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TES/Probabilte_statistiques/Loi_binomiale/1B_modelisation.pdf
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TES/Probabilte_statistiques/Loi_binomiale/1B_modelisation.pdf
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\title{Loi binomiale}
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\date{Février 2020}
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\begin{document}
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\section{Loi binomiale}
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Dans l'activité sur la simulation de sur-réservation d'un vol, la présence ou l'absence d'un passager peut être modélisé avec une \textbf{loi de Bernoulli}.
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\subsection*{Définition}
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La \textbf{loi de Bernoulli de paramètre $p$} notée $\mathcal{B}(p)$ est la loi de probabilité qui modélise les situations où il n'y a que 2 issues succès (qui a pour valeur 1) ou échec (qui a pour valeur 0). Le paramètre $p$ correspond à la probabilité d'un succès. Elle est donc définie par le tableau suivant
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\begin{center}
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\begin{tabular}{|c|*{2}{C{2cm}|}}
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\hline
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Valeurs & 1 & 0 \\
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\hline
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Probabilité & p & 1-p \\
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\hline
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\end{tabular}
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\end{center}
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\bigskip
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On a vu que pour simuler tout un vol, c'est à dire 53 passagers, il fallait répéter 53 fois la loi de Bernoulli. Les répétitions de loi de Bernoulli s'appellent \textbf{schéma de Bernoulli} et sont modéliser par la \textbf{binomiale}.
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\subsection*{Définition}
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La \textbf{loi Binomiale de paramètre $n$ et $p$} notée $\mathcal{B}(n;p)$ est la loi de probabilité qui modélise la répétition indépendantes et identiques de $n$ situations modélisées par une loi de Bernoulli de paramètre $p$.
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\bigskip
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Ces situations peuvent être représenté par un arbre de probabilité.
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\subsubsection*{Exemple}
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Dans une classe de 20 élèves, Sarah ne veut pas être interrogée sur son travail. Le professeur interroge au hasard 3 élèves qu'il choisit de façon indépendantes et identiques.
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On note $X$ le nombre de fois que Sarah est interrogée.
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\afaire{Quelle loi suit $X$? Représenter la situation avec un arbre de probabilité}
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\end{document}
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\title{Loi binomiale - Simulation surréservation \\ Tableur}
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\tribe{TESL}
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\date{Février 2020}
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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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\begin{exercise}[subtitle={Simuler une surréservation}]
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Pour obtenir un taux de remplissage convenable, les compagnies aériennes vendent régulièrement plus de place que n'en comporte l'avion car il arrive que des personnes ne se présentent pas au décollage. Si un passagers a réservé mais qu'il n'y a plus de place dans l'avion, il faudra par contre le dédommager. C'est pour cela qu'il faut évaluer le risque de surréservation.
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On considère une ligne aérienne entre deux villes pour laquelle:
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\begin{itemize}
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\item Tous les avions ont 50 places.
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\item 53 réservations sont vendues pour chaque vol (on supposera qu'elles sont toutes vendues)
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\item Chaque personne ayant réservé a 9 chance sur 10 de se présenter à l'embarquement ( donc 1 chance sur 10 de ne pas se présenter).
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\item Chaque personne ayant réservé une place se présente au non à l'embarquement indépendamment des autres personnes ayant réservé sur le même vol.
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\end{itemize}
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Pour évaluer les risques liés à une surréservation, nous allons \textbf{simuler} avec le tableur plusieurs vols sur cette ligne.
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\begin{enumerate}
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\item On commence par simuler un vol où 53 places ont été vendues.
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\begin{minipage}{0.6\textwidth}
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Pour savoir si une personne se présente ou non à l'embarquement, nous utiliserons la commande \ovalbox{\verb|=SI(ALEA()>0.9;0;1)|}. Cette commande renvoie:
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\begin{itemize}
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\item 0 si le passager ne s'est pas présenté
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\item 1 s'il s'est présenté.
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\end{itemize}
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\begin{enumerate}
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\item Réaliser la simulation pour le premier vol
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\item Combien de personnes se sont-elles présentées à l'embarquement?
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\item Quelle formule peut-on rentrer en \texttt{B56} pour calculer ce nombre?
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\end{enumerate}
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\end{minipage}
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\begin{minipage}{0.4\textwidth}
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\includegraphics[scale=0.27]{./fig/vol1}
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\end{minipage}
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\item
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\begin{enumerate}
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\item Réaliser cette simulation pour 100 vols de cette ligne.
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\hspace{-2cm}
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\includegraphics[scale=0.17]{./fig/vol100}
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\item Quelle formule doit-on entrer en \texttt{CX56} pour calculer la moyenne du nombre de passager?
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\item Pensez vous que le risque de surréservation est grand?
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\end{enumerate}
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\item On veut maintenant évaluer le risque de surréservation. Pour savoir si un vol est en surréservation, on utilise la commande \ovalbox{\verb|=SI(nbr_passagers > 50;1;0)|} (avec \verb|nbr_passagers| à remplacer le nom de la case)
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\begin{enumerate}
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\item Compléter le tableau pour connaître les vols en surréservation.
