag.cmd.push((function(){r89_pbjs.setTargetingForGPTAsync([el.id]),headerBidderBack("pbjs")}));try{const amazonVideoPlayer=document.querySelector('[id^="apsVideoDiv"]'),childAmazonIframe=amazonVideoPlayer.getElementsByTagName("iframe")[0];childAmazonIframe.id.includes("apstag")&&(amazonVideoPlayer.style.maxWidth="100%",childAmazonIframe.style.maxWidth="100%")}catch(err){r89.log("AMAZON_DID_NOT_WON")}}})}))),0===bidders.length?sendAdserverRequest():window.setTimeout((function(){sendAdserverRequest()}),1.5*timeout)},infiniteScroll:function(){r89.log("Bids: Request Infinite Scroll");const timeout=r89Data.config.timeouts.default,refreshItems=r89.infiniteScroll.placedItems.map((el=>el.slot)),slotIds=r89.infiniteScroll.placedItems.map((el=>el.id)),slotsAPS=r89.infiniteScroll.placedItems.map((el=>el.bidAPS)).filter(r89.helpers.filterUndefined),slotsPBJS=r89.infiniteScroll.placedItems.map((el=>el.bidPBJS)).filter(r89.helpers.filterUndefined),bidders=[];r89Data.config.integrations.aps&&slotsAPS.length>0&&bidders.push("aps"),r89Data.config.integrations.pbjs&&slotsPBJS.length>0&&bidders.push("pbjs");const requestManager={adserverRequestSent:!1};function headerBidderBack(bidder){!0!==requestManager.adserverRequestSent&&("aps"===bidder?requestManager.aps=!0:"pbjs"===bidder&&(requestManager.pbjs=!0),function allBiddersBack(){return bidders.map((function(bidder){return requestManager[bidder]})).filter(Boolean).length===bidders.length}()&&sendAdserverRequest())}function sendAdserverRequest(){!0!==requestManager.adserverRequestSent&&(requestManager.adserverRequestSent=!0,googletag.cmd.push((function(){googletag.pubads().refresh(refreshItems)})))}bidders.forEach((function(bidder){requestManager[bidder]=!1})),bidders.includes("aps")&&slotsAPS.length>0&&apstag.fetchBids({slots:slotsAPS,timeout:timeout},(function(){googletag.cmd.push((function(){apstag.setDisplayBids(),headerBidderBack("aps")}))})),bidders.includes("pbjs")&&slotsPBJS.length>0&&r89_pbjs.que.push((function(){r89_pbjs.requestBids({timeout:timeout,adUnits:slotsPBJS,bidsBackHandler:function(){googletag.cmd.push((function(){r89_pbjs.setTargetingForGPTAsync(slotIds),headerBidderBack("pbjs")}))}})})),0===bidders.length?sendAdserverRequest():window.setTimeout((function(){sendAdserverRequest()}),1.5*timeout)}}}(),load:function load({preload:preload,cmp:cmpConfig,preconnect:preconnect}){return{scrollLoaded:!1,scroll:function(){r89.load.scrollLoaded||(r89.load.scrollLoaded=!0,window.removeEventListener("scroll",r89.load.scroll),r89.load.contentLoaded())},contentLoadedState:!1,contentLoaded:function(){r89.load.preload(),r89.log("Load: ContentLoaded - "+document.readyState),"complete"===document.readyState||"interactive"===document.readyState?(r89.log("Load: ContentLoaded - Complete"),r89.load.contentLoadedState=!0,r89.load.main()):(r89.log("Load: ContentLoaded - Event Listener"),document.addEventListener("readystatechange",(event=>{r89.log("Load: Onreadystatechange - "+document.readyState),"interactive"!==document.readyState&&"complete"!==document.readyState||r89.load.contentLoadedState||(r89.load.contentLoadedState=!0,r89.load.main())})))},main:function(){if(r89.log("Load: Main"),r89.events.trackFunction("loadMain"),r89.load.setEnvironment(),r89.load.setCurrentUrl(),!r89.load.checkUrlFilters())return r89.log("Load: Canceled with URL filter"),!1;r89Data.config.is_mobile&&r89Data.config.scriptTimeout.active&&(r89Data.config.scriptTimeout.timeout=r89Data.config.scriptTimeout.options[r89.helpers.randomInt(0,r89Data.config.scriptTimeout.options.length-1)]),r89.adunits.main()},setEnvironment:function(){r89Data.config.viewport.width=Math.max(document.documentElement.clientWidth||0,window.innerWidth||0),r89Data.config.viewport.height=Math.max(document.documentElement.clientHeight||0,window.innerHeight||0),r89Data.config.viewport.width>=r89Data.config.desktop_width?(r89Data.config.is_mobile=0,r89Data.config.is_desktop=1):(r89Data.config.is_mobile=1,r89Data.config.is_desktop=0);let ranges=[[0,320],[320,360],[360,375],[375,414],[414,768],[768,992],[992,1024],[1024,1152],[1152,1280],[1280,1440],[1440,1680],[1680,1920]];r89Data.config.viewport.width>1920?r89Data.config.screenWidthKey="1920-plus":ranges.every((v=>!(r89Data.config.viewport.width2&&-1==r89Data.config.currentUrl.titleTags.indexOf(kv_tags[i2])&&r89Data.config.currentUrl.titleTags.push(kv_tags[i2])},checkUrlFilters:function(){if(void 0!==r89.pageConfig&&void 0!==r89.pageConfig.noAds&&1==r89.pageConfig.noAds)return!1;if(-1!==r89Data.config.urlFilters.exact.indexOf(r89Data.config.currentUrl.path))return!1;for(var i2=0;i2el.wrapper)).forEach((function(wrapper2){wrapper2.remove()}))}r89Data.config.outstream&&r89.outstream.adunit.wrapper&&r89.outstream.adunit.wrapper.remove(),this.resetKeys()},resetKeys:function(){r89.adunits.batches=[],r89.adunits.currentBatch=0,r89.adunits.placedItems={},clearInterval(r89.bids.trackViewableTimeInterval),clearInterval(r89.bids.trackRefreshInterval),this.resetSession()},resetSession:function(){r89.session.newPageview(),this.done()},done:function(){r89.load.main()},customTargetingKeys:function(){googletag.cmd.push((function(){if(void 0!