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+\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} + + +\pagestyle{empty} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} + +\end{document} diff --git a/TST/08_Loi_binomiale/exercises.tex b/TST/08_Loi_binomiale/exercises.tex index 0598c3c..1a0ff67 100644 --- a/TST/08_Loi_binomiale/exercises.tex +++ b/TST/08_Loi_binomiale/exercises.tex @@ -15,5 +15,64 @@ Proposer une expérience aléatoire qui pourrait être modélisée avec une loi binomiale. Vous détaillerez ensuite les paramètres et justifierez la modélisation. \end{exercise} +\begin{exercise}[subtitle={Jeux}, step={2}, origin={Création}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale}] + Bob joue à un jeu où il estime qu'il a 70\% de chance de gagner une partie. Entre 2 parties, il prend le temps de se reposer pour que la précédente partie n'influence pas la suivante. + + On note $V$ l'évènement "Bob gagne la partie". + + Bob fait 2 parties et on note $X$ la variable aléatoire qui compte le nombre de victoire. + \begin{enumerate} + \item Faire un arbre qui modélise la situation. + \item Déterminer la probabilité que Bob gagne une seule partie. + \item Avec quelle loi peut-on modéliser la variable aléatoire $X$? Préciser les paramètres. + \item Calculer et interpréter les probabilités suivantes + \[ + P(X = 0) \qquad P(X=2) + \] + \item Dresser le tableau de la loi de probabilités de $X$. + \item Calculer l'espérance de $X$. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Repas}, step={2}, origin={Création}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale}] + Bob adore manger des légumes. Chaque jour, il choisit au hasard un fruit dans une panière quotidiennement remplie par ses parents contenant 7 bananes, 5 pommes et 2 kiwi. + + Ses parents veulent essayer de prévoir la consommation en banane de Bob sur 3 jours. + + On note donc $X$ le nombre bananes mangées par Bob sur 3 jours et $B$ l'évènement "Bob mange une banane". + \begin{enumerate} + \item Faire un arbre qui modélise la situation. + \item Déterminer la probabilité que Bob gagne deux bananes. + \item Avec quelle loi peut-on modéliser la variable aléatoire $X$? Préciser les paramètres. + \item Calculer les probabilités suivantes + \[ + P(X = 1) \qquad P(X=0) \qquad P(X \leq 2) + \] + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Auto-école}, step={2}, origin={Création}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale}] + Dans une auto-école, à chaque session 75\% des candidats réussissent à avoir leur code. + + \begin{enumerate} + \item On interroge au hasard 4 candidats d'une session pour savoir s'ils ont eu leur code. On note $X$ variable aléatoire qui compte le nombre de réponse positive. + \begin{enumerate} + \item Avec quelle loi peut-on modéliser la variable aléatoire $X$? Préciser les paramètres. + \item Calculer les probabilités suivantes + \[ + P(X = 1) \qquad \qquad + P(X = 4) \qquad \qquad + P(X \leq 1) + \] + \item Quelle est la probabilité qu'au moins un candidat ait répondu positivement. + \end{enumerate} + \item Cette fois-ci, on choisit un candidat et on note $Y$ le nombre de sessions qu'il a du passer avant d'avoir code. + \begin{enumerate} + \item Faire un arbre pour représenter la situation. + \item Peut-on modéliser $Y$ avec une loi binomiale? Si oui, préciser les paramètres. + \end{enumerate} + \end{enumerate} +\end{exercise} + \collectexercisesstop{banque}