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\hspace{-2cm}
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\includegraphics[scale=0.15]{./fig/vol100_overbooking}
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\item Quelle formule doit-on rentrer en \texttt{CX59} pour compter le nombre de vols en surréservation?
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\item Quelle formule doit-on entrer dans \texttt{CX61} pour calculer la fréquence des vols en surréservation?
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\end{enumerate}
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\item Pensez vous que la compagnie va devoir souvent dédommager des voyageurs?
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\item On suppose qu'un billet coûte 100\euro. Quand un client doit être dédommagé, cela coûte 250\euro à la compagnie. Expliquer pourquoi il est intéressant pour la compagnie de vendre des billets en surréservation.
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\end{enumerate}
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\end{exercise}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "master"
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%%% End:
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BIN
TES/Probabilte_statistiques/Loi_binomiale/2E_modelisation.pdf
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TES/Probabilte_statistiques/Loi_binomiale/2E_modelisation.pdf
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\title{Loi binomiale - modélisation}
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\tribe{TESL}
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\date{Février 2020}
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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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\begin{exercise}[subtitle={Temps de trajet}]
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Pour aller au travail, je croise 3 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.
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On note $X$ la variable aléatoire qui compte le nombre de feux rouges que je rencontre en allant travailler.
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\begin{enumerate}
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\item Modéliser la situation avec un arbre.
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\item Construire la loi de probabilité associée à $X$ (c'est un tableau reliant les valeurs possibles prisent par $X$ et la probabilité associée).
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\item Quelle est la probabilité que je rencontre plus de 2 feux rouges?
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\item Combien de feux rouge vais-je avoir en moyenne quand je vais au travail?
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\end{enumerate}
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\end{exercise}
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\begin{exercise}[subtitle={Absentéisme - pas bien!}]
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Une entreprise emploie 20 personnes. Une étude statistique a montré qu'un jour donné, la probabilité qu'un employé soit absent est de 0.05. On admet que l'absence d'un employé est indépendante de l'absence des autres.
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\begin{enumerate}
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\item On s'intéresse aux absences sur une seule journée. On note $X$ la variable aléatoire qui compte le nombre d'employés absents.
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\begin{enumerate}
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\item Quelle loi de probabilité peut modéliser les valeurs prises par $X$?
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\item Quelle est la probabilité que 2 employés soient absents?
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\item Quelle est la probabilité que plus de 10 employés soient absents?
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\end{enumerate}
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\item On veut maintenant étudier ces absences sur un mois de 31 jours. On note $Y$ la variable aléatoire qui compte le nombre de jours-employés absents sur le mois.
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\begin{enumerate}
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\item Justifier que $Y$ suit une loi binomiale dont on précisera les paramètres.
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\item Quelle est la probabilité que cette entreprise ait 10jours-employés d'absences?
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\item Même question pour plus de 5jours-employés.
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\item Combien de jours-employés absent peut-on envisager en moyenne sur un mois?
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\end{enumerate}
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\end{enumerate}
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\end{exercise}
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\vfill
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\printexercise{exercise}{1}
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\printexercise{exercise}{2}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "master"
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%%% End:
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BIN
TES/Probabilte_statistiques/Loi_binomiale/3E_annales.pdf
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TES/Probabilte_statistiques/Loi_binomiale/3E_annales.pdf
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TES/Probabilte_statistiques/Loi_binomiale/3E_annales.tex
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TES/Probabilte_statistiques/Loi_binomiale/3E_annales.tex
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\title{Loi binomiale - modélisation}
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\tribe{TESL}
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\date{Février 2020}
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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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\begin{exercise}[subtitle={Suite - Liban 2018}]
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80 personnes s'apprêtent à passer le portique de sécurité. On suppose que pour chaque personne la probabilité que le portique sonne est égale à $\np{0,02192}$.
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Soit $X$ la variable aléatoire donnant le nombre de personnes faisant sonner le portique, parmi les 80 personnes de ce groupe.
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end\begin{enumerate}
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\item Justifier que $X$ suit une loi binomiale dont on précisera les paramètres.
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\item Calculer l'espérance de $X$ et interpréter le résultat.
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\item
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\begin{list}{\textbullet}{Sans le justifier, donner la valeur arrondie à $10^{-3}$ de:}
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\item la probabilité qu'au moins une personne du groupe fasse sonner le portique;
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\item la probabilité qu'au maximum 5 personnes fassent sonner le portique.
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\end{list}
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\item Sans le justifier, donner la valeur du plus petit entier $n$ tel que $P(X \leqslant n) \geqslant 0,9$.
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\end{enumerate}
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\end{exercise}
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\begin{exercise}[subtitle={Antille-Guyane 2018}]
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Victor a téléchargé un jeu sur son téléphone. Le but de ce jeu est d'affronter des obstacles à l'aide de personnages qui peuvent être de trois types: \og Terre \fg, \og Air\fg{} ou \og Feu \fg.