==r89.pageConfig&&"object"==typeof r89.pageConfig.targetingKeys)for(var targetingKey in r89.pageConfig.targetingKeys)googletag.pubads().setTargeting(targetingKey,r89.pageConfig.targetingKeys[targetingKey])}))}}}(),adunits:function adunits({adunits:adunitsConfig,batches:batches,inimage_adunits:inImageAdsConfig,inimage_bidders:inImageBiddersConfig,video_overlay:videoOverlayConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},items:adunitsConfig,performance:{},batchNames:batches,batches:[],currentBatch:0,allBatchesPlaced:0,placedItems:{},lazyLoadDistances:[0,250,500,500,500,750,1e3,1250,1500,1750,2e3],main:function(){r89.log("Adnits: Main"),r89.events.trackFunction("adunitsMain"),r89Data.config.stickySidebar&&r89.stickySidebar.main(),this.prepareBatches(),this.placeWrappers()},pickLazyLoadDistance:function(){return this.lazyLoadDistances[Math.floor(Math.random()*this.lazyLoadDistances.length)]},prepareBatches:function(){this.batchNames.forEach((function(item){"video_overlay"!==item&&(this.placedItems[item]=[],this.batches.push(item))}),this)},placeWrappers:function(){r89.log("Adunits: PlaceWrappers"),this.setStatus("placeWrappers"),this.items.forEach(this.placeWrapper,this),this.attachObserversOnWrappers(),this.placeWrappersDone()},attachObserversOnWrappers:function(){const observer=new IntersectionObserver((function(entries,observer2){entries.forEach((entry=>{let targetId=entry.target.id.replace("-wrapper",""),el=r89.adunits.searchBatches(targetId);el&&(el.isIntersectingAt50=entry.isIntersecting)}))}),{root:null,threshold:.5});document.querySelectorAll('[id^=r89-][id$="-wrapper"]').forEach((element=>{try{"undefined"!==element.id&&observer.observe(document.getElementById(element.id))}catch(err){r89.log("FAILED",element)}}))},placeWrapper:function(el,index){if(!r89.helpers.checkScreenSize(r89Data,el))return!1;r89.log("Adunits: Attempt to place "+el.div_id);let selector_divs=[];if(el.selector_all)selector_divs=document.querySelectorAll(el.selector);else{let selector_div=null;const selector_split=el.selector.split(",");for(let i2=0;i20?selector_divs.forEach((function(selector_div){const id=r89Data.config.prefix+el.div_id+"-"+count;++count;const adunit=document.createElement("div");adunit.id=id;const wrapper2=document.createElement("div");if(wrapper2.id=id+"-wrapper",el.wrapper_style_parsed&&Object.keys(el.wrapper_style_parsed).forEach((key=>{wrapper2.style[key]=el.wrapper_style_parsed[key]})),el.show_label){const ad_unit_label=document.createElement("div");ad_unit_label.className="r89-ad-label",ad_unit_label.innerHTML=r89Data.config.ad_label_text,r89Data.config.labelStyle&&Object.keys(r89Data.config.labelStyle).forEach((key=>{ad_unit_label.style[key]=r89Data.config.labelStyle[key]})),ad_unit_label.style.display="none";const ad_unit_label_wrapper=document.createElement("div");el.wrapper_sticky&&(ad_unit_label_wrapper.style.position="sticky",ad_unit_label_wrapper.style.top=r89Data.config.adunit_wrapper_sticky_top),ad_unit_label_wrapper.appendChild(ad_unit_label),wrapper2.appendChild(ad_unit_label_wrapper)}if(el.wrapper_sticky){const stickyWrapper=document.createElement("div");stickyWrapper.style.position="sticky",stickyWrapper.style.top=r89Data.config.adunit_wrapper_sticky_top,wrapper2.appendChild(stickyWrapper),stickyWrapper.appendChild(adunit)}else wrapper2.appendChild(adunit);if(r89Data.config.positionNames.includes(el.position.name)){if("insertBefore"===el.position.name?selector_div.parentNode.insertBefore(wrapper2,selector_div):"insertAfter"===el.position.name?null===selector_div.nextElementSibling?selector_div.parentNode.appendChild(wrapper2):selector_div.parentNode.insertBefore(wrapper2,selector_div.nextElementSibling):"appendChild"===el.position.name&&selector_div.appendChild(wrapper2),el.id=id,el.wrapper=wrapper2,el.rendered=!1,el.viewable=!1,el.viewableTime=0,el.visible=!1,el.doRefresh=!1,el.refreshed=!1,el.isEmpty=!0,el.isIntersectingAt50=!1,el.failedCount=0,el.has_amazon&&(el.bidAPS={slotID:id,sizes:el.aps_sizes,slotName:el.slot_name}),el.bidder&&el.bidder in r89Data.bidders){const mediaTypes={};el.pbjs_sizes.length>0&&(mediaTypes.banner={sizes:el.pbjs_sizes}),el.has_native&&(mediaTypes.native={type:"image"}),el.bidPBJS={code:id,mediaTypes:mediaTypes,bids:r89Data.bidders[el.bidder],adSlotId:el.ad_slot_id}}el.lazy_load_ab_testing&&(el.lazy_load_distance=this.pickLazyLoadDistance()),this.placedItems[el.batch].push({...el})}}),this):r89.log("Adunits: Selector not found "+el.div_id)},placeWrappersDone:function(){this.setStatus("placeWrappersDone"),"uncalled"!==r89.reset.getStatus()?