122 lines
2.9 KiB
TeX
122 lines
2.9 KiB
TeX
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\title{Variable aléatoires - Arbres}
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\tribe{1ST}
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\date{Décembre 2019}
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\pagestyle{empty}
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%\geometry{left=15mm,right=15mm, bottom=8mm, top=5mm}
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\begin{document}
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\section*{Arbres de probabilité}
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Quand on répète plusieurs fois une expérience aléatoire. On a vu qu'il pouvait être intéressante de faire un \textbf{arbre de probabilité}.
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On note la probabilité sur les branches et les issues sur les noeuds.
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Un arbre de probabilité doit respecter une règle:
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\begin{center}
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Somme des probabilités des branches issues d'un noeud doit être égale à 1
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\end{center}
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Pour calculer la probabilité d'un chemin:
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\begin{center}
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On multiplie les probabilités des branches parcourues.
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\end{center}
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\subsection{De l'arbre vers la loi de probabilité}
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On note $X$ la variable aléatoire qui compte le nombre de $S$ dans l'arbre suivant.
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\begin{tikzpicture}[yscale=0.8]
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\node {$\bullet$}
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child {node {$S$}
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child {node {$S$}
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child {node {$S$}
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edge from parent
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node[above] {0.1}
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}
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child[missing] {}
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child {node {$E$}
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edge from parent
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node[above] {0.9}
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}
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edge from parent
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node[above] {0.1}
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}
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child[missing] {}
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child[missing] {}
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child {node {$E$}
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child {node {$S$}
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edge from parent
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node[above] {0.1}
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}
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child[missing] {}
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child {node {$E$}
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edge from parent
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node[above] {0.9}
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}
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edge from parent
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node[above] {0.9}
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}
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edge from parent
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node[above] {0.1}
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}
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child[missing] {}
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child[missing] {}
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child[missing] {}
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child[missing] {}
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child[missing] {}
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child {node {$E$}
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child {node {$S$}
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child {node {$S$}
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edge from parent
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node[above] {0.1}
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}
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child[missing] {}
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child {node {$E$}
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edge from parent
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node[above] {0.9}
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}
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edge from parent
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node[above] {0.1}
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}
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child[missing] {}
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child[missing] {}
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child {node {$E$}
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child {node {$S$}
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edge from parent
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node[above] {0.1}
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}
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child[missing] {}
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child {node {$E$}
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edge from parent
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node[above] {0.9}
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}
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edge from parent
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node[above] {0.9}
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}
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edge from parent
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node[above] {0.9}
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}
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;
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\end{tikzpicture}
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On peut en déduire la loi de probabilité
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\begin{center}
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\begin{tabular}{|c|*{4}{p{2cm}|}}
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\hline
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Nombre de $S$ &&&&\\
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\hline
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$p_i$ &&&& \\
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\hline
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\end{tabular}
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\end{center}
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\afaire{Compléter la loi de probabilité}
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\end{document}
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