From 2a7d2b9dba699439cf82876d17dd3d22dbfa4acd Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Sun, 7 Feb 2021 08:49:15 +0100 Subject: [PATCH] Feat: fin du chapitre sur la loi binomiale pour les TST --- TST/08_Loi_binomiale/2B_esperance.tex | 24 +------ TST/08_Loi_binomiale/3B_coef_bino.pdf | Bin 0 -> 59777 bytes TST/08_Loi_binomiale/3B_coef_bino.tex | 61 ++++++++++++++++++ ...bis_loi_binomiale.pdf => 3E_coef_bino.pdf} | Bin ...bis_loi_binomiale.tex => 3E_coef_bino.tex} | 0 TST/08_Loi_binomiale/4B_triangle_pascal.pdf | Bin 0 -> 58010 bytes TST/08_Loi_binomiale/4B_triangle_pascal.tex | 58 +++++++++++++++++ TST/08_Loi_binomiale/4E_triangle_pascal.pdf | Bin 0 -> 36207 bytes TST/08_Loi_binomiale/4E_triangle_pascal.tex | 30 +++++++++ TST/08_Loi_binomiale/5E_application.pdf | Bin 0 -> 34385 bytes TST/08_Loi_binomiale/5E_application.tex | 28 ++++++++ TST/08_Loi_binomiale/exercises.tex | 58 +++++++++++++++++ TST/08_Loi_binomiale/index.rst | 43 ++++++++++-- 13 files changed, 276 insertions(+), 26 deletions(-) create mode 100644 TST/08_Loi_binomiale/3B_coef_bino.pdf create mode 100644 TST/08_Loi_binomiale/3B_coef_bino.tex rename TST/08_Loi_binomiale/{2Ebis_loi_binomiale.pdf => 3E_coef_bino.pdf} (100%) rename TST/08_Loi_binomiale/{2Ebis_loi_binomiale.tex => 3E_coef_bino.tex} (100%) create mode 100644 TST/08_Loi_binomiale/4B_triangle_pascal.pdf create mode 100644 TST/08_Loi_binomiale/4B_triangle_pascal.tex create mode 100644 TST/08_Loi_binomiale/4E_triangle_pascal.pdf create mode 100644 TST/08_Loi_binomiale/4E_triangle_pascal.tex create mode 100644 TST/08_Loi_binomiale/5E_application.pdf create mode 100644 TST/08_Loi_binomiale/5E_application.tex diff --git a/TST/08_Loi_binomiale/2B_esperance.tex b/TST/08_Loi_binomiale/2B_esperance.tex index b581eb1..a053f0a 100644 --- a/TST/08_Loi_binomiale/2B_esperance.tex +++ b/TST/08_Loi_binomiale/2B_esperance.tex @@ -11,27 +11,7 @@ \maketitle -\setcounter{2} -\subsection*{Formule pour calculer des probabilité} - -\begin{propriete} - Soit $X \sim \mathcal{B} (n; p)$ une variable aléatoire, alors on peut calculer la probabilité avec la formule suivante - \\[2cm] -\end{propriete} - -\paragraph{Exemples} - -Soit $X \sim \mathcal{B}(3; 0.9)$ la variable aléatoire utiliser pour modéliser l'exemple précédent. - -\[ - P(X = 0) = -\] -\[ - P(X = 2) = -\] -\afaire{} - - +\setcounter{section}{2} \subsection*{Espérance de la loi binomiale} \begin{propriete} @@ -47,6 +27,6 @@ Soit $X \sim \mathcal{B}(3; 0.9)$. L'espérance de $X$ est alors \[ E[X] = \] -\afaire{} +\afaire{Faire le calcul et interpréter le résultat dans le cadre du contexte expliqué dans l'exemple précédent.} \end{document} diff --git a/TST/08_Loi_binomiale/3B_coef_bino.pdf b/TST/08_Loi_binomiale/3B_coef_bino.pdf new file mode 100644 index 0000000000000000000000000000000000000000..72fea9c1a6e5f40fefb2c1afd31fd6bd87f9abf5 GIT binary patch literal 59777 zcmce-Q?M}2wk5i3+qP}nw!OY(+qS=D+qP}nwr%ylPxpy?9y;Ro%ia}H6^U6HHDb(~ zH8V#Rse*_YEh8N(6zSa3@EQ~g0Rw@ZktGxl4-~zMhrKBQy@tGzrKzzq6uq*Gk@J7G zh}k*XK+y{c*|`&FGyG%E$namME)@O0dj9J!Mksn&LuW^ee=HdPvG}L|U)O&uluVuM zTpW!}od{U}X;Ffrmoc?9cm6}b$V%{kj(F*UX`F@@segK~0q zG&Qt=a^F~lpHZNg*#iOq{|3B+GPO1NKRy4H{`KhW{|x+}fxz+ag24G7gTVMNa{ete zF#e13{|pa|O#e0z|2b!3{I_`%)4w?W&oIICFVg=l?!?6Wp9Ar)Ao%YA!S&pYO!~3ExcY%pCtae*gc6?