Feat: Bilans sur les proba conditionnelles
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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{février 2021}
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\pagestyle{empty}
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\begin{document}
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\maketitle
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\end{document}
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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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\section{Notations}
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\subsection*{Les ensembles}
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Soit $E$ un ensemble et $A$ et $B$ deux sous ensemble de $E$.
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\begin{center}
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\includegraphics[ scale=0.6 ]{./fig/ensembles}
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\end{center}
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\begin{itemize}
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\item \textbf{Complémentaire de $A$} contient tous les éléments qui n'ont pas les caractéristiques de $A$.
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\begin{center}
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\includegraphics[ scale=0.6 ]{./fig/Abar}
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\end{center}
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\item \textbf{Intersection de $A$ et $B$} contient tous les éléments qui ont les caractéristiques de $A$ \textbf{ET} de $B$.
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\begin{center}
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\includegraphics[ scale=0.6 ]{./fig/inter}
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\end{center}
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\item \textbf{Union de $A$ et $B$} contient tous les éléments qui ont les caractéristiques de $A$ \textbf{OU} de $B$.
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\begin{center}
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\includegraphics[ scale=0.6 ]{./fig/union}
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\end{center}
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\end{itemize}
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\pagebreak
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\subsection*{Les probabilités}
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\begin{definition}{Probabilités conditionnelles}
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Soit $A$ et $B$ deux ensembles d'un population totale $E$ avec $A$ un ensemble non vide.
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\begin{itemize}
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\item Probabilités de l'évènement $A$
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\[
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P(A) = \frac{\mbox{Effectif de $A$}}{\mbox{Effectif total}}
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\]
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\item Probabilités de l'évènement $B$ sachant $A$
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\[
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P_A(B) = \frac{\mbox{Effectif des éléments qui sont dans $A$ et $B$}}{\mbox{Effectifs dees éléments qui sont dans $A$}}
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\]
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\begin{center}
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\includegraphics[ scale=0.6 ]{./fig/condi_A}
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\end{cente}
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\end{itemize}
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\end{definition}
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\paragraph{Exemple}~\\
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\begin{minipage}{0.5\linewidth}
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\begin{tabular}{|*{4}{c|}}
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\hline
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& Homme & Femme & Total \\
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\hline
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Employé & 10 & 15 & 25 \\
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\hline
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Vacataire & 14 & 17 & 31 \\
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\hline
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Total & 24 & 32 & 56 \\
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\hline
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\end{tabular}
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\end{minipage}
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\begin{minipage}{0.5\linewidth}
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On note
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\[
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A = \left\{ \mbox{Homme} \right\} \qquad
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\]
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\[
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B = \left\{ \mbox{Employé} \right\} \qquad
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\]
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\end{minipage}
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\bigskip
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\[
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P(A) =
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\]
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Interprétation:
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\[
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P_A(B) =
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\]
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Interprétation:
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\bigskip
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\afaire{}
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\end{document}
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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}{1}
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\section{Arbre et probabilité conditionnelles}
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Les probabilités conditionnelles peuvent se représenter sous forme d'arbre de probabilité.
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Soit $A$ deux évènements de $E$ avec $P(A) \neq 0$ et $B$, $C$ et $D$ trois autres évènements de $E$. Alors on peut considérer l'arbre de probabilité ci-contre et on obtient les propriétés suivantes:
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\begin{minipage}{0.3\textwidth}
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\begin{tikzpicture}[grow=right, sloped, xscale=2, yscale=1.5]
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\node {.}
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child [red] {node {$A$}
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child {node {$B$}
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edge from parent
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node[above] {$P_A(B)$}
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}
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child [black] {node {$C$}
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edge from parent
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node[above] {$P_A(C)$}
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}
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child [black] {node {$D$}
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edge from parent
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node[above] {$P_A(D)$}
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}
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edge from parent
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node[above] {$P(A)$}
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}
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child[missing] {}
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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] {$P_{\overline{A}}(B)$}
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}
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child [black] {node {$C$}
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edge from parent
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node[above] {$P_{\overline{A}}(C)$}
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}
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child [black] {node {$D$}
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edge from parent
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node[above] {$P_{\overline{A}}(D)$}
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}
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edge from parent
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node[above] {$P(\overline{A})$}
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}%
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;
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\end{tikzpicture}
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\end{minipage}
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\hfill
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\begin{minipage}{0.6\textwidth}
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\begin{itemize}
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\item La somme des probabilités des branches issues d'un même noeud est égale à 1.
