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z{OVGzyN;{6PB6Jn!Cp|4eed2ab8htXDlnay>%adCo)8NUi0=zngTeoTN(f-k(aEUE GV*D>kZDx}I literal 0 HcmV?d00001 diff --git a/Complementaire/01_Binomiale_et_echantillonnage/3B_coefBino_formule.tex b/Complementaire/01_Binomiale_et_echantillonnage/3B_coefBino_formule.tex new file mode 100644 index 0000000..75c5c91 --- /dev/null +++ b/Complementaire/01_Binomiale_et_echantillonnage/3B_coefBino_formule.tex @@ -0,0 +1,82 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Binomiale et echantillonnage - Cours} +\date{Novembre 2020} + +\pagestyle{empty} + +\begin{document} + +\maketitle + +\setcounter{section}{3} +\section{Coefficients binomiaux} + +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Définition} + Soit $n$ et $k$ deux entiers naturels tels que $0 \leq k \leq n$. + + \textbf{Le coefficient binomial} $\coefBino{n}{k}$, se lit "$k$ parmi $n$", et le nombre de façon d'obtenir $k$ succès quand on fait $n$ répétitions. + + Par convention, $\coefBino{0}{0} = 1$. +\end{bclogo} + +\paragraph{Exemples}% +\afaire{Tracer l'arbre qui correspond à une loi binomiale $\mathcal{B}(3, 0.1)$. Lister le nombre succès possibles et le nombre de chemins qui y mène puis faire lien avec les coefficients binomiaux.} + +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Propriétés} + Soit $n$ et $k$ deux entiers naturels tels que $0 \leq k \leq n$. + \[ + \coefBino{n}{0} = \coefBino{n}{n} = 1 \qquad \qquad \coefBino{n-1}{k-1} + \coefBino{n-1}{k} = \coefBino{n}{k} + \] + Il est possible de calculer ces coefficients binomiaux grâce au triangle de Pascale. + + \begin{center} + \begin{tabular}{|*{6}{c|}} + \hline + n \verb|\| k & 1 & 2 & 3 & 4 & 5 \\ + \hline + 1 & & & & & \\ + \hline + 2 & & & & & \\ + \hline + 3 & & & & & \\ + \hline + 4 & & & & & \\ + \hline + 5 & & & & & \\ + \hline + \end{tabular} + \end{center} + \afaire{Compléter le tableau en utilisant les règles de calculs.} +\end{bclogo} + +\paragraph{Exemples}% +Nombre de façon de d'avoir 4 succès en 5 répétitions $\coefBino{...}{...} = ...$ +\afaire{à compléter} + +\section{Formules des probabilités pour la loi binomiale} + +\begin{bclogo}[barre=none, arrondi=0.1, logo=]{Propriétés} + Soit $X \sim \mathcal{B}(n, p)$ alors pour tout entier naturel $k$ inférieur à $n$ + \[ + P(X = k) = \coefBino{n}{k} p^k (1-p)^{n-k} + \] +\end{bclogo} + + +\paragraph{Exemples}% +Soit $X \sim \mathcal{B}(5, 0.1)$ alors +\[ + P(X = 3) = +\] +\afaire{à compléter} + + + + + + + +\end{document} diff --git a/Complementaire/01_Binomiale_et_echantillonnage/exercises.tex b/Complementaire/01_Binomiale_et_echantillonnage/exercises.tex index bd1ea0f..3479c4c 100644 --- a/Complementaire/01_Binomiale_et_echantillonnage/exercises.tex +++ b/Complementaire/01_Binomiale_et_echantillonnage/exercises.tex @@ -149,7 +149,6 @@ Questions coup-de-pouce pour étudier chaque question: \item Faire un arbre qui modélise la situation. \item Déterminer la probabilité que Bob gagne une seule partie. \item Avec quelle loi peut-on modéliser la variable aléatoire $X$? Préciser les paramètres. - \item Démontrer que l'espérance de $X$ est de 1,4. \item Si Bob joue tous les jours deux parties, combien en moyenne peut-il espérer en gagner quotidiennement? \end{enumerate} \end{exercise} diff --git a/Complementaire/01_Binomiale_et_echantillonnage/index.rst b/Complementaire/01_Binomiale_et_echantillonnage/index.rst index 21fe2d6..f050e41 100644 --- a/Complementaire/01_Binomiale_et_echantillonnage/index.rst +++ b/Complementaire/01_Binomiale_et_echantillonnage/index.rst @@ -49,6 +49,11 @@ Plusieurs situations pouvant être modélisées ou pas par une loi binomiale où Cours: formule de calcul de probabilité pour la loi binomiale et graphique pour les représenter. +.. image:: ./3B_coefBino_formule.pdf + :height: 200px + :alt: Cours sur les coefficients binomiaux et la formule de calculs de probabiltés + + Étape 4: Augmenter le nombre de répétitions ===========================================