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+\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 $A$ + \[ + P_A(B) = \frac{\mbox{Effectif des éléments qui sont dans $A$ et $B$}}{\mbox{Effectifs des éléments qui sont dans $A$}} + \] + \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 +On choisit au hasard une personne de cette entreprise. + +\[ + P(A) = + \] +Interprétation: +\[ + P_A(B) = + \] +Interprétation: + \bigskip + +\afaire{} + + + +\end{document} diff --git a/Complementaire/02_Inference_Baysienne/1E_probabilite_conditionnelle.pdf b/Complementaire/02_Inference_Baysienne/1E_probabilite_conditionnelle.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c8acd8c2c832f87a6fb4f2437f48d50eb9faf845 GIT binary patch literal 58979 zcmce-V~lRywk=#-WqZ}Cs;6w*wriDb+qP}nwr$(CZR5WC+y4gltVMiiiFrHrgi98CfA%z*#%{IiRhrK6EOfL6p(&(TQ8$iT+X2%3in z+QHG@NY4t|b$tP0T8?ab7X%363-}h=$lCD#%>2{(*U?%3+4w&Lf$iS~f&D)Qf&O3Q z{99E zgwet}qcg^}K*ZW_+=9nw9Tv;cfP7(_yft0_Qowpa&seaeWv4mDGvdw! zK^Ytcf+vJUjkBrFhI-JX-ui>C?CrL(42>43o~EW*$F_L?Et@`Nd&9~_m^<;KocNM9 zN<1gA|3Khz()b9OM;YaOq%4ILj6dRA&tLL`1;e9`ade3GC@wm5{Q-W@6FgVCHvVlN zZ&6K2EItsrM3q5zf&RGp-JtqwLVfbfNn?}xc!*|ImpyHYHDk&8ft9yb{oMj76dR^O zdQoi~UQ^^}~m=Moph)Jr6+xEa(o~op9Wsp;( zKo@G!c-S_d4Ugfs8FOBHbJ{d&a{A;h#eE(YpQWa4S=IF>q@(zxLbzP*j;$KpUnO>d 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zB_3iv&yH=Jy?4K(5cvK*nKrh&jy2=g>>~Ew@bW4jv(zd2KR6=1(Qq}B(ENIt$ zf0yGMd@799cT{ESMDyVs-0hT*k-n3N*zBHRwAtp}5qeJ-9)7B>n#7#*8Hbs9k+PGr zg`7CnZKkam!+}$H%}4l+fgO*qJR0uq9I0(^i@%-D7Qr$(|oWO zl)!Mg@5B0}e|#MbyYZLto2o2X>%UkqY8bnvH)Jw{DH+?Da!U7eM1Yrz@?GJu`JVQ0 zqI(`rEJGDJmBBiN2P{!rUaS8F%lvx{_P-KweNCtpE>O|I66ygkdm}3= z0G5*ClL5;L$Vv;z2nfmwhyng@lgCm1rBe(5gZ_(=8}JV{u!BG16`-8mBPf!AS~yxl z|3MZm@JCQ(kLxi@ic$niYfy(Cxr~a0tjS7w$ejAp`+xtTiP zu=K0{h}*n*?st~e{\columncolor{highlightbg}}c|*{2}{p{4cm}|}|p{4cm}|} + \hline + \rowcolor{highlightbg} + Bac \backslash Bac Blanc & Moins de la moyenne & Plus de moyenne & Total\\ + \hline + Moins de la moyenne & 30 & 300 & 60\\ + \hline + Plus de la moyenne & 20 & 70 & 90\\ + \hline + Total & 50 & 100 & 150\\ + \hline + \end{tabular} + \end{center} + + On choisit au hasard un jeune parmi ceux interrogés. On note $A = \left\{ \mbox{Plus de la moyenne au bac blanc} \right\}$ et $B = \left\{ \mbox{Moins de la moyenne au bac blanc} \right\}$. + + Pour chacune des phrases suivantes, justifier si elles sont vraies ou fausses. + \begin{enumerate} + \item La probabilité qu'un élève ait plus de moyenne au bac blanc est supérieur à 30\%. + \item La probabilité qu'il ait la moyenne au bac blanc mais pas au au bac est de moins de 0.1. + \item La probabilité qu'il n'ait pas la moyenne au bac et pas la moyenne au bac blanc est de $\frac{1}{5}$. + \item La probabilité qu'il ait la moyenne au bac blanc ou au bac est de plus de 70\%. + \item La probabilité qu'un élève qui a eu la moyenne au bac blanc ait ensuite la moyenne au bac est de 0.6. + \item La probabilité qu'un élève qui n'a pas eu la moyenne au bac blanc ne l'ait toujours pas au bac est de 50\%. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Moyen de paiement}, step={1}, origin={Création}, topics={Probabilités conditionnelles}, tags={probabilité, simulation}] + Le gérant d'une grande enseigne de distribution a commandé une étude statistique des moyens de paiement de ses clients. Les résultats ont été représenté dans l'arbre ci-dessous. + + \bigskip + \begin{tikzpicture}[sloped] + \node {.} + child {node {Moins de 20\euro} + child {node {Espèce} + edge from parent + node[above] {0.6} + } + child[missing] {} + child {node {Paiement sans contact} + edge from parent + node[above] {0.3} + } + child[missing] {} + child {node {Carte bleu} + edge from parent + node[above] {0.1} + } + edge from parent + node[above] {0.4} + } + child[missing] {} + child[missing] {} + child[missing] {} + child[missing] {} + child[missing] {} + child { node {Plus de 20\euro} + child {node {Espèce} + edge from parent + node[above] {0.4} + } + child[missing] {} + child {node {Paiement sans contact} + edge from parent + node[above] {0.1} + } + child[missing] {} + child {node {Carte bleu} + edge from parent + node[above] {0.5} + } + edge from parent + node[above] {0.6} + } ; + \end{tikzpicture} + \bigskip + + On sélectionne un client de cette enseigne au hasard. On note + \[ + M = \left\{ \mbox{ Moins de 20\euro } \right\} \qquad E = \left\{ \mbox{ Espèce }\right\} \qquad B = \left\{ \mbox{ carte bleu }\right\} + \] + Pour chacune des phrases suivantes, justifier si elles sont vraies ou fausses. + \begin{enumerate} + \item La probabilité qu'un client achète pour plus de 20\euro est de 0.6. + \item Si l'achat est de moins de 20\euro, il y a une probabilité de 10\% qu'il soit fait en paiement sans contact. + \item Si les achats sont de plus de 20\euro, la probabilité qu'il ait été fait avec de l'espèce est de 0.4. + \item La probabilité qu'un achat soit de plus de 20\euro et payé avec une carte bleu est de 90\%. + \item La probabilité qu'un achat soit de moins de 20\euro et payé avec une carte bleu est de 0.2. + \item La probabilité qu'un achat soit de moins de 20\euro et payé avec de l'espère est de 1. + \item La probabilité ait été payé avec le paiement sans contact est de 72\%. + \end{enumerate} +\end{exercise} +\collectexercisesstop{banque} 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b/Complementaire/02_Inference_Baysienne/index.rst index 1bcd8f1..2e5cf4f 100644 --- a/Complementaire/02_Inference_Baysienne/index.rst +++ b/Complementaire/02_Inference_Baysienne/index.rst @@ -13,8 +13,16 @@ Inférence Bayésienne Vrai/faux sur des données représentées sous forme d'un tableau puis d'un arbre +.. image:: ./1E_probabilite_conditionnelle.pdf + :height: 200px + :alt: Vrai/Faux sur un tableau et un arbre + Bilan: Notations ensemblistes et calculs de probabilités. +.. image:: ./1B_notation.pdf + :height: 200px + :alt: Bilan sur les notations ensemblistes et les calculs de probabilités + Étape 2: Étonnement sur les tests =================================