(r89.log("Adunits: PlaceWrappersDone - Reset"),r89.adunits.placeAds()):(r89.log("Adunits: PlaceWrappersDone"),r89.cmp.load(),0===Object.values(r89.adunits.placedItems).flat().length&&r89.events.push("pageview",{type:"noWrappers"}))},placeAds:function(){r89.log("Adunits: PlaceAds"),this.setStatus("placeAds"),r89.events.trackFunction("adunitsPlaceAds"),this.batches=this.batches.filter((function(batch){return"outstream"===batch&&r89Data.config.outstream?1:this.placedItems[batch].length}),this),1===inImageAdsConfig.active&&inImageBiddersConfig["InImage-R89"].length>0&&r89.inImageR89.main(),videoOverlayConfig.active&&r89.video_overlay.main(),this.placeBatch()},placeBatch:function(){if(r89.log("Adunits: PlaceBatch"),this.setStatus("placeBatch"),this.batches.length>this.currentBatch){const batch=this.batches[this.currentBatch];r89.log("Adunits: PlaceBatch "+batch),this.placedItems[batch].forEach(this.placeAd,this),"lazy"===batch?this.lazyLoadObservers(batch):"outstream"===batch?r89.outstream.main():r89.bids.request(batch)}else this.placeBatchesDone()},placeBatchesDone:function(){r89.log("Adunits: PlaceBatchesDone"),this.setStatus("placeBatchesDone"),this.allBatchesPlaced?r89.log("Adunits: PlaceBatchesDone command ran twice. Stopping further script."):(this.allBatchesPlaced=1,r89Data.config.refresh.active&&(r89.bids.trackViewableTime(),r89.bids.trackRefresh()),r89.inImage.main())},placeAd:function(el){googletag.cmd.push((function(){r89.log("Adunits: PlaceAd "+el.id);const slot=googletag.defineSlot(el.slot_name,el.gpt_sizes,el.id);slot.setTargeting("ad_slot",el.ad_slot.name);let floor=r89.helpers.getGAMFloor(el.ad_slot_id);floor&&slot.setTargeting("flr",floor),"object"==typeof r89.adunits.performance[el.gam_ad_unit]&&(slot.setTargeting("au_vb",r89.adunits.performance[el.gam_ad_unit].au_vb),slot.setTargeting("au_cb",r89.adunits.performance[el.gam_ad_unit].au_cb)),el.lazy_load&&slot.setTargeting("lazy_load_dist",el.lazy_load_distance.toString()),slot.addService(googletag.pubads()),googletag.display(el.id),el.slot=slot}))},updateCurrentBatch:function(){++this.currentBatch},searchBatches:function(id){for(let key in r89.adunits.placedItems){let items=r89.adunits.placedItems[key].filter((function(item){return item.id===id}));if(1===items.length)return items[0]}return!1},lazyLoadObservers:function(batch){r89.log("Adunits: LazyLoadObservers"),this.placedItems[batch].forEach(this.lazyLoadObserver,this),this.updateCurrentBatch(),this.placeBatch()},lazyLoadObserver:function(el){el.lazy_load_ab_testing&&r89.events.trackEvent("lazyLoadDistance",el.gam_ad_unit+";"+el.lazy_load_distance);const options={rootMargin:"0px 0px "+el.lazy_load_distance+"px 0px",threshold:0};new IntersectionObserver((function(entries,observer2){entries.forEach((entry=>{entry.isIntersecting&&(r89.log("Adunit: Observed "+el.id),"requestIdleCallback"in window?requestIdleCallback((function(){r89.bids.lazy(el)}),{timeout:250}):r89.bids.lazy(el),observer2.unobserve(entry.target))}))}),options).observe(el.wrapper)}}}(r89Data),initialize:function initialize(){r89.log("Init: PublisherCall Initialize"),"done"===r89.init.getStatus()?r89.reset.main():r89.load.contentLoaded()},stickySidebar:function stickySidebar({sticky_sidebar:stickySidebarConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},sidebar:stickySidebarConfig,main:function(){if(this.sidebar.active&&r89Data.config.viewport.width>=this.sidebar.screen_width&&(wrapper=document.body.querySelector(this.sidebar.selector),null!==wrapper)){let wrapper_height=wrapper.clientHeight-this.sidebar.decrease_height;this.sidebar.minimum_elements>1&&wrapper_height0){let widgets_inserted=!1;if(widgets.forEach((function(widget){let selector_div=null;const selector_split=widget.selector.split(",");for(let i2=0;i2{entry.isIntersecting&&(r89.log("Outbrain: Lazy load insert."),"requestIdleCallback"in window?requestIdleCallback((function(){document.head.appendChild(scriptElm)}),{timeout:250}):document.head.appendChild(scriptElm),observer2.unobserve(entry.target))}))}),options).observe(r89.outbrain.wrapper)}else document.head.appendChild(scriptElm)}}}this.done()},done:function(){r89.customAdunits.main()}}}(r89Data),outstream:function outstream({outstream:outstreamConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},adunit:outstreamConfig.adunit,deal_id_exception:outstreamConfig.deal_id_exception,line_item_exception:outstreamConfig.line_item_exception,ad_slot_id:outstreamConfig.ad_slot_id,main:function(){r89.log("Outstream: Main"),r89Data.config.outstream?this.adunit.is_desktop&&r89Data.config.is_desktop?(this.adunit.bidder=this.adunit.desktop_bidder,this.adunit.sizes=this.adunit.desktop_sizes,this.adunit.gpt_sizes=this.adunit.desktop_gpt_sizes,this.adunit.aps_sizes=this.adunit.desktop_aps_sizes,this.adunit.pbjs_sizes=this.adunit.desktop_pbjs_sizes,this.placeWrapper(this.adunit)):this.adunit.is_mobile&&r89Data.config.is_mobile?