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Au début de chaque partie, Victor obtient de façon aléatoire un personnage d'un des trois types et peut, en cours de partie, conserver ce personnage ou changer une seule fois de type de personnage.
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Le jeu a été programmé de telle sorte que :
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\begin{itemize}
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\item[$\bullet~~$] la probabilité que la partie débute avec un personnage de type \og Terre \fg{} est $0,3$ ;
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\item[$\bullet~~$] la probabilité que la partie débute avec un personnage de type \og Air \fg{} est $0,5$ ;
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\item[$\bullet~~$] si la partie débute avec un personnage de type \og Terre \fg, la probabilité que celui-ci soit conservé est $0,5$ ;
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\item[$\bullet~~$] si la partie débute avec un personnage de type \og Air\fg, la probabilité que celui-ci soit conservé est $0,4$ ;
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\item[$\bullet~~$] si la partie débute avec un personnage de type \og Feu \fg, la probabilité que celui-ci soit conservé est $0,9$.
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\end{itemize}
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On note les évènements suivants :
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\begin{itemize}
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\item[$\bullet~~$] $T$ : la partie débute avec un personnage de type \og Terre \fg{} ;
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\item[$\bullet~~$] $A$ : la partie débute avec un personnage de type \og Air \fg{} ;
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\item[$\bullet~~$] $F$ : la partie débute avec un personnage de type \og Feu \fg{} ;
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\item[$\bullet~~$] $C$ : Victor conserve le même personnage tout au long de la partie.
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\end{itemize}
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\medskip
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\begin{enumerate}
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\item Recopier et compléter l'arbre de probabilités ci-dessous.
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\begin{center}
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\begin{tikzpicture}[sloped]
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\node {.}
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child {node {$T$}
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child {node {$S$}
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edge from parent
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node[above] {...}
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}
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child {node {$\overline{C}$}
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edge from parent
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node[above] {...}
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}
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edge from parent
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node[above] {...}
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}
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child[missing] {}
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child {node {$A$}
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child {node {$C$}
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edge from parent
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node[above] {...}
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}
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child {node {$\overline{C}$}
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edge from parent
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node[above] {...}
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}
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edge from parent
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node[above] {...}
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}
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child[missing] {}
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child { node {$F$}
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child {node {$C$}
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edge from parent
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node[above] {...}
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}
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child {node {$\overline{C}$}
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edge from parent
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node[above] {...}
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}
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edge from parent
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node[above] {...}
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} ;
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\end{tikzpicture}
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\end{center}
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\item Calculer la probabilité que Victor obtienne et conserve un personnage de type \og Air \fg.
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\item Justifier que la probabilité que Victor conserve le personnage obtenu en début de partie est $0,53$.
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\item On considère une partie au cours de laquelle Victor a conservé le personnage obtenu en début de partie.
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Quelle est la probabilité que ce soit un personnage de type \og Air\fg{} ?
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\end{enumerate}
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\bigskip
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\textbf{Partie B}
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\medskip
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On considère $10$ parties jouées par Victor, prises indépendamment les unes des autres.
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On rappelle que la probabilité que Victor obtienne un personnage de type \og Terre \fg{} est $0,3$.
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$Y$ désigne la variable aléatoire qui compte le nombre de personnages de type \og Terre \fg{} obtenus au début de ses $10$ parties.
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\medskip
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\begin{enumerate}
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\item Justifier que cette situation peut être modélisée par une loi binomiale dont on précisera les paramètres.
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\item Calculer la probabilité que Victor ait obtenu exactement 3 personnages de type \og Terre \fg{} au début de ses $10$ parties.
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\item Calculer la probabilité que Victor ait obtenu au moins une fois un personnage de type
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\og Terre \fg{} au début de ses $10$ parties.
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\end{enumerate}
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\end{exercise}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "master"
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%%% End:
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TES/Probabilte_statistiques/Loi_binomiale/fig/vol1.pdf
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"
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id="image5322"
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x="4134.0459"
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y="1710.1553" />
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</g>
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</svg>
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After Width: | Height: | Size: 57 KiB |
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47
TES/Probabilte_statistiques/Loi_binomiale/index.rst
Normal file
47
TES/Probabilte_statistiques/Loi_binomiale/index.rst
Normal file
@@ -0,0 +1,47 @@
|
||||
Loi binomiale vers loi normale pour l'année 2019-2020 en terminale ES
|
||||
#####################################################################
|
||||
|
||||
:date: 2020-02-18
|
||||
:modified: 2020-02-18
|
||||
:authors: Bertrand Benjamin
|
||||
:category: TESL
|
||||
:tags: Probabilité, Variable aléatoire, Binomiale, Normale
|
||||
:summary: Loi binomiale vers la loi normale pour l'année 2019-2020 en terminale ESL
|
||||
|
||||
|
||||
Étape 1: Simulation d'overbooking
|
||||
=================================
|
||||
|
||||
Activité avec le tableur pour étudier une situation d'overbooking d'une compagnie aérienne. Cette activité est l'occasion de reparler de simulation aléatoire avec le tableur, de re-expliquer comment générer un 0 ou un 1 et de revoir la commande `SI`.