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On a alors
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\[
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P(A) + P(\overline{ A }) = 1
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\]
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ou encore
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\[
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P_A(B) + P_A(C) + P_A(D) = 1
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\]
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\item La probabilité d'un chemin est égale au produit des probabilités des branches parcourues.
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On a alors (chemin rouge)
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\[
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P(A \cap B) = P(A) \times P_A(B)
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\]
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Ou encore la formule de Bayes
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\[
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P_A(B) = \frac{P(A \cap B)}{ P(A) }
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\]
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\item La probabilité d'un évènement est égale à la somme des probabilités des chemins qui conduisent à cet évènement.
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C'est la loi des probabilités totale qui peut se traduire dans notre exemple par
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\[
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P(B) = P(A\cap B) + P(\overline{A} \cap B)
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\]
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ou
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\[
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P(C) = P(A\cap C) + P(\overline{A} \cap C)
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\]
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\end{itemize}
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\end{minipage}
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\paragraph{Exemple}~\\
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\begin{tabular}{|*{4}{p{2cm}|}c|}
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\hline
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& Moins de 20ans & entre 20 et 50 ans & Plus de 50ans & Total \\
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\hline
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Guéris & 20 & 16 & 30 & 66\\
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\hline
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Malade & 24 & 10 & 5 & 39\\
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\hline
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Total & 44 & 26 & 35 & 105\\
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\hline
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\end{tabular}
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On note
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\[
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A = \left\{ \mbox{Malade} \right\} \qquad P = \left\{ \mbox{Plus de 50ans} \right\} \qquad
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E = \left\{ \mbox{Entre 20 et 50ans } \right\} \qquad M = \left\{ \mbox{Moins de 20ans} \right\} \qquad
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\]
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\begin{center}
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\begin{tikzpicture}[sloped, xscale=2, yscale=1.5]
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\node {.}
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child [red] {node {$A$}
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child {node {$P$}
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edge from parent
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node[above] {...}
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child [black] {node {$E$}
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}
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node[above] {...}
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}
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node[above] {...}
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}
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child[missing] {}
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child[missing] {}
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child { node {$\overline{A}$}
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child {node {$P$}
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edge from parent
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node[above] {...}
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}
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child [black] {node {$E$}
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edge from parent
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node[above] {...}
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}
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child [black] {node {$M$}
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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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;
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\end{tikzpicture}
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\end{center}
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\afaire{Compléter l'arbre avec les probabilités}
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\end{document}
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@ -2,7 +2,7 @@ Probabilités conditionnelles
|
||||||
############################
|
############################
|
||||||
|
|
||||||
:date: 2021-02-07
|
:date: 2021-02-07
|
||||||
:modified: 2021-03-08
|
:modified: 2021-03-10
|
||||||
:authors: Benjamin Bertrand
|
:authors: Benjamin Bertrand
|
||||||
:tags: Probabilité, Simulation
|
:tags: Probabilité, Simulation
|
||||||
:category: TST
|
:category: TST
|
||||||
|
|
@ -25,6 +25,10 @@ Le workflow de la séance:
|
||||||
|
|
||||||
Bilan: Notations ensemblistes et probabilistes
|
Bilan: Notations ensemblistes et probabilistes
|
||||||
|
|
||||||
|
.. image:: ./1B_notation.pdf
|
||||||
|
:height: 200px
|
||||||
|
:alt: Notations ensemblistes et probabilistes
|
||||||
|
|
||||||
Étape 2: Vrai/Faux à partir d'un arbre
|
Étape 2: Vrai/Faux à partir d'un arbre
|
||||||
======================================
|
======================================
|
||||||
|
|
||||||
|
|
@ -36,6 +40,10 @@ Travail similaire à l'étape 1 mais cette fois-ci à partir d'un arbre.
|
||||||
|
|
||||||
Bilan: représentation d'une situation avec des arbres
|
Bilan: représentation d'une situation avec des arbres
|
||||||
|
|
||||||
|
.. image:: ./2B_arbre.pdf
|
||||||
|
:height: 200px
|
||||||
|
:alt: Arbre de probabilités conditionnelles
|
||||||
|
|
||||||
Étape 3: Construction d'un arbre
|
Étape 3: Construction d'un arbre
|
||||||
================================
|
================================
|
||||||
|
|
||||||
|
|
|
||||||
Loading…
Reference in New Issue