(this.adunit.selector=this.adunit.selector_mobile,this.adunit.bidder=this.adunit.mobile_bidder,this.adunit.sizes=this.adunit.mobile_sizes,this.adunit.gpt_sizes=this.adunit.mobile_gpt_sizes,this.adunit.aps_sizes=this.adunit.mobile_aps_sizes,this.adunit.pbjs_sizes=this.adunit.mobile_pbjs_sizes,this.placeWrapper(this.adunit)):this.done():this.done()},placeWrapper:function(el){let selector_div=null,selector_split=el.selector.split(",");for(let i2=0;i20&&el.bidPBJS.push({code:id,mediaTypes:{banner:{sizes:el.pbjs_sizes}},bids:banner_bidders,adSlotId:r89.outstream.ad_slot_id}),el.bidPBJS.push({code:id,mediaTypes:{video:{context:"outstream",playerSize:[640,360],mimes:["video/mp4"],protocols:[1,2,3,4,5,6,7,8],playbackmethod:[2],skip:0,minduration:3,maxduration:120,linearity:1,api:[1,2],plcmt:4,renderer:{url:"https://tags.refinery89.com/video/js/renderV2.js",render:function(bidz){bidz.renderer.push((()=>{bidz.adLabel=1==r89.outstream.adunit.show_label?r89Data.config.ad_label_text:"",bidz.show_close_button=r89.outstream.adunit.show_close_button,bidz.scroll_position_mobile=r89.outstream.adunit.mobile_scrolling_position,bidz.scroll_position_desktop=r89.outstream.adunit.desktop_scrolling_position,bidz.is_scrollable=r89.outstream.adunit.corner_scroll,bidz.is_refresh=!1,bidz.deal_id_exceptions=outstreamConfig.deal_id_exception,bidz.line_item_exceptions=outstreamConfig.line_item_exception,r89VideoJS.renderAds(bidz)}))}}}},bids:video_bidders,adSlotId:r89.outstream.ad_slot_id})}googletag.cmd.push((function(){var slot=googletag.defineSlot(el.slot_name,el.gpt_sizes,el.id);slot.setTargeting("ad_slot","Outstream-Video");let floor=r89.helpers.getGAMFloor(r89.outstream.ad_slot_id);floor&&slot.setTargeting("flr",floor),slot.addService(googletag.pubads()),googletag.display(el.id),el.slot=slot})),r89.bids.outstream(el)}}else this.done()},done:function(){r89.adunits.updateCurrentBatch(),r89.adunits.placeBatch()}}}(r89Data),inImage:function inImage(){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},checkDevice:function(){return 1==r89Data.config.in_image.is_mobile&&1==r89Data.config.is_mobile||1==r89Data.config.in_image.is_desktop&&1==r89Data.config.is_desktop},checkSelector:function(){if(r89Data.config.in_image.selector){return null!==window.document.querySelector(r89Data.config.in_image.selector)}return!0},main:function(){if(r89.log("InImage: Main"),"uncalled"==r89.reset.getStatus()&&r89Data.config.in_image.active&&this.checkDevice()&&this.checkSelector()){r89.log("InImage: Insert");const script=document.createElement("script");script.src=r89Data.config.in_image.script,script.type="text/javascript",script.async=!0,window.document.head.appendChild(script),this.setStatus("loaded")}this.done()},done:function(){r89.log("InImage: Done"),this.setStatus("done"),r89.elasticNative.main()}}}(),elasticNative:function elasticNative({elastic_native:elasticNativeConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},adunit:elasticNativeConfig,viewTime:0,inView:!1,measureCalled:!1,intervalTime:2500,measuredTime:0,label:{text:"▼ Gesponsord",style:{color:"#999999",textAlign:"center",fontSize:"12px",lineHeight:"16px",margin:"0px 0px 15px 0px",padding:"4px 0 6px 5px",fontFamily:"Arial",borderBottom:"1px solid #ccc"}},main:function(){if(this.setStatus="called",this.adunit){r89.log("Elastic Native: Insert");let selector_div=null;const selector_split=this.adunit.selector.split(",");for(let i2=0;i2{entry.isIntersecting?r89.elasticNative.inView=!0:r89.elasticNative.inView=!1}))}),{rootMargin:"0px 0px 0px 0px",threshold:0}).observe(r89.elasticNative.adunit.wrapper);var inViewInterval=setInterval((function(){r89.elasticNative.inView&&(r89.elasticNative.measuredTime+=r89.elasticNative.intervalTime),!1===r89.elasticNative.measureCalled&&r89.elasticNative.measuredTime>=r89.elasticNative.viewTime&&(r89.elasticNative.callMeasure(),clearInterval(inViewInterval))}),r89.elasticNative.intervalTime)},callMeasure:function(){r89.log("Elastic: Refresh Tracking Slot"),googletag.cmd.push((function(){googletag.pubads().refresh([r89.elasticNative.gamMeasureSlot])})),r89.elasticNative.measureCalled=!0}}}(r89Data),customAdunits:function customAdunits({custom_adunits:customAdunitsConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},adunits:customAdunitsConfig,main:function(){r89.log("CustomAdunits: Main"),r89.events.trackFunction("customAdunitsCalled"),"uncalled"==r89.reset.getStatus()&&"uncalled"==this.getStatus()&&this.adunits.length>0&&r89.customAdunits.insert(),this.done()},insert:function(){if(r89.customAdunits.adunits.length>0){const adunits2=r89.customAdunits.adunits.filter((function(adunit){return!(!adunit.is_desktop||!r89Data.config.is_desktop)||!