`v(%xUDus-&}pdtEkQZ)alQ);T{f6Y>0#Ngs|Kzg#LjB+~j$^O3_oO$;`%3nWb%wDF4C3KFevt@(>-R9rpCq}a$v zBvRC%sAHiBOK|x~`JJA2PZy`>hh`?!QW2|~ipJsV$NT#Bs5he_yB^|vTGh)SC5}-7k#$cLGriOIBOAm!I=&>Br 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\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}{|*{8}{p{0.8cm}|}} + \hline + n \verb|\| k & 0 & 1 & 2 & 3 & 4 & 5 & 6\\ + \hline + 0 & 1 & & & & & &\\ + \hline + 1 & & & & & & &\\ + \hline + 2 & & & & & & &\\ + \hline + 3 & & & & & & &\\ + \hline + 4 & & & & & & &\\ + \hline + 5 & & & & & & &\\ + \hline + 6 & & & & & & &\\ + \hline + \end{tabular} + \end{center} + \afaire{Compléter le tableau en utilisant les règles de calculs.} +\end{propriete} + +\paragraph{Exemples}% +\begin{itemize} + \item Nombre de façon de d'avoir 4 succès en 5 répétitions $\coefBino{...}{...} = ...$ +\afaire{à compléter} + \item Soit $X\sim \mathcal{B}(5, 0.3)$. + \[ + P(X = 4) = + \] +\afaire{à compléter en utilisant les coefficients binomiaux.} +\end{itemize} + +\end{document} diff --git a/TST/08_Loi_binomiale/4E_triangle_pascal.pdf 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zK(DT7VhT+!X=`laMxez&z{#cqO|N9(Ve+#FO|M3v#Yn(J!1!}a(az3^fb&1fGZp{C z+|-WXfBFXhF%Tip;$dT87ZDTW5dK*^3kz`wv$3;@vapD7GW=9)a<3h@#AzfFFw z^1n(x37DAvcP%Hu{~LmgJ1fm3{0Aj?uh}kvS_B|W5=5ON*fso48^i>Zoy2&ebea*s zX^s(1I0QPi`_ek1_Fb{X>0D$OhGJ)c&0Qj*{PaY<*lNf8ej|qV*o9e+QUq$E9AJXw zGb_dV6nmYejF9sAfGcfLw;6d&DEXy}H@Mo=cKM8i!Vp0;kvwJb*o>i*YMo)1%&)Sp#0yTpQDojz{%~$bAo1O=451rCM6Y>7lZyE D^5sNy literal 0 HcmV?d00001 diff --git a/TST/08_Loi_binomiale/5E_application.tex b/TST/08_Loi_binomiale/5E_application.tex new file mode 100644 index 0000000..d48f210 --- /dev/null +++ b/TST/08_Loi_binomiale/5E_application.tex @@ -0,0 +1,28 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Loi binomiale - Cours} +\date{Février 2021} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=5, +} + +\setcounter{exercise}{5} + + +\pagestyle{empty} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} +\vfill +\printcollection{banque} +\vfill +\printcollection{banque} +\vfill + +\end{document} diff --git a/TST/08_Loi_binomiale/exercises.tex b/TST/08_Loi_binomiale/exercises.tex index f132d49..dbe9522 100644 --- a/TST/08_Loi_binomiale/exercises.tex +++ b/TST/08_Loi_binomiale/exercises.tex @@ -93,4 +93,62 @@ \end{enumerate} \end{exercise} +\begin{exercise}[subtitle={Triangle de Pascal}, step={4}, origin={Création}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale}] + \noindent + \begin{minipage}{0.4\linewidth} + Dans cet exercice, $n$ représente le nombre de répétitions et $k$ le nombre de succès. + \begin{enumerate} + \item En vous aidant de ce qui a été fait à l'exercice précédent, compléter le tableau ci-dessous avec les coefficients binomiaux. + \item Quelles sont les cases qui seront toujours vide? + \item Quelles sont les cases qu'il est "facile" de remplir? + \item Conjecturer une façon de calculer les autres. + \end{enumerate} + \end{minipage} + \hfill + \begin{minipage}{0.5\linewidth} + \begin{tabular}{|*{8}{p{0.8cm}|}} + \hline + n \verb|\| k & 0 & 1 & 2 & 3 & 4 & 5 & 6\\ + \hline + 0 & 1 & & & & & &\\ + \hline + 1 & & & & & & &\\ + \hline + 2 & & & & & & &\\ + \hline + 3 & & & & & & &\\ + \hline + 4 & & & & & & &\\ + \hline + 5 & & & & & & &\\ + \hline + 6 & & & & & & &\\ + \hline + \end{tabular} + \end{minipage} +\end{exercise} + +\begin{exercise}[subtitle={Vaccination des chiots}, step={5}, origin={Indice Math Complémentaire 84p178}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale}] + Dans un chenil, on vaccine 5 chiots de façon indépendante. Lors des vaccinations précédente, on avait constaté que le chiot avait une chance sur cinq d'avoir une réaction forte au vaccin. + + On note $X$ le nombre de chiots qui auront une réaction forte au vaccin. + \begin{enumerate} + \item Quelle est la loi suivie par $X$? Préciser les paramètres. + \item Calculer $P(X=1)$. Interpréter le résultat. + \item Quelle est la probabilité que 5 chiots aient une réaction forte?? + \item Calculer l'espérance de $X$. Interpréter le résultat. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Temps de trajet}, step={5}, origin={Création}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale}] + Pour aller au travail, je croise 6 feux. En interrogeant les employés municipaux en charge de la voirie, j'ai appris que ces feux étaient indépendants les uns des autres et qu'ils étaient rouges 70\% du temps. + + On note $X$ la variable aléatoire qui compte le nombre de feux rouges que je rencontre en allant travailler. + \begin{enumerate} + \item Quelle est la loi suivie par $X$? Préciser les paramètres. + \item Calculer $P(X=2)$. Interpréter le résultat. + \item Quelle est la probabilité que je rencontre 5 feux rouges ou plus? + \item Combien de feux rouge vais-je avoir en moyenne quand je vais au travail? + \end{enumerate} +\end{exercise} \collectexercisesstop{banque} diff --git a/TST/08_Loi_binomiale/index.rst b/TST/08_Loi_binomiale/index.rst index 93bb26c..6ec104c 100644 --- a/TST/08_Loi_binomiale/index.rst +++ b/TST/08_Loi_binomiale/index.rst @@ -2,7 +2,7 @@ Loi binomiale ############# :date: 2021-01-20 -:modified: 2021-01-25 +:modified: 2021-02-07 :authors: Benjamin Bertrand :tags: Probabilité, Binomiale, Tableur :category: TST @@ -37,11 +37,46 @@ Lors de la lecture du bilan, on donnera la méthode pour calculer des probabilit :height: 200px :alt: Exercices où l'on utilise les arbres pour calculer des probabilités avec la loi binomiale -Cours/Bilan: formule pour calculer des probabilités et l'espérance d'une loi binomiale +Cours/Bilan: L'espérance d'une loi binomiale. .. image:: ./2B_esperance.pdf :height: 200px :alt: formule pour calculer des probabilités et l'espérance d'une loi binomiale -Étape 3: Simulation avec python -=============================== +Étape 3: Étude des nombres de chemins +===================================== + +On commence par un travail technique où il est demandé construire le tableau de la loi de probabilités à partir des paramètres de la loi binomiale. On demande aux élèves de rentrer les calculatrices et d'écrire le calcul qu'ils demanderaient à la calculatrice. On précise aussi que dès que c'est possible, ils doivent essayer de ne pas faire l'arbre de probabilités pour construire leur tableau. + +.. image:: ./3E_coef_bino.pdf + :height: 200px + :alt: Tracer des arbres de probabilités et découverte de la formule pour calculer des probabilités. + + +Bilan du premier exercice: La formule des probabilités est donnée sans évoquer les coefficients binomiaux. Puis on explique que le "nombre de chemins" est appelé coefficient binomial et l'on réécrit la formule de probabilités. + +.. image:: ./3B_coef_bino.pdf + :height: 200px + :alt: formule de probabilité et introduction des coefficients binomiaux. + +Étape 4: Triangle de Pascal +=========================== + +Rangement des coefficients binomiaux dans un tableau et réflexion sur comment calculer les suivants + +.. image:: ./4E_triangle_pascal.pdf + :height: 200px + :alt: Triangle de Pascal + +Bilan: formules de calculs, triangle de Pascal et calcul de probabilité. + +.. image:: ./4B_triangle_pascal.pdf + :height: 200px + :alt: Bilan sur le triangle de Pascal. + +Étape 5: Exercices d'applications +================================= + +.. image:: ./5E_application.pdf + :height: 200px + :alt: Application de la loi binomiale.