|
||||
|
||||
.. image:: 1E_simulation_overbooking_tbl.pdf
|
||||
:height: 200px
|
||||
:alt: Activité avec tableur pour simuler de l'overbooking sur plusieurs vols.
|
||||
|
||||
En bilan, on expliquera la modélisation d'un vol avec une loi binomiale.
|
||||
|
||||
.. image:: 1B_modelisation.pdf
|
||||
:height: 200px
|
||||
:alt: Bilan sur la modélisation avec la loi binomiale
|
||||
|
||||
|
||||
Étape 2: Modélisation avec une binomiale et arbre
|
||||
=================================================
|
||||
|
||||
Situations à modéliser avec une loi binomiale où l'on augmente petit à petit le nombre de répétitions pour aboutir à un calcul nécessaire avec la calculatrice.
|
||||
|
||||
.. image:: 2E_modelisation.pdf
|
||||
:height: 200px
|
||||
:alt: Situations modélisables avec la loi binomiale
|
||||
|
||||
Étape 3: Exercices d'annales
|
||||
============================
|
||||
|
||||
.. image:: 3E_annales.pdf
|
||||
:height: 200px
|
||||
:alt: Annales avec la loi binomiale
|
||||
|
||||
.. image:: Texas_Probabilites_Loi_Binomiale.pdf
|
||||
:height: 200px
|
||||
:alt: Pense bête sur l'utilisation de la calculatrice avec la loi binomiale
|
||||
|
||||
BIN
TES/Probabilte_statistiques/Loi_densite/1E_graphs.pdf
Normal file
BIN
TES/Probabilte_statistiques/Loi_densite/1E_graphs.pdf
Normal file
Binary file not shown.
115
TES/Probabilte_statistiques/Loi_densite/1E_graphs.tex
Normal file
115
TES/Probabilte_statistiques/Loi_densite/1E_graphs.tex
Normal file
@@ -0,0 +1,115 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\title{Loi à densité - }
|
||||
\tribe{Terminale ES}
|
||||
\date{Février 2020}
|
||||
|
||||
\pagestyle{empty}
|
||||
\geometry{left=10mm,right=10mm, top=10mm}
|
||||
|
||||
\usepackage{pgfplots}
|
||||
\pgfplotsset{compat=1.7}
|
||||
\pgfmathdeclarefunction{gauss}{2}{%
|
||||
\pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
|
||||
}
|
||||
|
||||
\newcommand{\graphs}{%
|
||||
\begin{tabular}{cc}
|
||||
|
||||
X \sim \mathcal{B}(40;0.5) & Y \sim \mathcal{N}(20; 10) \\
|
||||
\begin{tikzpicture}[
|
||||
declare function={binom(\k,\n,\p)=\n!/(\k!*(\n-\k)!)*\p^\k*(1-\p)^(\n-\k);},
|
||||
]
|
||||
\begin{axis}[
|
||||
yscale=0.5,
|
||||
samples at={0,...,40},
|
||||
yticklabel style={
|
||||
/pgf/number format/fixed,
|
||||
/pgf/number format/fixed zerofill,
|
||||
/pgf/number format/precision=2
|
||||
},
|
||||
enlargelimits=false, clip=false, axis on top,
|
||||
ybar=0pt, bar width=1
|
||||
]
|
||||
%\addplot [fill=cyan, fill opacity=0.5] {binom(x,40,0.2)}; \addlegendentry{$p=0.2$}
|
||||
\addplot [] {binom(x,40,0.5)};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
&
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[
|
||||
yscale=0.5,
|
||||
no markers, domain=0:40, samples=100,
|
||||
yticklabel style={
|
||||
/pgf/number format/fixed,
|
||||
/pgf/number format/fixed zerofill,
|
||||
/pgf/number format/precision=2
|
||||
},
|
||||
%axis lines*=left, xlabel=$x$, ylabel=$y$,
|
||||
%every axis y label/.style={at=(current axis.above origin),anchor=south},
|
||||
%every axis x label/.style={at=(current axis.right of origin),anchor=west},
|
||||
%height=5cm, width=12cm,
|
||||
enlargelimits=false,
|
||||
clip=false, axis on top,
|
||||
%grid = major
|
||||
]
|
||||
\addplot [very thick,cyan!50!black] {gauss(20,3.16)};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\\
|
||||
X \sim \mathcal{B}(40;0.4) & Y \sim \mathcal{N}(...; ...) \\
|
||||
\begin{tikzpicture}[
|
||||
declare function={binom(\k,\n,\p)=\n!/(\k!*(\n-\k)!)*\p^\k*(1-\p)^(\n-\k);},
|
||||
]
|
||||
\begin{axis}[
|
||||
yscale=0.5,
|
||||
samples at={0,...,40},
|
||||
yticklabel style={
|
||||
/pgf/number format/fixed,
|
||||
/pgf/number format/fixed zerofill,
|
||||
/pgf/number format/precision=2
|
||||
},
|
||||
enlargelimits=false, clip=false, axis on top,
|
||||
ybar=0pt, bar width=1
|
||||
]
|
||||
