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KI%!oIl>Y@pdzPC3 literal 0 HcmV?d00001 diff --git a/TST/03_Variables_aleatoires/2E_esperance.tex b/TST/03_Variables_aleatoires/2E_esperance.tex new file mode 100644 index 0000000..b8140a0 --- /dev/null +++ b/TST/03_Variables_aleatoires/2E_esperance.tex @@ -0,0 +1,82 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Espérance - Exercices} +\date{Octobre 2020} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} + +\pagestyle{empty} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} +\vfill +\printcollection{banque} + +\pagebeak + +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Coup de pouce} + \begin{itemize} + \item La fréquence peut être interprétée comme une probabilité. + \item Lister les gains possibles à chaque confiscation. + \item Faire le tableau qui relie les gains possibles et les probabilités associées. + \item Calculer l'espérance. + \end{itemize} +\end{bclogo} +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Coup de pouce} + \begin{itemize} + \item La fréquence peut être interprétée comme une probabilité. + \item Lister les gains possibles à chaque confiscation. + \item Faire le tableau qui relie les gains possibles et les probabilités associées. + \item Calculer l'espérance. + \end{itemize} +\end{bclogo} +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Coup de pouce} + \begin{itemize} + \item La fréquence peut être interprétée comme une probabilité. + \item Lister les gains possibles à chaque confiscation. + \item Faire le tableau qui relie les gains possibles et les probabilités associées. + \item Calculer l'espérance. + \end{itemize} +\end{bclogo} +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Coup de pouce} + \begin{itemize} + \item La fréquence peut être interprétée comme une probabilité. + \item Lister les gains possibles à chaque confiscation. + \item Faire le tableau qui relie les gains possibles et les probabilités associées. + \item Calculer l'espérance. + \end{itemize} +\end{bclogo} +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Coup de pouce} + \begin{itemize} + \item La fréquence peut être interprétée comme une probabilité. + \item Lister les gains possibles à chaque confiscation. + \item Faire le tableau qui relie les gains possibles et les probabilités associées. + \item Calculer l'espérance. + \end{itemize} +\end{bclogo} +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Coup de pouce} + \begin{itemize} + \item La fréquence peut être interprétée comme une probabilité. + \item Lister les gains possibles à chaque confiscation. + \item Faire le tableau qui relie les gains possibles et les probabilités associées. + \item Calculer l'espérance. + \end{itemize} +\end{bclogo} +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Coup de pouce} + \begin{itemize} + \item La fréquence peut être interprétée comme une probabilité. + \item Lister les gains possibles à chaque confiscation. + \item Faire le tableau qui relie les gains possibles et les probabilités associées. + \item Calculer l'espérance. + \end{itemize} +\end{bclogo} + + +\end{document} diff --git a/TST/03_Variables_aleatoires/exercises.tex b/TST/03_Variables_aleatoires/exercises.tex index a7bbca9..00fcd16 100644 --- a/TST/03_Variables_aleatoires/exercises.tex +++ b/TST/03_Variables_aleatoires/exercises.tex @@ -1,10 +1,61 @@ \collectexercises{banque} -\begin{exercise}[subtitle={<++>}, step={1}, origin={<++>}, topics={Variables aléatoires}, tags={Probabilité, Simulation, Programmation, Tableur}] - <++> +\begin{exercise}[subtitle={Défauts}, step={2}, origin={Indice TST}, topics={Variables aléatoires}, tags={Probabilité}] + Une entreprise produit des écrans. Ces écrans peuvent présenter deux défauts: un défaut de dimension et un défaut de couleur. + + La probabilité qu'un écran pris au hasard ait un seul défaut est de 0,08 et la probabilité qu'il ait 2 défauts est 0.01. + + On note $X$ la variable aléatoire qui compte le nombre de défauts d'un écran pris au hasard. + \begin{enumerate} + \item Construire le tableau décrivant la loi de probabilité de $X$. + \item Calculer la probabilité qu'un écran ait au moins au défaut. + \item Calculer l'espérance de $X$ puis interpréter le résultat. + \end{enumerate} \end{exercise} -\begin{solution} - <++> -\end{solution} +\begin{exercise}[subtitle={Dépannages}, step={2}, origin={Indice TST}, topics={Variables aléatoires}, tags={Probabilité}] + Un garage veut étudier ses dépannages extérieurs. Pour cela, il note $X$ la variable aléatoire qui compte le nombre de dépannages extérieurs en une journée. La loi de cette variable aléatoire est donnée par le tableau suivant -\collectexercisesstop{banque} \ No newline at end of file + \begin{minipage}{0.4\textwidth} + \begin{tabular}{|c|*{5}{c|}} + \hline + $x_i$ & 1 & 2 & 3 & 4 & 5 \\ + \hline + $P(X=x_i)$ & 0,35 & 0,25 & 0,2 & 0,12 & 0,05 \\ + \hline + \end{tabular} + \end{minipage} + \begin{minipage}{0.6\textwidth} + \begin{enumerate} + \item Vérifier que ce tableau est bien celui d'une variable aléatoire. + \item Calculer les quantités suivantes $P(X < 2)$, $P(X \leq 2)$, $P(X > 5)$ + \item Calculer l'espérance de $X$ puis interpréter. + \end{enumerate} + \end{minipage} +\end{exercise} + +\begin{exercise}[subtitle={Marché noir}, step={2}, origin={Création}, topics={Variables aléatoires}, tags={Probabilité}] + À force de confisquer les téléphones portables de ses élèves, un professeur a pu établir le tableau suivant + \begin{center} + \footnotesize + \begin{tabular}[h]{|p{3cm}| *{6}{c|}} + \hline + Type de portable & Vieux & À clapet & Smartphone & Téléphone satellite & Tablette\\ + \hline + Fréquence (en \%)& 10 & 5 & 50 & 5 & 30\\ + \hline + \end{tabular} + \end{center} + Il décide alors de ne plus les rendre en fin de cours mais de les vendre au marché noir. Il se renseigne alors sur les prix de vente: + \begin{center} + \footnotesize + \begin{tabular}[h]{|p{3cm}| *{6}{c|}} + \hline + Type de portable & Vieux & À clapet & Smartphone & Téléphone satellite & Tablette \\ \hline + Prix de revente (en \euro) & 11 & 11 & 150 & 200 & 250 \\ \hline + \end{tabular} + \end{center} + Combien peut-il espérer gagner en moyenne à chaque fois qu'il confisque un téléphone? +\end{exercise} + + +\collectexercisesstop{banque}