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file mode 100644 index 0000000..77a4537 --- /dev/null +++ b/TST/10_Probabilites_conditionnelles/3E_construction_arbre.tex @@ -0,0 +1,24 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Probabilités conditionnelles - Exercice} +\date{Mars 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=3, +} + +\pagestyle{empty} +\geometry{left=10mm,right=10mm, top=5mm, bottom=5mm} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} +\vfill +\printcollection{banque} +\vfill + +\end{document} diff --git a/TST/10_Probabilites_conditionnelles/exercises.tex b/TST/10_Probabilites_conditionnelles/exercises.tex index 509949c..fa63350 100644 --- a/TST/10_Probabilites_conditionnelles/exercises.tex +++ b/TST/10_Probabilites_conditionnelles/exercises.tex @@ -96,4 +96,117 @@ \item La probabilité ait été payé avec de l'espèce est de 72\%. \end{enumerate} \end{exercise} + +\begin{exercise}[subtitle={Neuf ou occasion?}, step={3}, origin={Création}, topics={Probabilités conditionnelles}, tags={probabilité, simulation}] + Un concessionnaire automobile vend chaque année 65\% de véhicules neufs. Une étude montre que parmi les acheteurs de véhicules neufs, 40\% adhèrent à un contrat d'assurance. Par ailleurs, 7\% des acheteurs qui ont acquis un véhicule d'occasion, ont adhéré à un contrat de maintenance. + + \begin{minipage}{0.6\textwidth} + On choisit un client au hasard parmi les clients de ce concessionnaire et on considère les évènements suivants: + \begin{itemize} + \item $N = \left\{ \mbox{ Le client achète un véhicule neuf } \right\}$ + \item $M = \left\{ \mbox{ Le client souscrit à un contrat de maintenance } \right\}$ + \end{itemize} + \begin{enumerate} + \item Traduire les données de l'énoncé en terme de probabilité en utilisant les évènements $N$ et $M$. + \item À partir des données de l'énoncé, compléter l'arbre de probabilité traduisant la situation. + + \end{enumerate} + + \end{minipage} + \hfill + \begin{minipage}{0.3\textwidth} + \begin{tikzpicture}[sloped] + \node {.} + child {node {$N$} + child {node {$M$} + edge from parent + node[above] {...} + } + child {node {$\overline{M}$} + edge from parent + node[above] {...} + } + edge from parent + node[above] {...} + } + child[missing] {} + child { node {$\overline{N}$} + child {node {$M$} + edge from parent + node[above] {...} + } + child {node {$\overline{M}$} + edge from parent + node[above] {...} + } + edge from parent + node[above] {...} + } ; + \end{tikzpicture} + \end{minipage} + \begin{enumerate} + \setcounter{enumi}{2} + \item Traduire en français les probabilités suivantes, les calculer puis les placer sur l'arbre. + \begin{multicols}{3} + \begin{enumerate} + \item $P(\overline{N})$ + \item $P_N(\overline{M})$ + \item $P(M \cap N)$ + \item $P(M \cap \overline{N})$ + \item $P_{\overline{N}}(M)$ + \item $P_{\overline{N}}(\overline{M})$ + \end{enumerate} + \end{multicols} +\end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Sécurité}, step={3}, origin={?annale?}, topics={Probabilités conditionnelles}, tags={probabilité, simulation}] + Dans un aéroport, les portiques de sécurité servent à détecter les objets métalliques que peuvent emporter les voyageurs. + + On choisit au hasard un voyageur franchissant un portique. Et on note $S$ l'événement "le voyageur fait sonner le portique" , $M$ l'événement " le voyageur porte un objet métallique" + + On considère qu'un voyageur sur 500 porte sur lui un objet métallique. Et on note que + + \begin{minipage}{0.6\textwidth} + \begin{itemize} + \item Lorsqu'un voyageur franchit le portique avec un objet métallique, la probabilité que le portique sonne est égale à $0,98$; + \item Lorsqu'un voyageur franchit le portique sans objet métallique, la probabilité que le portique ne sonne pas est aussi égale à $0,98$. + \end{itemize} + \begin{enumerate} + \item À l'aide des données de l'énoncé, préciser les valeurs de $P(M)$, $P_{M}(S)$ et $P_{\overline{M}}(\overline{S})$. + \item Recopier et compléter l'arbre pondéré ci-contre illustrant cette situation. + \end{enumerate} + \end{minipage} + \hfill + \begin{minipage}{0.3\textwidth} + \begin{tikzpicture}[sloped] + \node {.} + child {node {$M$} + child {node {$S$} + edge from parent + node[above] {...} + } + child {node {$\overline{S}$} + edge from parent + node[above] {...} + } + edge from parent + node[above] {...} + } + child[missing] {} + child { node {$\overline{M}$} + child {node {$S$} + edge from parent + node[above] {...} + } + child {node {$\overline{S}$} + edge from parent + node[above] {...} + } + edge from parent + node[above] {...} + } ; + \end{tikzpicture} + \end{minipage} +\end{exercise} \collectexercisesstop{banque} diff --git a/TST/10_Probabilites_conditionnelles/index.rst b/TST/10_Probabilites_conditionnelles/index.rst index 1214333..a2f8c11 100644 --- a/TST/10_Probabilites_conditionnelles/index.rst +++ b/TST/10_Probabilites_conditionnelles/index.rst @@ -2,7 +2,7 @@ Probabilités conditionnelles ############################ :date: 2021-02-07 -:modified: 2021-03-10 +:modified: 2021-03-26 :authors: Benjamin Bertrand :tags: Probabilité, Simulation :category: TST @@ -51,6 +51,10 @@ Bilan: représentation d'une situation avec des arbres - Phrases + arbre à construire - Des formules et un arbre à construire +.. image:: ./3E_construire_arbre.pdf + :height: 200px + :alt: Construction d'un arbre de probabilités + Bilan: multiplication des probabilités sur les branches pour obtenir la probabilité du chemin. Étape 4: Calculer des probabilités avec un arbre