%\addplot [fill=cyan, fill opacity=0.5] {binom(x,40,0.2)}; \addlegendentry{$p=0.2$}
|
||||
\addplot [] {binom(x,40,0.4)};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
&
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[
|
||||
yscale=0.5,
|
||||
no markers, domain=0:40, samples=100,
|
||||
yticklabel style={
|
||||
/pgf/number format/fixed,
|
||||
/pgf/number format/fixed zerofill,
|
||||
/pgf/number format/precision=2
|
||||
},
|
||||
%axis lines*=left, xlabel=$x$, ylabel=$y$,
|
||||
%every axis y label/.style={at=(current axis.above origin),anchor=south},
|
||||
%every axis x label/.style={at=(current axis.right of origin),anchor=west},
|
||||
%height=5cm, width=12cm,
|
||||
enlargelimits=false,
|
||||
clip=false, axis on top,
|
||||
%grid = major
|
||||
]
|
||||
\addplot [very thick,cyan!50!black] {gauss(16,3)};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
|
||||
\end{tabular}
|
||||
|
||||
}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\graphs
|
||||
\vfill
|
||||
\graphs
|
||||
\vfill
|
||||
\graphs
|
||||
\vfill
|
||||
|
||||
\end{document}
|
||||
BIN
TES/Probabilte_statistiques/Loi_densite/1P_tableur.pdf
Normal file
BIN
TES/Probabilte_statistiques/Loi_densite/1P_tableur.pdf
Normal file
Binary file not shown.
128
TES/Probabilte_statistiques/Loi_densite/1P_tableur.tex
Normal file
128
TES/Probabilte_statistiques/Loi_densite/1P_tableur.tex
Normal file
@@ -0,0 +1,128 @@
|
||||
\documentclass[10pt,xcolor=table]{classPres}
|
||||
%\usepackage{myXsim}
|
||||
|
||||
|
||||
\usepackage{pgfplots}
|
||||
\pgfplotsset{compat=1.7}
|
||||
\pgfmathdeclarefunction{gauss}{2}{%
|
||||
\pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
|
||||
}
|
||||
|
||||
\title{}
|
||||
\author{}
|
||||
\date{Févier 2020}
|
||||
|
||||
\begin{document}
|
||||
\begin{frame}{Représentation de la loi binomiale}
|
||||
\begin{enumerate}
|
||||
\item Rédiger une situation modélisable avec une loi binomiale.
|
||||
\item Construire avec le tableur le tableau des valeurs de cette loi.
|
||||
\begin{center}
|
||||
\texttt{LOI.BINOMIALE(k; n; p; 0)}
|
||||
\end{center}
|
||||
\item Représenter avec un diagramme barre ces probabilités
|
||||
\item Reporter dessus l'espérance et l'écart-type.
|
||||
\end{enumerate}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[fragile]{Loi binomiale vers la loi normale}
|
||||
Dans ces 2 représentations $n = 40$.
|
||||
|
||||
\begin{tikzpicture}[
|
||||
declare function={binom(\k,\n,\p)=\n!/(\k!*(\n-\k)!)*\p^\k*(1-\p)^(\n-\k);},
|
||||
]
|
||||
\begin{axis}[
|
||||
xscale=0.65,
|
||||
samples at={0,...,40},
|
||||
yticklabel style={
|
||||
/pgf/number format/fixed,
|
||||
/pgf/number format/fixed zerofill,
|
||||
/pgf/number format/precision=1
|
||||
},
|
||||
ybar=0pt, bar width=1
|
||||
]
|
||||
%\addplot [fill=cyan, fill opacity=0.5] {binom(x,40,0.2)}; \addlegendentry{$p=0.2$}
|
||||
\node[draw=black,fill=white,anchor=north west] at (rel axis cs:0,1) {$p=0.5$};
|
||||
\addplot [fill=orange, fill opacity=0.5] {binom(x,40,0.5)};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\hfill
|
||||
\begin{tikzpicture}[
|
||||
declare function={binom(\k,\n,\p)=\n!/(\k!*(\n-\k)!)*\p^\k*(1-\p)^(\n-\k);},
|
||||
]
|
||||
\begin{axis}[
|
||||
xscale=0.65,
|
||||
samples at={0,...,40},
|
||||
yticklabel style={
|
||||
/pgf/number format/fixed,
|
||||
/pgf/number format/fixed zerofill,
|
||||
/pgf/number format/precision=1
|
||||
},
|
||||
ybar=0pt, bar width=1
|
||||
]
|
||||
\node[draw=black,fill=white,anchor=north west] at (rel axis cs:0,1) {$p=0.2$};
|
||||
\addplot [fill=cyan, fill opacity=0.5] {binom(x,40,0.2)};
|
||||
%\addplot [fill=orange, fill opacity=0.5] {binom(x,40,0.5)}; \addlegendentry{$p=0.5$}
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[fragile]{Calculer probabilité}
|
||||
Que représente $P(X > 10)$?
|
||||
\hfill
|
||||
Comment interpréter $P(Y>10)$?
