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b/TST/10_Probabilites_conditionnelles/4B_Bayes.tex new file mode 100644 index 0000000..fc53c62 --- /dev/null +++ b/TST/10_Probabilites_conditionnelles/4B_Bayes.tex @@ -0,0 +1,120 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Probabilités conditionnelles - Cours} +\date{Mars 2021} + +\pagestyle{empty} + +\begin{document} + +\maketitle + +\setcounter{section}{2} +\subsection*{Exemple de l'utilisation des formules - test anti-covid} + +On s'intéresse aux tests anti-covid et on considère les évènements suivants + +\[ + A = \left\{ \mbox{Malade du covid} \right\} +\] +\[ + B = \left\{ \mbox{Testé positif} \right\} +\] + +On considère que l'on a 1\% de chance d'être malade du covid. + +Les tests anti-covid ne sont pas fiables à 100\% (aucun test ne peut l'être). Voici les données du fabriquant + +\begin{itemize} + \item \textbf{Sensibilité} - probabilité qu'une personne malade soit testée positive - vrai positifs + \[ + 70\% + \] + \item \textbf{Spécificité} - probabilité qu'une personne saine soit testée négative - vrai négatifs + \[ + 95\% + \] +\end{itemize} + +\begin{center} + \textbf + On est testé positif, quelle est la probabilité que l'on soit malade? +\end{center} + +Complétons l'arbre de probabilité de gauche avec les données. + +\begin{center} + \begin{tikzpicture}[sloped, grow=right] + \node {.} + child {node {$A$} + child {node {$B$} + edge from parent + node[above] {...} + } + child {node {$\overline{B}$} + edge from parent + node[above] {...} + } + edge from parent + node[above] {...} + } + child[missing] {} + child { node {$\overline{A}$} + child {node {$B$} + edge from parent + node[above] {...} + } + child {node {$\overline{B}$} + edge from parent + node[above] {...} + } + edge from parent + node[above] {...} + } ; + \end{tikzpicture} + \hspace{3cm} + \begin{tikzpicture}[sloped, grow=left] + \node {.} + child {node {$B$} + child {node {$A$} + edge from parent + node[above] {...} + } + child {node {$\overline{A}$} + edge from parent + node[above] {...} + } + edge from parent + node[above] {...} + } + child[missing] {} + child { node {$\overline{B}$} + child {node {$A$} + edge from parent + node[above] {...} + } + child {node {$\overline{A}$} + edge from parent + node[above] {...} + } + edge from parent + node[above] {...} + } ; + \end{tikzpicture} +\end{center} + +Calculons la probabilité d'être testé positif. +\[ + P(\cdots) = +\] + +Calculons la probabilité d'être malade sachant que l'on est testé positif. +\[ + P_{\cdots}(\cdots) = +\] + +\afaire{compléter les pointillés et faire les calculs} + +\end{document} diff --git a/TST/10_Probabilites_conditionnelles/index.rst b/TST/10_Probabilites_conditionnelles/index.rst index 7aff1d3..8698268 100644 --- a/TST/10_Probabilites_conditionnelles/index.rst +++ b/TST/10_Probabilites_conditionnelles/index.rst @@ -67,3 +67,7 @@ Exercices de calculs de probabilités tirés d'annales STMG. Les exercices ont Bilan: Utilisation de la formule des probabilités totales et de la formule de Bayes. +.. image:: ./4B_Bayes.pdf + :height: 200px + :alt: Exemple de l'utilisation de la formule de Bayes +