(!adunit.is_mobile||!r89Data.config.is_mobile)}));if(adunits2.length>0){adunits2.map((el=>el.code)).forEach((function(code){const script=document.createElement("script");script.innerHTML=code,document.head.appendChild(script)})),r89.events.trackFunction("customAdunitsLoaded")}}r89.customAdunits.setStatus("done")},done:function(){r89.log("CustomAdunits: Done"),this.setStatus("done"),r89.fallbackAdunits.main()}}}(r89Data),fallbackAdunits:function fallbackAdunits({fallbacks:fallbackAdUnitsConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},items:fallbackAdUnitsConfig,insertCount:0,main:function(){if(r89.log("Fallback: Main"),this.items.length>0){const configVars=[["minScreenWidth","min_screen_width"],["maxScreenWidth","max_screen_width"],["minScreenHeight","min_screen_height"],["maxScreenHeight","max_screen_height"],["lazyLoad","lazy_load_distance"],["sizes","sizes"]];r89.pushAd=function(divId,adunit,config){r89.log("Fallback: Insert "+adunit+" in #"+divId);const source=r89.fallbackAdunits.items.find((element=>element.div_id==adunit));if(!source)return void r89.log("Fallback: Ad unit not found.");r89.log("Fallback: Ad unit found.");const el=Object.assign({},source);el.selector=divId.replace("#",""),"object"==typeof config&&configVars.forEach((configVar=>{configVar[0]in config&&(el[configVar[1]]=config[configVar[0]],"lazy_load_distance"==configVar[1]&&config[configVar[0]]&&(el.lazy_load=1),"sizes"==configVar[1]&&(el.gpt_sizes=el.sizes,el.aps_sizes=el.sizes,el.pbjs_sizes=el.sizes))})),r89.fallbackAdunits.placeWrapper(el)},"object"==typeof r89.callAds&&r89.callAds.forEach((item=>{r89.pushAd(item[0],item[1],item[2])}))}this.done()},placeWrapper:function(el){let place=!1;if(null===el.min_screen_width&&null!==el.max_screen_width&&el.max_screen_width>=r89Data.config.viewport.width&&(place=!0),null===el.max_screen_width&&null!==el.min_screen_width&&el.min_screen_width<=r89Data.config.viewport.width&&(place=!0),null!==el.min_screen_width&&null!==el.max_screen_width&&el.min_screen_width<=r89Data.config.viewport.width&&el.max_screen_width>=r89Data.config.viewport.width&&(place=!0),!place)return r89.log("Fallback: Screen resolution requirements not met."),!1;r89.log("Fallback: Attempt to place "+el.div_id);const selector_div=document.getElementById(el.selector);if(!selector_div)return r89.log("Fallback: Selector div not found."),!1;r89.fallbackAdunits.insertCount++;const id=r89Data.config.prefix+el.div_id+"-"+r89.fallbackAdunits.insertCount,adunit=document.createElement("div");adunit.id=id;const wrapper2=document.createElement("div");if(wrapper2.id=id+"-wrapper","wrapper_style_parsed"in el&&Object.keys(el.wrapper_style_parsed).forEach((key=>{wrapper2.style[key]=el.wrapper_style_parsed[key]})),wrapper2.appendChild(adunit),selector_div.appendChild(wrapper2),el.id=id,el.wrapper=wrapper2,el.rendered=!1,el.viewable=!1,el.viewableTime=0,el.visible=!1,el.doRefresh=!1,el.refreshed=!1,el.isEmpty=!0,el.failedCount=0,el.has_amazon&&(el.bidAPS={slotID:id,sizes:el.aps_sizes,slotName:el.slot_name}),el.bidder&&el.bidder in r89Data.bidders){const mediaTypes={};el.pbjs_sizes.length>0&&(mediaTypes.banner={sizes:el.pbjs_sizes}),el.has_native&&(mediaTypes.native={type:"image"}),el.bidPBJS={code:id,mediaTypes:mediaTypes,bids:r89Data.bidders[el.bidder]}}googletag.cmd.push((function(){r89.log("Fallback: Place ad "+el.id);const slot=googletag.defineSlot(el.slot_name,el.gpt_sizes,el.id);slot.setTargeting("ad_slot",el.ad_slot.name),slot.addService(googletag.pubads()),googletag.display(el.id),el.slot=slot,r89.adunits.placedItems.fallback.push({...el}),el.lazy_load?r89.fallbackAdunits.lazyLoad(el):r89.fallbackAdunits.request(el)}))},request:function(el){"requestIdleCallback"in window?requestIdleCallback((function(){r89.bids.fallback(el)}),{timeout:250}):r89.bids.fallback(el)},lazyLoad:function(el){const options={rootMargin:"0px 0px "+el.lazy_load_distance+"px 0px",threshold:0};new IntersectionObserver((function(entries,observer2){entries.forEach((entry=>{entry.isIntersecting&&(r89.log("Fallback: Observed "+el.id),r89.fallbackAdunits.request(el),observer2.unobserve(entry.target))}))}),options).observe(el.wrapper)},done:function(){r89.log("Fallback: Done"),this.setStatus("done")}}}(r89Data),infiniteScroll:function infiniteScroll({adunits_infinite_scroll:infiniteScrollConfig,inimage_adunits:inImageConfig,inimage_bidders:inImageBiddersConfig}){return{status:"uncalled",setStatus:function(status){this.status=status},getStatus:function(){return this.status},selector:null,items:infiniteScrollConfig,placedItems:[],count:0,main:function(){r89.log("Infinite Scroll: Main"),this.load()},load:function(selector){if(r89.log("Infinite Scroll: Load"),++this.count,void 0===selector&&r89Data.config.infinite_scroll)this.selector=document.querySelector(r89Data.config.infinite_scroll.selector);else if("string"==typeof selector)this.selector=document.querySelector(selector);else{if("object"!=typeof selector)return void r89.log("Infinite Scroll: Selector not defined");this.selector=selector}null!=this.selector?