|
||||
|
||||
|
||||
\[
|
||||
X \sim \mathcal{B}(40;0.5) \qquad Y \sim \mathcal{N}(20; 10)
|
||||
\]
|
||||
|
||||
\begin{tikzpicture}[
|
||||
declare function={binom(\k,\n,\p)=\n!/(\k!*(\n-\k)!)*\p^\k*(1-\p)^(\n-\k);},
|
||||
]
|
||||
\begin{axis}[
|
||||
xscale=0.6,
|
||||
yscale=0.9,
|
||||
samples at={0,...,40},
|
||||
yticklabel style={
|
||||
/pgf/number format/fixed,
|
||||
/pgf/number format/fixed zerofill,
|
||||
/pgf/number format/precision=2
|
||||
},
|
||||
enlargelimits=false, clip=false, axis on top,
|
||||
ybar=0pt, bar width=1
|
||||
]
|
||||
%\addplot [fill=cyan, fill opacity=0.5] {binom(x,40,0.2)}; \addlegendentry{$p=0.2$}
|
||||
\addplot [fill=orange, fill opacity=0.5] {binom(x,40,0.5)};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\hfill
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[
|
||||
xscale=0.6,
|
||||
yscale=0.9,
|
||||
no markers, domain=0:40, samples=100,
|
||||
yticklabel style={
|
||||
/pgf/number format/fixed,
|
||||
/pgf/number format/fixed zerofill,
|
||||
/pgf/number format/precision=2
|
||||
},
|
||||
%axis lines*=left, xlabel=$x$, ylabel=$y$,
|
||||
%every axis y label/.style={at=(current axis.above origin),anchor=south},
|
||||
%every axis x label/.style={at=(current axis.right of origin),anchor=west},
|
||||
%height=5cm, width=12cm,
|
||||
enlargelimits=false,
|
||||
clip=false, axis on top,
|
||||
%grid = major
|
||||
]
|
||||
\addplot [very thick,cyan!50!black] {gauss(20,3)};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\end{frame}
|
||||
\end{document}
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "master"
|
||||
%%% End:
|
||||
|
||||
BIN
TES/Probabilte_statistiques/Loi_densite/2E_loi_normale.pdf
Normal file
BIN
TES/Probabilte_statistiques/Loi_densite/2E_loi_normale.pdf
Normal file
Binary file not shown.
73
TES/Probabilte_statistiques/Loi_densite/2E_loi_normale.tex
Normal file
73
TES/Probabilte_statistiques/Loi_densite/2E_loi_normale.tex
Normal file
@@ -0,0 +1,73 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\title{Loi à densité - Loi normale}
|
||||
\tribe{Terminale ES}
|
||||
\date{Février 2020}
|
||||
|
||||
\pagestyle{empty}
|
||||
\geometry{left=10mm,right=10mm, top=10mm}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{exercise}[subtitle={Technique}]
|
||||
\begin{multicols}{2}
|
||||
Soit $X \sim \mathcal{N}(3; 1)$. Calculer les quantités suivantes
|
||||
\begin{multicols}{2}
|
||||
\begin{enumerate}
|
||||
\item $P(X > 1)$
|
||||
\item $P(X > 0)$
|
||||
\item $P(X \leq 3)$
|
||||
\item $P(0 \leq X \leq 3)$
|
||||
\end{enumerate}
|
||||
\end{multicols}
|
||||
|
||||
Soit $Y \sim \mathcal{N}(1; 16)$. Calculer les quantités suivantes
|
||||
\begin{multicols}{2}
|
||||
\begin{enumerate}
|
||||
\item $P(Y > 1)$
|
||||
\item $P(Y < -2)$
|
||||
\item $P(0 \leq Y \leq 3)$
|
||||
\item $P(Y = 3)$
|
||||
\end{enumerate}
|
||||
\end{multicols}
|
||||
\end{multicols}
|
||||
|
||||
\end{exercise}
|
||||
|
||||
\begin{exercise}[subtitle={Vignoble}]
|
||||
Dans un vignoble, une étude statistique a établi que la probabilité qu'un pied de vigne soit atteind du maladie est de 0,4. On observe 600 pieds de vignes ainsi, on peut considérer qu'il s'agit d'un tirage avec remise).
|
||||
|
||||
\begin{enumerate}
|
||||
\item On note $X$ la variable aléatoire qui compte le nombre d'arbres malades.
|
||||
\begin{enumerate}
|
||||
\item Avec quelle loi de probabilité peut-on modéliser $X$?
|
||||
\item Calculer l'espérance et l'écart-type de $X$.
|
||||
\item Calculer les probabilités suivantes $\qquad P(X \leq 323) \qquad \qquad P(X \geq 256) \qquad \qquad P(240 < X < 252)$
|
||||
\end{enumerate}
|
||||
\item On veut approcher le nombre de pied malade avec une loi normale ayant les même caractéristiques que $X$. On note cette variable aléatoire $Y$.
|
||||
\begin{enumerate}
|
||||
\item Pourquoi peut-on considérer de faire cette approximation?
