121 lines
3.0 KiB
TeX
121 lines
3.0 KiB
TeX
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\author{Benjamin Bertrand}
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\title{Probabilités conditionnelles - Cours}
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\date{Mars 2021}
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\pagestyle{empty}
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\begin{document}
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\maketitle
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\setcounter{section}{2}
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\subsection*{Exemple de l'utilisation des formules - test anti-covid}
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On s'intéresse aux tests anti-covid et on considère les évènements suivants
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\[
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A = \left\{ \mbox{Malade du covid} \right\}
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\]
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\[
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B = \left\{ \mbox{Testé positif} \right\}
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\]
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On considère que l'on a 1\% de chance d'être malade du covid.
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Les tests anti-covid ne sont pas fiables à 100\% (aucun test ne peut l'être). Voici les données du fabriquant
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\begin{itemize}
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\item \textbf{Sensibilité} - probabilité qu'une personne malade soit testée positive - vrai positifs
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\[
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70\%
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\]
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\item \textbf{Spécificité} - probabilité qu'une personne saine soit testée négative - vrai négatifs
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\[
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95\%
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\]
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\end{itemize}
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\begin{center}
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\textbf
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On est testé positif, quelle est la probabilité que l'on soit malade?
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\end{center}
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Complétons l'arbre de probabilité de gauche avec les données.
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\begin{center}
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\begin{tikzpicture}[sloped, grow=right]
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\node {.}
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child {node {$A$}
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child {node {$B$}
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edge from parent
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node[above] {...}
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}
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child {node {$\overline{B}$}
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edge from parent
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node[above] {...}
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}
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edge from parent
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node[above] {...}
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}
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child[missing] {}
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child { node {$\overline{A}$}
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child {node {$B$}
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edge from parent
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node[above] {...}
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}
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child {node {$\overline{B}$}
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edge from parent
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node[above] {...}
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}
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edge from parent
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node[above] {...}
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} ;
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\end{tikzpicture}
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\hspace{3cm}
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\begin{tikzpicture}[sloped, grow=left]
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\node {.}
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child {node {$B$}
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child {node {$A$}
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edge from parent
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node[above] {...}
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}
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child {node {$\overline{A}$}
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edge from parent
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node[above] {...}
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}
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edge from parent
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node[above] {...}
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}
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child[missing] {}
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child { node {$\overline{B}$}
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child {node {$A$}
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edge from parent
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node[above] {...}
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}
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child {node {$\overline{A}$}
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edge from parent
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node[above] {...}
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}
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edge from parent
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node[above] {...}
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} ;
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\end{tikzpicture}
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\end{center}
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Calculons la probabilité d'être testé positif.
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\[
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P(\cdots) =
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\]
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Calculons la probabilité d'être malade sachant que l'on est testé positif.
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\[
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P_{\cdots}(\cdots) =
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\]
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\afaire{compléter les pointillés et faire les calculs}
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\end{document}
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