(this.placedItems=[],this.items.forEach(this.placeWrapper,this),this.placedItems.forEach(this.placeAd,this),1===inImageConfig.active&&inImageBiddersConfig["InImage-R89"].length>0&&r89.inImageR89.main(!0),r89.bids.infiniteScroll()):r89.log("Infinite Scroll: Selector not found")},placeWrapper:function(el,index){if(!r89.helpers.checkScreenSize(r89Data,el))return!1;r89.log("Adunits: Attempt to place "+el.div_id);let selector_divs=[];if(el.selector_all)selector_divs=this.selector.querySelectorAll(el.selector);else{let selector_div=null;const selector_split=el.selector.split(",");for(let i2=0;i20?selector_divs.forEach((function(selector_div){const id=r89Data.config.prefix+el.div_id+"-infinite-"+this.count+"-"+count;++count;const adunit=document.createElement("div");adunit.id=id;const wrapper2=document.createElement("div");if(wrapper2.id=id+"-wrapper",el.wrapper_style_parsed&&Object.keys(el.wrapper_style_parsed).forEach((key=>{wrapper2.style[key]=el.wrapper_style_parsed[key]})),el.show_label){const ad_unit_label=document.createElement("div");ad_unit_label.className="r89-ad-label",ad_unit_label.innerHTML=r89Data.config.ad_label_text,r89Data.config.labelStyle&&Object.keys(r89Data.config.labelStyle).forEach((key=>{ad_unit_label.style[key]=r89Data.config.labelStyle[key]}));const ad_unit_label_wrapper=document.createElement("div");el.wrapper_sticky&&(ad_unit_label_wrapper.style.position="sticky",ad_unit_label_wrapper.style.top=r89Data.config.adunit_wrapper_sticky_top),ad_unit_label_wrapper.appendChild(ad_unit_label),wrapper2.appendChild(ad_unit_label_wrapper)}if(el.wrapper_sticky){const stickyWrapper=document.createElement("div");stickyWrapper.style.position="sticky",stickyWrapper.style.top=r89Data.config.adunit_wrapper_sticky_top,wrapper2.appendChild(stickyWrapper),stickyWrapper.appendChild(adunit)}else wrapper2.appendChild(adunit);if(r89Data.config.positionNames.includes(el.position.name)){if("insertBefore"==el.position.name?selector_div.parentNode.insertBefore(wrapper2,selector_div):"insertAfter"==el.position.name?null===selector_div.nextElementSibling?selector_div.parentNode.appendChild(wrapper2):selector_div.parentNode.insertBefore(wrapper2,selector_div.nextElementSibling):"appendChild"==el.position.name&&selector_div.appendChild(wrapper2),el.id=id,el.wrapper=wrapper2,el.rendered=!1,el.viewable=!1,el.viewableTime=0,el.visible=!1,el.doRefresh=!1,el.isIntersectingAt50=!1,el.failedCount=0,el.refreshed=!1,el.isEmpty=!0,el.has_amazon&&(el.bidAPS={slotID:id,sizes:el.aps_sizes,slotName:el.slot_name}),el.bidder&&el.bidder in r89Data.bidders){const mediaTypes={};el.pbjs_sizes.length>0&&(mediaTypes.banner={sizes:el.pbjs_sizes}),el.has_native&&(mediaTypes.native={type:"image"}),el.bidPBJS={code:id,mediaTypes:mediaTypes,bids:r89Data.bidders[el.bidder]}}this.placedItems.push({...el})}}),this):r89.log("Infinite Scroll: Selector not found "+el.div_id)},placeAd:function(el){googletag.cmd.push((function(){r89.log("Adunits: PlaceAd "+el.id);const slot=googletag.defineSlot(el.slot_name,el.gpt_sizes,el.id);slot.setTargeting("ad_slot",el.ad_slot.name),"object"==typeof r89.adunits.performance[el.gam_ad_unit]&&(slot.setTargeting("au_vb",r89.adunits.performance[el.gam_ad_unit].au_vb),slot.setTargeting("au_cb",r89.adunits.performance[el.gam_ad_unit].au_cb)),slot.addService(googletag.pubads()),googletag.display(el.id),el.slot=slot}))}}}(r89Data),inImageR89:function inImageR89({inimage_adunits:inImageAdsConfig,inimage_bidders:inImageBiddersConfig,seller:seller,gam_ad_unit:gamAdunit,gam_network_id:gamNetworkId}){return{main:function(infiniteScroll2=!1){try{if(r89.inimageR89={},!1===infiniteScroll2){r89.inImageR89.ads=[];let css_scripts=["https://tags.refinery89.com/static/css/inimage.css"];for(let i2=0;i20){let imageOrientation2=function(width,height){let category="horizontal";return category=width/height>1?["horizontal",horizontal_image_class]:["vertical",vertical_image_class],category};var adImages=[],inimageDivs=[];r89.inimageR89.FAILSAFE_TIMEOUT=2e3,r89.inimageR89.inimageAdunits=[],r89.inimageR89.targetingDivs="",r89.inimageR89.inimageAdunits=[],infiniteScroll2?r89.inimageR89.counter++:r89.inimageR89.counter=images.length;var horizontal_image_class=inImageAdsConfig.ad_units[0].horizontal_class_name,vertical_image_class=inImageAdsConfig.ad_units[0].vertical_class_name;vertical_image_class||(vertical_image_class="middle"),horizontal_image_class||(horizontal_image_class="bottom-right");for(let i2=0;i2=468?horizontalSizes:verticalSizes;let adSizes=[];for(let i2=0;i2\n\t\t\t\t\t\t \n\t\t\t\t\t\t\t \n\t\t\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t ';return inImageAdsConfig.ad_units[0].show_label||(htmlLabel='\n\t\t\t\t\t\t
    \n\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t\t \n\t\t\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\t
