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zV_A`*;Xi~k1Vf5Jgq-?!y_>GB4s36LfYfFC-_iK{;bvA63DsJJM1&;eV6voz0+SUF z<&=D`X#+)VIF;i*dP?#c*g|}O^(Y$Ms|D>8-u;Fd&y-Mly;^PrW0b+jwYO^sRj)Mr zLtikJ!u_J|NszCE5G;ZLVv7N!;#@F`u|CL?0VV_ytPNJ6!I&U{*k2GBMi`@C!48!D z$TEW%Q+>2k{XDdN#I%Co6(!!wJ1y3hx*pZ(HFn*qvd;{zvqr`PjVgH<1g&CsLvKT7Q(j_pCyOdm~OPM literal 0 HcmV?d00001 diff --git a/Complementaire/02_Inference_Baysienne/5E_annales.tex b/Complementaire/02_Inference_Baysienne/5E_annales.tex new file mode 100644 index 0000000..c001bc9 --- /dev/null +++ b/Complementaire/02_Inference_Baysienne/5E_annales.tex @@ -0,0 +1,21 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Probabilités conditionnelles - Exercice} +\date{Mars 2021} + +\pagestyle{empty} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=5, +} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} +\vfill + +\end{document} diff --git a/Complementaire/02_Inference_Baysienne/exercises.tex b/Complementaire/02_Inference_Baysienne/exercises.tex index ba65239..9201527 100644 --- a/Complementaire/02_Inference_Baysienne/exercises.tex +++ b/Complementaire/02_Inference_Baysienne/exercises.tex @@ -151,4 +151,136 @@ On découvrit que les rapports d'autopsie des deux enfants avaient ignoré des Explication vidéo \url{https://peertube.noussommes.org/videos/watch/playlist/cf21ffd6-9b0d-4339-9c22-db71afbf8d4d?playlistPosition=29} \end{exercise} + +\begin{exercise}[subtitle={Aéroport}, step={5}, origin={Création}, 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. + + \begin{itemize} + \item $S$ l'événement \og le voyageur fait sonner le portique \fg{}; + \item $M$ l'événement \og le voyageur porte un objet métallique \fg{}. + \end{itemize} + + On considère qu'un voyageur sur 500 porte sur lui un objet métallique. Et on note que + + \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{minipage}{0.6\textwidth} + \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. + \item Montrer que: $P(S)=\np{0,02192}$. + \item En déduire la probabilité qu'un voyageur porte un objet métallique sachant qu'il a fait sonner le portique. (On arrondira le résultat à $10^{-3}$.) + \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} + +\begin{exercise}[subtitle={Sponsort}, step={5}, origin={Création}, topics={Probabilités conditionnelles}, tags={probabilité, simulation}] + Un navigateur s'entraîne régulièrement dans le but de battre le record du monde de + traversée de l'Atlantique à la voile. + + \emph{Dans cet exercice, les résultats seront arrondis au millième si nécessaire.} + + Une entreprise nommée \og Régate \fg, s'intéresse aux résultats de ce navigateur. + + La probabilité qu'il réalise la traversée en moins de 6 jours est de 0,16. + + Si le navigateur réalise la traversée en moins de 6 jours, l'entreprise le sponsorise avec une probabilité de 0,95. + + Sinon, l'entreprise hésite et le sponsorise avec une probabilité de 0,50. + + On note $M$ l'évènement \og la traversée est réalisée par le navigateur en moins de 6 jours \fg et $F$ l'évènement \og l'entreprise sponsorise le navigateur \fg. + + \begin{enumerate} + \item Représenter cette situation à l'aide d'un arbre pondéré. + \item Montrer que la probabilité que l'entreprise ne sponsorise pas le navigateur à la + prochaine course est $0,428$. + \item L'entreprise a finalement choisi de ne pas financer le navigateur. + + Calculer la probabilité que le navigateur ait tout de même réalisé la traversée en moins + de $6$ jours. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Paiements}, step={5}, origin={Création}, topics={Probabilités conditionnelles}, tags={probabilité, simulation}] + Un commerçant dispose dans sa boutique d'un terminal qui permet à ses clients, s'ils souhaitent + régler leurs achats par carte bancaire, d'utiliser celle-ci en mode sans contact (quand le montant de + la transaction est inférieur ou égal à 30~\euro) ou bien en mode code secret (quel que soit le montant + de la transaction). + + Il remarque que 80\,\% de ses clients règlent des sommes inférieures ou égales à 30~\euro. Parmi eux : + \begin{itemize} + \item 40\,\% paient en espèces; + \item 40\,\% paient avec une carte bancaire en mode sans contact ; + \item les autres paient avec une carte bancaire en mode code secret. + \end{itemize} + Et que 20\,\% de ses clients règlent des sommes strictement supérieures à 30~\euro. Parmi eux : + \begin{itemize} + \item 70\,\% paient avec une carte bancaire en mode code secret ; + \item les autres paient en espèces. + \end{itemize} + On interroge au hasard un client qui vient de régler un achat dans la boutique. + + On considère les évènements suivants : + + $V$ : \og le client a réglé un montant inférieur ou égal à 30~\euro \fg ; + + $E$ : \og le client a réglé en espèces\fg ; + + $C$ : \og le client a réglé avec sa carte bancaire en mode code secret\fg ; + + $S$ : \og le client a réglé avec sa carte bancaire en mode sans contact \fg. + + \medskip + + \begin{enumerate} + \item + \begin{enumerate} + \item Donner la probabilité de l'évènement $V$, notée $P(V)$, ainsi que la probabilité de $S$ sachant + $V$ notée $P_V(S)$. + \item Traduire la situation de l'énoncé à l'aide d'un arbre pondéré. + \end{enumerate} + \item + \begin{enumerate} + \item Calculer la probabilité que pour son achat, le client ait réglé un montant inférieur ou égal à $30$~\euro{} et qu'il ait utilisé sa carte bancaire en mode sans contact. + \item Montrer que la probabilité de l'évènement: \og pour son achat, le client a réglé avec sa carte + bancaire en utilisant l'un des deux modes\fg{} est égale à $0,62$. + \end{enumerate} + \end{enumerate} +\end{exercise} \collectexercisesstop{banque} diff --git a/Complementaire/02_Inference_Baysienne/index.rst b/Complementaire/02_Inference_Baysienne/index.rst index 6ea2788..dcf487b 100644 --- a/Complementaire/02_Inference_Baysienne/index.rst +++ b/Complementaire/02_Inference_Baysienne/index.rst @@ -2,7 +2,7 @@ Inférence Bayésienne #################### :date: 2021-03-15 -:modified: 2021-03-31 +:modified: 2021-04-01 :authors: Benjamin Bertrand :tags: Probabilité, Bayes :category: Complementaire @@ -67,7 +67,11 @@ On a 6 dés avec 4, 6, 8, 10, 12 et 20 faces. Quelqu'un en choisi un secrètemen :alt: Retrouver le dé choisi. -Étape 5: Optimisation des paramètres -==================================== +Étape 5: Retour sur des exercices classiques +============================================ -Retour sur les tests et recherche de l'optimisation des paramètres. +Séries d'exercices d'annales de terminale ES sur les probabilités conditionnelles. On s'écarte du thème mais des élèves ont des concours à passer et c'est certainement ce qu'ils retrouveront l'année prochaine. + +.. image:: ./5E_annales.pdf + :height: 200px + :alt: Exercices d'annales du bac ES sur les probabilités conditionnelles