Feat: Bilans sur les proba conditionnelles
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\documentclass[a4paper,10pt]{article}
\usepackage{myXsim}
\author{Benjamin Bertrand}
\title{Probabilités conditionnelles - Cours}
\date{février 2021}
\pagestyle{empty}
\begin{document}
\maketitle
\end{document}

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\documentclass[a4paper,10pt]{article}
\usepackage{myXsim}
\author{Benjamin Bertrand}
\title{Probabilités conditionnelles - Cours}
\date{Mars 2021}
\pagestyle{empty}
\begin{document}
\maketitle
\section{Notations}
\subsection*{Les ensembles}
Soit $E$ un ensemble et $A$ et $B$ deux sous ensemble de $E$.
\begin{center}
\includegraphics[ scale=0.6 ]{./fig/ensembles}
\end{center}
\begin{itemize}
\item \textbf{Complémentaire de $A$} contient tous les éléments qui n'ont pas les caractéristiques de $A$.
\begin{center}
\includegraphics[ scale=0.6 ]{./fig/Abar}
\end{center}
\item \textbf{Intersection de $A$ et $B$} contient tous les éléments qui ont les caractéristiques de $A$ \textbf{ET} de $B$.
\begin{center}
\includegraphics[ scale=0.6 ]{./fig/inter}
\end{center}
\item \textbf{Union de $A$ et $B$} contient tous les éléments qui ont les caractéristiques de $A$ \textbf{OU} de $B$.
\begin{center}
\includegraphics[ scale=0.6 ]{./fig/union}
\end{center}
\end{itemize}
\pagebreak
\subsection*{Les probabilités}
\begin{definition}{Probabilités conditionnelles}
Soit $A$ et $B$ deux ensembles d'un population totale $E$ avec $A$ un ensemble non vide.
\begin{itemize}
\item Probabilités de l'évènement $A$
\[
P(A) = \frac{\mbox{Effectif de $A$}}{\mbox{Effectif total}}
\]
\item Probabilités de l'évènement $B$ sachant $A$
\[
P_A(B) = \frac{\mbox{Effectif des éléments qui sont dans $A$ et $B$}}{\mbox{Effectifs dees éléments qui sont dans $A$}}
\]
\begin{center}
\includegraphics[ scale=0.6 ]{./fig/condi_A}
\end{cente}
\end{itemize}
\end{definition}
\paragraph{Exemple}~\\
\begin{minipage}{0.5\linewidth}
\begin{tabular}{|*{4}{c|}}
\hline
& Homme & Femme & Total \\
\hline
Employé & 10 & 15 & 25 \\
\hline
Vacataire & 14 & 17 & 31 \\
\hline
Total & 24 & 32 & 56 \\
\hline
\end{tabular}
\end{minipage}
\begin{minipage}{0.5\linewidth}
On note
\[
A = \left\{ \mbox{Homme} \right\} \qquad
\]
\[
B = \left\{ \mbox{Employé} \right\} \qquad
\]
\end{minipage}
\bigskip
\[
P(A) =
\]
Interprétation:
\[
P_A(B) =
\]
Interprétation:
\bigskip
\afaire{}
\end{document}

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\documentclass[a4paper,10pt]{article}
\usepackage{myXsim}
\author{Benjamin Bertrand}
\title{Probabilités conditionnelles - Cours}
\date{Mars 2021}
\pagestyle{empty}
\begin{document}
\maketitle
\setcounter{section}{1}
\section{Arbre et probabilité conditionnelles}
Les probabilités conditionnelles peuvent se représenter sous forme d'arbre de probabilité.
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:
\begin{minipage}{0.3\textwidth}
\begin{tikzpicture}[grow=right, sloped, xscale=2, yscale=1.5]
\node {.}
child [red] {node {$A$}
child {node {$B$}
edge from parent
node[above] {$P_A(B)$}
}
child [black] {node {$C$}
edge from parent
node[above] {$P_A(C)$}
}
child [black] {node {$D$}
edge from parent
node[above] {$P_A(D)$}
}
edge from parent
node[above] {$P(A)$}
}
child[missing] {}
child[missing] {}
child { node {$\overline{A}$}
child {node {$B$}
edge from parent
node[above] {$P_{\overline{A}}(B)$}
}
child [black] {node {$C$}
edge from parent
node[above] {$P_{\overline{A}}(C)$}
}
child [black] {node {$D$}
edge from parent
node[above] {$P_{\overline{A}}(D)$}
}
edge from parent
node[above] {$P(\overline{A})$}
}%
;
\end{tikzpicture}
\end{minipage}
\hfill
\begin{minipage}{0.6\textwidth}
\begin{itemize}
\item La somme des probabilités des branches issues d'un même noeud est égale à 1.
On a alors
\[
P(A) + P(\overline{ A }) = 1
\]
ou encore
\[
P_A(B) + P_A(C) + P_A(D) = 1
\]
\item La probabilité d'un chemin est égale au produit des probabilités des branches parcourues.
On a alors (chemin rouge)
\[
P(A \cap B) = P(A) \times P_A(B)
\]
Ou encore la formule de Bayes
\[
P_A(B) = \frac{P(A \cap B)}{ P(A) }
\]
\item La probabilité d'un évènement est égale à la somme des probabilités des chemins qui conduisent à cet évènement.
C'est la loi des probabilités totale qui peut se traduire dans notre exemple par
\[
P(B) = P(A\cap B) + P(\overline{A} \cap B)
\]
ou
\[
P(C) = P(A\cap C) + P(\overline{A} \cap C)
\]
\end{itemize}
\end{minipage}
\paragraph{Exemple}~\\
\begin{tabular}{|*{4}{p{2cm}|}c|}
\hline
& Moins de 20ans & entre 20 et 50 ans & Plus de 50ans & Total \\
\hline
Guéris & 20 & 16 & 30 & 66\\
\hline
Malade & 24 & 10 & 5 & 39\\
\hline
Total & 44 & 26 & 35 & 105\\
\hline
\end{tabular}
On note
\[
A = \left\{ \mbox{Malade} \right\} \qquad P = \left\{ \mbox{Plus de 50ans} \right\} \qquad
E = \left\{ \mbox{Entre 20 et 50ans } \right\} \qquad M = \left\{ \mbox{Moins de 20ans} \right\} \qquad
\]
\begin{center}
\begin{tikzpicture}[sloped, xscale=2, yscale=1.5]
\node {.}
child [red] {node {$A$}
child {node {$P$}
edge from parent
node[above] {...}
}
child [black] {node {$E$}
edge from parent
node[above] {...}
}
child [black] {node {$M$}
edge from parent
node[above] {...}
}
edge from parent
node[above] {...}
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child[missing] {}
child[missing] {}
child { node {$\overline{A}$}
child {node {$P$}
edge from parent
node[above] {...}
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child [black] {node {$E$}
edge from parent
node[above] {...}
}
child [black] {node {$M$}
edge from parent
node[above] {...}
}
edge from parent
node[above] {...}
}%
;
\end{tikzpicture}
\end{center}
\afaire{Compléter l'arbre avec les probabilités}
\end{document}

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@ -2,7 +2,7 @@ Probabilités conditionnelles
############################
:date: 2021-02-07
:modified: 2021-03-08
:modified: 2021-03-10
:authors: Benjamin Bertrand
:tags: Probabilité, Simulation
:category: TST
@ -25,6 +25,10 @@ Le workflow de la séance:
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
======================================
@ -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
.. image:: ./2B_arbre.pdf
:height: 200px
:alt: Arbre de probabilités conditionnelles
Étape 3: Construction d'un arbre
================================