    '),htmlLabel},attachCloseButtonListeners:function(){document.querySelectorAll("[id^=remove-inimage-close-overlay-]").forEach((function(el){el.addEventListener("click",(function(){let identifier=el.id.slice(-1);document.getElementById("overlay-inimage-ad-"+identifier).remove(),el.closest(".r89-ad-label").remove()}))}))},destroyAdSlots:function(inimageSlot){googletag.destroySlots(inimageSlot)},attachLazyLoadObserver:function(el,infiniteScroll2){let options={};options=infiniteScroll2?{rootMargin:"0px 0px 20px 0px",threshold:0}:{rootMargin:"0px 0px 100px 0px",threshold:0};new IntersectionObserver((function(entries,observer2){entries.forEach((entry=>{if(entry.isIntersecting){const img=entry.target.getElementsByTagName("img");if(img.length){const min_img_height=inImageAdsConfig.ad_units[0].min_image_height,min_img_width=inImageAdsConfig.ad_units[0].min_image_width,imageSelected=img[0],identifier=entry.target.id.split("-")[2],imageHeight=imageSelected.offsetHeight,imageWidth=imageSelected.offsetWidth;if(imageWidth>min_img_width||imageHeight>min_img_height){let adSizes=r89.inImageR89.generateAdSizes(imageWidth,imageHeight);if(adSizes.length){if(0==inImageAdsConfig.ad_units[0].max_ads||r89.inImageR89.adsPlaced{try{let number=entry.target.id.split("-")[2];entry.isIntersecting?r89.inImageR89.ads["overlay-inimage-ad-"+number].intersecting=!0:r89.inImageR89.ads["overlay-inimage-ad-"+number].intersecting=!1,r89.log("PREPARED_SLOTS",r89.inImageR89.ads)}catch(err){r89.log(err)}}))}),{rootMargin:"0px",threshold:.5}).observe(document.querySelector("#r89-inimage-"+i2))},refreshInImageR89:function(){setInterval((function(){let keys=Object.keys(r89.inImageR89.ads);for(let i2=0;i2{event.slot.getSlotElementId().includes(slotId)&&event.isEmpty&&(r89.inImageR89.ads[event.slot.getSlotElementId()].viewable=!0)})),googletag.pubads().addEventListener("impressionViewable",(function(event){event.slot.getSlotElementId().includes(slotId)&&(r89.inImageR89.ads[event.slot.getSlotElementId()].viewable=!0)}))}}}(r89Data),video_overlay:function videoOverlay({video_overlay:videoOverlayConfig,gam_ad_unit:gamAdunit,gam_network_id:gamNetworkId}){return{main:function(){let player2,youtube_element=document.querySelectorAll('iframe[src*="youtube"][src*="/embed/"]');if(youtube_element.length>0){let bidForInstreamAndDisplayInstream2=function(auto_play,ad_type,ev=!1,i3){r89_pbjs.que.push((function(){r89_pbjs.addAdUnits(instreamAdUnitBids[0]),r89_pbjs.setConfig({userSync:{syncEnabled:!1},deviceAccess:!0,debug:!1,cache:{url:"https://prebid.adnxs.com/pbc/v1/cache"},instreamTracking:{enabled:!0},disableAjaxTimeout:!0,bidderTimeout:3e4,consentManagement:{gdpr:{cmpApi:"iab",timeout:1e4,allowAuctionWithoutConsent:!0}}}),r89_pbjs.requestBids({bidsBackHandler:function(){r89_pbjs.getHighestCpmBids(gamAdunit+"-Video-Overlay-Preroll-"+i3);const randNum=Math.floor(9999999*Math.random()).toString(),currentUrl=window.location.href,video_param={iu:"/"+gamNetworkId+"/"+gamAdunit+"/"+gamAdunit+"-Video-Overlay-Preroll",output:"vast",correlator:randNum,vpos:"preroll",wta:1,vpmute:0,vpa:"click",description_url:currentUrl,gdfp_req:1,env:"vp",sz:"640x480|640x360",cust_params:{tier:r89Data.config.demandTier,ad_slot:"Overlay-Video",website_id:r89Data.config.website_id.toString()}},videoUrl=r89_pbjs.adServers.dfp.buildVideoUrl({adUnit:instreamAdUnitBids[0],params:video_param});""!==videoUrl&&("mid-roll"===ad_type&&ev.target.pauseVideo(),initVideo(i3,videoUrl,auto_play,ad_type))}})}))},checkScript2=function(){if(void 0!==this.window.loaded||"undefined"==typeof videojs)setTimeout((function(){checkScript2()}),100);else for(let i3=0;i3{try{entry.isIntersecting?(r89_overlay_skipped_finished.includes(gamAdunit+"-Video-Overlay-Preroll-"+i3)&&(r89.log("STOP_OBSERVING",gamAdunit+"-Video-Overlay-Preroll-"+i3),observer2.unobserve(document.querySelector("#"+gamAdunit+"-Video-Overlay-Preroll-"+i3))),r89_overlay_played.includes(gamAdunit+"-Video-Overlay-Preroll-"+i3)?r89.log("ALREADY_ADDED",r89_overlay_played):(r89_overlay_played.push(gamAdunit+"-Video-Overlay-Preroll-"+i3),bidForInstreamAndDisplayInstream2(!1,"pre-roll",!1,i3))):r89.log("NOT_IN_VIEW")}catch(err){r89.log(err)}}))}),{rootMargin:"0px",threshold:1}).observe(document.querySelector("#"+gamAdunit+"-Video-Overlay-Preroll-"+i3))},assignExistingClasses2=function(element,classList){return element.className+=classList},setVideoAdUnit2=function(i3,ad_unit_name){if(instreamAdUnitCode.push(ad_unit_name+"-"+i3),r89.log("YT_ELEMENT",youtube_element[i3]),youtube_element[i3].previousElementSibling||youtube_element[i3].nextElementSibling){const grand_pa=youtube_element[i3].parentNode;var youtube_parent=document.createElement("div");youtube_element[i3].nextElementSibling?(grand_pa.insertBefore(youtube_parent,youtube_element[i3].nextElementSibling),youtube_parent.append(youtube_element[i3])):youtube_element[i3].previousElementSibling&&(grand_pa.insertBefore(youtube_parent,youtube_element[i3]),youtube_parent.append(youtube_element[i3]))}else youtube_parent=youtube_element[i3].parentNode;if(youtube_parent.id=instreamAdUnitCode[i3],0==youtube_parent.classList.length){const iframeWidth=youtube_element[i3].offsetWidth,iframeHeight=youtube_element[i3].offsetHeight;youtube_parent.style.width=iframeWidth+"px",youtube_parent.style.height=iframeHeight+"px"}youtube_parent.className+=" containerz-instream ";const video_div=document.createElement("div");video_div.id="video-instream"+i3,youtube_parent.appendChild(video_div);const match=youtube_element[i3].src.match(/.*(?:youtu.be\/|v\/|u\/\w\/|embed\/|watch\?v=)([^#\&\?]*).