|
||||
\item Quels sont les paramètres de cette loi normale?
|
||||
\item Calculer les probabilités précédement calculées et compararer les résultats.
|
||||
\item Que pensez-vous de cette approximation?
|
||||
\end{enumerate}
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\begin{exercise}[subtitle={Pots de confiture}]
|
||||
Une entreprise conditionne des pots de confiture de 300g. À cause des aléas de production, le poid des pots n'est jamais exactement de 300g mais suit une loi normale d'espérance $\mu=300$ et d'écart-type $\sigma=2$. L'entreprise ne commercialise pas les pôts dont l'écart poids avec ce qui est voulu est supérieur à 4g.
|
||||
\begin{enumerate}
|
||||
\item On prélève un pot au hasard, quelle est la probabilité qu'il soit commercialisé?
|
||||
\item Déterminer $a$ tel que $P(X<a) = 0.01$.
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\vfill
|
||||
\printexercise{exercise}{1}
|
||||
\printexercise{exercise}{2}
|
||||
\printexercise{exercise}{3}
|
||||
\vfill
|
||||
|
||||
|
||||
\end{document}
|
||||
BIN
TES/Probabilte_statistiques/Loi_densite/3E_ecart_type.pdf
Normal file
BIN
TES/Probabilte_statistiques/Loi_densite/3E_ecart_type.pdf
Normal file
Binary file not shown.
42
TES/Probabilte_statistiques/Loi_densite/3E_ecart_type.tex
Normal file
42
TES/Probabilte_statistiques/Loi_densite/3E_ecart_type.tex
Normal file
@@ -0,0 +1,42 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\title{Loi à densité - Loi normale}
|
||||
\tribe{Terminale ES}
|
||||
\date{Février 2020}
|
||||
|
||||
\pagestyle{empty}
|
||||
\geometry{left=10mm,right=10mm, top=10mm}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{exercise}[subtitle={Taille des bébés}]
|
||||
Pour suivre la croissance des bébés, on retrouve des graphiques réalisés sur les statistiques de la croissance. Les données statistiques permettent de modéliser à chaque âge la taille des enfants par une loi normale.
|
||||
|
||||
Répondre aux questions suivantes en utilisant ce graphique.
|
||||
\begin{enumerate}
|
||||
\item Quelle est la taille moyenne et l'écart type d'un enfant à la naissance?
|
||||
\item Même question à 1an puis 2ans.
|
||||
\item Comment varie la taille moyenne avec l'âge? Est-ce étonnant?
|
||||
\item Comment varie l'écart-type des tailles avec l'âge? Commenter cette évolution.
|
||||
\item On choisit au hasard un enfant de 36mois et on note $X$ sa taille.
|
||||
\begin{enumerate}
|
||||
\item Quelle est la loi que suit $X$?
|
||||
\item Calculer et interpréter $P(X\geq 100)$.
|
||||
\end{enumerate}
|
||||
\item En prenant quelques exemples répondre aux questions suivantes
|
||||
\begin{enumerate}
|
||||
\item Quel que soit l'âge, quelle est la probabilité qu'un enfant choisi au hasard ait une taille comprise entre les courbes en gros pointillés.
|
||||
\item Même questions pour une taille comprise dans la zone violette.
|
||||
\item Même questions pour les pointillés fins.
|
||||
\end{enumerate}
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\vfill
|
||||
\printexercise{exercise}{1}
|
||||
\vfill
|
||||
\printexercise{exercise}{1}
|
||||
\vfill
|
||||
|
||||
\end{document}
|
||||
BIN
TES/Probabilte_statistiques/Loi_densite/3E_graph.pdf
Normal file
BIN
TES/Probabilte_statistiques/Loi_densite/3E_graph.pdf
Normal file
Binary file not shown.
19
TES/Probabilte_statistiques/Loi_densite/3E_graph.tex
Normal file
19
TES/Probabilte_statistiques/Loi_densite/3E_graph.tex
Normal file
@@ -0,0 +1,19 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\title{Loi à densité - Loi normale}
|
||||
\tribe{Terminale ES}
|
||||
\date{Février 2020}
|
||||
|
||||
\pagestyle{empty}
|
||||
\geometry{left=10mm,right=10mm, top=10mm}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\vfill
|
||||
\includegraphics[scale=0.9]{./fig/filles_tailles}
|
||||
\vfill
|
||||
\includegraphics[scale=0.9]{./fig/filles_tailles}
|
||||
\vfill
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
BIN
TES/Probabilte_statistiques/Loi_densite/fig/filles_tailles.png
Normal file
BIN
TES/Probabilte_statistiques/Loi_densite/fig/filles_tailles.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 193 KiB |
53
TES/Probabilte_statistiques/Loi_densite/index.rst
Normal file
53
TES/Probabilte_statistiques/Loi_densite/index.rst
Normal file
@@ -0,0 +1,53 @@
|
||||
Lois à densité pour l'année 2019-2020 en terminale ES
|
||||
#####################################################
|
||||
|
||||
:date: 2020-02-18
|
||||
:modified: 2020-02-18
|
||||
:authors: Bertrand Benjamin
|
||||
:category: TESL
|
||||
:tags: Probabilité, Variable aléatoire
|
||||
:summary: Loi à densité pour l'année 2019-2020 en terminale ESL
|
||||
|
||||
Ce chapitre suit directement le chapitre sur la loi binomiale, il y aura uniquement une différence de nom de chapitre dans le cours.
