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+\date{Mars 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} + +\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 a58868f..9ff0c41 100644 --- a/TST/10_Probabilites_conditionnelles/exercises.tex +++ b/TST/10_Probabilites_conditionnelles/exercises.tex @@ -1,10 +1,99 @@ \collectexercises{banque} -\begin{exercise}[subtitle={<++>}, step={1}, origin={<++>}, topics={Probabilités conditionnelles}, tags={probabilité, simulation}] - <++> +\begin{exercise}[subtitle={Vrai-Faux}, step={1}, origin={Création}, topics={Probabilités conditionnelles}, tags={probabilité, simulation}] + On interroge un échantillon de \np{1500} jeunes ayant entre 14 et 18ans pour savoir s'ils fument et si au moins l'un de leurs parents fume. + + Les résultats de l'enquête sont consignés dans le tableau suivant. + \begin{center} + \begin{tabular}{|>{\columncolor{highlightbg}}c|*{2}{p{4cm}|}|p{4cm}|} + \hline + \rowcolor{highlightbg} + & Fumeur & Non fumeur & Total\\ + \hline + Au moins un parent fumeur & 300 & 300 & 600\\ + \hline + Aucun parent fumeur & 200 & 700 & 900\\ + \hline + Total &500 & \np{1000} & \np{1500}\\ + \hline + \end{tabular} + \end{center} + + On choisit au hasard un jeune parmi ceux interrogés. On note $A = \left\{ \mbox{Fumeur} \right\}$ et $B = \left\{ \mbox{Au moins un parent fumeur} \right\}$. + + Pour chacune des phrases suivantes, justifier si elles sont vraies ou fausses. + \begin{enumerate} + \item La probabilité qu'il soit fumeur est de plus de 30\%. + \item La probabilité qu'il soit fumeur et qu'aucun parent ne soit fumeur est de moins de 0.1. + \item La probabilité qu'au moins un de ses parents soit fumeur et qu'il ne le soit pas est de $\frac{1}{5}$. + \item La probabilité qu'il soit fumeur ou qu'un de ses parents le soit est de plus de 70\%. + \item Sachant qu'il est fumeur, la probabilité que ses parents le soit aussi est de 0.6. + \item Sachant qu'aucun de ses parents ne soit fumeur, la probabilité qu'il ne soit pas aussi est de 50\%. + \end{enumerate} \end{exercise} -\begin{solution} - <++> -\end{solution} +\begin{exercise}[subtitle={Moyen de paiement}, step={2}, origin={Création}, topics={Probabilités conditionnelles}, tags={probabilité, simulation}] + Le gérant d'une grande enseigne de distribution a commandé une étude statistique des moyens de paiement de ses clients. Les résultats ont été représenté dans l'arbre ci-dessous. + \bigskip + \begin{tikzpicture}[sloped] + \node {.} + child {node {Moins de 20\euro} + child {node {Espèce} + edge from parent + node[above] {0.6} + } + child[missing] {} + child {node {Paiement sans contact} + edge from parent + node[above] {0.3} + } + child[missing] {} + child {node {Carte bleu} + edge from parent + node[above] {0.1} + } + edge from parent + node[above] {0.4} + } + child[missing] {} + child[missing] {} + child[missing] {} + child[missing] {} + child[missing] {} + child { node {Plus de 20\euro} + child {node {Espèce} + edge from parent + node[above] {0.4} + } + child[missing] {} + child {node {Paiement sans contact} + edge from parent + node[above] {0.1} + } + child[missing] {} + child {node {Carte bleu} + edge from parent + node[above] {0.5} + } + edge from parent + node[above] {0.6} + } ; + \end{tikzpicture} + \bigskip + + On sélectionne un client de cette enseigne au hasard. On note + \[ + M = \left\{ \mbox{ Moins de 20\euro } \right\} \qquad E = \left\{ \mbox{ Espèce }\right\} \qquad B = \left\{ \mbox{ carte bleu }\right\} + \] + Pour chacune des phrases suivantes, justifier si elles sont vraies ou fausses. + \begin{enumerate} + \item La probabilité qu'un client achète pour plus de 20\euro est de 0.6. + \item Parmi les achats de moins de 20\euro, il y a un probabilité de 10\% qu'il soit fait en paiement sans contact. + \item Parmi les achats de plus de 20\euro, la probabilité qu'il ait été fait avec de l'espèce est de 0.4. + \item La probabilité qu'un achat soit de plus de 20\euro et payé avec une carte bleu est de 90\%. + \item La probabilité qu'un achat soit de moins de 20\euro et payé avec une carte bleu est de 0.2. + \item La probabilité qu'un achat soit de moins de 20\euro et payé avec de l'espère est de 1. + \item La probabilité ait été payé avec le paiement sans contact est de 72\M. + \end{enumerate} +\end{exercise} \collectexercisesstop{banque} diff --git a/TST/10_Probabilites_conditionnelles/index.rst b/TST/10_Probabilites_conditionnelles/index.rst index b915e7f..ae737fb 100644 --- a/TST/10_Probabilites_conditionnelles/index.rst +++ b/TST/10_Probabilites_conditionnelles/index.rst @@ -2,22 +2,51 @@ Probabilités conditionnelles ############################ :date: 2021-02-07 -:modified: 2021-02-07 +:modified: 2021-03-08 :authors: Benjamin Bertrand :tags: Probabilité, Simulation :category: TST :summary: Première approche des probabilités conditionnelles -Étape 1: Tableau croisé et probabilité -====================================== +Étape 1: Vrai/Faux à partir d'un tableau +======================================== Vrai/faux à partir d'un tableau croisé. Après les premières réponses ont introduira les notations ensemblistes et on poussera les élèves à les utiliser. -Bilan: Notations et représentation du tableau avec un arbre. +.. image:: ./1E_tableau_croise.pdf + :height: 200px + :alt: Vrai-Faux à partir d'un tableau des effectifs -Étape 2: Construction d'arbre -============================= +Le workflow de la séance: -Étape 3: Calculs de probabilités avec les arbres +- Élève individuellement: avoir un avis (pas forcement justifié) sur chacune des phrases +- En groupe: mettre en commun les avis et construire les justifications (avec un calcul) +- En plénière: correction de certains éléments et notation probabilistes + +Bilan: Notations ensemblistes et probabilistes + +Étape 2: Vrai/Faux à partir d'un arbre +====================================== + +Travail similaire à l'étape 1 mais cette fois-ci à partir d'un arbre. + +.. image:: ./2E_lecture_arbre.pdf + :height: 200px + :alt: Vrai-Faux à partir d'un arbre de probabilités + +Bilan: représentation d'une situation avec des arbres + +Étape 3: Construction d'un arbre +================================ + +- Phrases + arbre vide à compléter +- Phrases + arbre à construire +- Des formules et un arbre à construire + +Bilan: multiplication des probabilités sur les branches pour obtenir la probabilité du chemin. + +Étape 4: Calculer des probabilités avec un arbre ================================================ +Exercices de calculs de probabilités +