*/);match&&11==match[1].length&&match[1],prepareVideoBidders2(instreamAdUnitCode[i3])},prepareVideoBidders2=function(code){instreamAdUnitBids.push({code:code,divId:code,mediaTypes:{video:{context:"instream",playerSize:[640,360],mimes:["video/mp4","application/javascript"],protocols:[1,2,3,4,5,6,7],playbackmethod:[1,2,3],skip:1,linearity:1,api:[1,2,7,8,9],maxduration:30,minduration:0,startdelay:0,plcmt:1,placement:1,w:640,h:480}},bids:videoOverlayConfig.bidders})};const js_scripts=["https://imasdk.googleapis.com/js/sdkloader/ima3.js","https://tags.refinery89.com/video/js/video1.min.js","https://tags.refinery89.com/video/js/video2.min.js","https://tags.refinery89.com/video/js/video3.js","https://www.youtube.com/iframe_api"],css_scripts=["https://tags.refinery89.com/video/css/video2.min.css","https://tags.refinery89.com/video/css/video3.css"];for(var i2=0;i2\n \n \n `;document.getElementById(instreamAdUnitCode[i3]).insertAdjacentHTML("afterbegin",playerHTML);const real_height=document.getElementById("removeable-instream-div-"+i3).offsetHeight,real_width=youtube_element[i3].offsetWidth;real_width>0&&(document.getElementById("removeable-instream-div-"+i3).style.width=real_width+"px"),document.getElementById("removeable-instream-div-"+i3).style.height=real_height>0?real_height+"px":"100%";try{let onPlayerStateChange2=function(event2){if(Math.floor(player2.getDuration())>30){if(videoOverlayConfig["mid-roll"].active&&1===event2.r89Data){const total_length=Math.floor(player2.getDuration()),mid_point=Math.floor(total_length/2);var interval=setInterval((function(){Math.floor(player2.getCurrentTime())===mid_point?(clearInterval(interval),bidForInstreamAndDisplayInstream2(!0,"mid-roll",event2)):r89.log("WAITING")}),1e3)}videoOverlayConfig["post-roll"].active&&0===event2.r89Data&&bidForInstreamAndDisplayInstream2(!0,"post-roll")}else r89.log("VIDEO_NOT_LONG_ENOUGH_FOR_MID_ROLL_AND_POST_ROLL")},onPlayerReady2=function(event2){null===event2.r89Data&&0==videoOverlayConfig.autoplay&&(r89.log("PLAYING"),event2.target.playVideo())};const player=videojs("content_instream_video_"+i3);player.ima({id:"content_instream_video_"+i3,adTagUrl:vastXML,debug:!1}),player.playsinline(!0);let startEvent="click";(navigator.userAgent.match(/iPhone/i)||navigator.userAgent.match(/iPad/i)||navigator.userAgent.match(/Android/i))&&(startEvent="touchend"),document.getElementsByClassName("placeholder-instream-"+i3)[0].addEventListener("click",(function(){const wrapperDiv=document.getElementById("content_instream_video_"+i3);wrapperDiv.addEventListener(startEvent,initAdDisplayContainer(wrapperDiv))}));var initAdDisplayContainer=function(wrapperDiv=!1){player.ima.initializeAdDisplayContainer(),wrapperDiv.removeEventListener(startEvent,initAdDisplayContainer)};player.src("https://tags.refinery89.com/video/video/ads.mp4"),(auto_play||videoOverlayConfig.autoplay&&"pre-roll"==ad_type)&&player.on("adsready",(function(){"pre-roll"===ad_type&&player.muted(!0);const promise=player.play();void 0!==promise&&promise.then((_=>{})).catch((error=>{r89.log("AUTO_PLAY_ERROR",error)}))}));const contentPlayer=document.getElementById("content_instream_video_"+i3+"_html5_api");(navigator.userAgent.match(/iPad/i)||navigator.userAgent.match(/iPhone/i)||navigator.userAgent.match(/Android/i))&&contentPlayer.hasAttribute("controls")&&contentPlayer.removeAttribute("controls"),player.on("adserror",(function(response){r89_overlay_skipped_finished.push(gamAdunit+"-Video-Overlay-Preroll-"+i3),document.getElementById("removeable-instream-div-"+i3).remove(),"pre-roll"===ad_type&&(player2=new YT.Player("video-instream"+i3,{height:youtube_element[0].height,width:youtube_element[0].width,videoId:video_id,playerVars:{autoplay:0,origin:window.location.origin},events:{onStateChange:onPlayerStateChange2}})),youtube_element[i3].className.length>0&&assignExistingClasses2(document.getElementById("video-instream"+i3),youtube_element[i3].className),youtube_element[i3].remove(),!1===player.isDisposed()&&player.dispose()})),player.on("ads-manager",(function(response){const adsManager=response.adsManager;adsManager.addEventListener(google.ima.AdEvent.Type.COMPLETE,(function(){adsManager.destroy(),r89_overlay_skipped_finished.push(gamAdunit+"-Video-Overlay-Preroll-"+i3),document.getElementById("removeable-instream-div-"+i3).remove(),"pre-roll"===ad_type&&(player2=new YT.Player("video-instream"+i3,{height:youtube_element[0].height,width:youtube_element[0].width,videoId:video_id,playerVars:{autoplay:0,origin:window.location.origin},events:{onStateChange:onPlayerStateChange2,onReady:onPlayerReady2}})),youtube_element[i3].className.length>0&&assignExistingClasses2(document.getElementById("video-instream"+i3),youtube_element[i3].className),youtube_element[i3].remove(),0==player.isDisposed()&&player.dispose()})),adsManager.addEventListener(google.ima.AdEvent.Type.SKIPPED,(function(){adsManager.destroy(),r89_overlay_skipped_finished.push(gamAdunit+"-Video-Overlay-Preroll-"+i3),document.getElementById("removeable-instream-div-"+i3).remove(),"pre-roll"===ad_type&&(player2=new 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