|
||||
|
||||
Étape 1: Prolongement de la binomiale
|
||||
=====================================
|
||||
|
||||
.. image:: 1P_tableur.pdf
|
||||
:height: 200px
|
||||
:alt: Consignes orales
|
||||
|
||||
Création d'une situation modélisable avec un loi binomiale et ajout de questions à poser en termes de probabilité sur la situation. On demandera aux élèves que le nombre de répétition soit grand (plus que 30).
|
||||
|
||||
Utilisation du tableur pour calculer toutes les valeurs prises et tracé du diagramme associé. Si on a de la chance, on est dans une salle où l'on peut imprimer..!
|
||||
|
||||
.. image:: 1E_graphs.pdf
|
||||
:height: 200px
|
||||
:alt: Graphes pour comparaison
|
||||
|
||||
|
||||
Sur ce graphique, on demande aux élèves de calculer et repérer l'espérance et l'écart type.
|
||||
|
||||
Explication en cours magistrale du "prolongement" vers une loi continue.
|
||||
|
||||
Étape 2: Calculs de probabilité avec la loi normale
|
||||
===================================================
|
||||
|
||||
.. image:: 2E_loi_normale.pdf
|
||||
:height: 200px
|
||||
:alt: Exercices utilisant la loi normale
|
||||
|
||||
Modélisation et calculs de probabilités avec la loi normale.
|
||||
|
||||
Étape 3: Probabilité et écart-type
|
||||
==================================
|
||||
|
||||
Exercice de remobilisation de la loi normale.
|
||||
|
||||
.. image:: 3E_ecart_type.pdf
|
||||
:height: 200px
|
||||
:alt: travail sur les courbes de tailles
|
||||
|
||||
.. image:: 3E_graph.pdf
|
||||
:height: 200px
|
||||
:alt: Courbes de tailles
|
||||
Binary file not shown.
BIN
TES/Probabilte_statistiques/Loi_uniforme/1B_marche_noir.pdf
Normal file
BIN
TES/Probabilte_statistiques/Loi_uniforme/1B_marche_noir.pdf
Normal file
Binary file not shown.
94
TES/Probabilte_statistiques/Loi_uniforme/1B_marche_noir.tex
Normal file
94
TES/Probabilte_statistiques/Loi_uniforme/1B_marche_noir.tex
Normal file
@@ -0,0 +1,94 @@
|
||||
\documentclass[a4paper,10pt, landscape, twocolumn]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\title{Bilan - marché noir}
|
||||
\tribe{Terminale ES}
|
||||
\date{Septembre 2019}
|
||||
|
||||
\pagestyle{empty}
|
||||
|
||||
\newcommand\exemple{%
|
||||
\paragraph{Exemple - Marché noir}
|
||||
|
||||
$X$ la variable aléatoire décrivant les gains du professeur après avoir confisquer puis revendu un téléphone.
|
||||
|
||||
$X$ a pour loi
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}[h]{|p{2cm}| *{6}{c|}}
|
||||
\hline
|
||||
Valeur & 10 & 40 & 150 & 200 & 250 \\ \hline
|
||||
Probabilité& 0.10 & 0.05 & 0.50 & 0.05 & 0.30\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
|
||||
\begin{tikzpicture}[scale=0.8]
|
||||
\draw[very thin, gray] (0,0) grid (10,4);
|
||||
|
||||
\draw[line width=2mm,color=red] plot[ycomb] coordinates {(0.4, 0.4) (1.6, 0.2) (6, 2) (8, 0.2) (10, 1.2)};
|
||||
|
||||
\draw[->, very thick] (-0.5,0) -- (10.5,0);
|
||||
\draw[->, very thick] (0,-0.5) -- (0,4.2);
|
||||
|
||||
\draw (0, 4.1) node [above] {Probabilité};
|
||||
\draw (0, 4) node [left] {1};
|
||||
\draw (0, 2) node [left] {0.5};
|
||||
\draw (10, 0) node [above right] {Valeur};
|
||||
\draw (10, 0) node [below] {250};
|
||||
\draw (1, 0) node [below] {25};
|
||||
|
||||
|
||||
% \draw[very thick, color=red, |-|] (0.4, 0) -- (0.4, 0.4);
|
||||
% \draw[very thick, color=red, |-|] (1.6, 0) -- (1.6, 0.2);
|
||||
% \draw[very thick, color=red, |-|] (6, 0) -- (6, 2);
|
||||
% \draw[very thick, color=red, |-|] (8, 0) -- (8, 0.2);
|
||||
% \draw[very thick, color=red, |-|] (10, 0) -- (10, 1.2);
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
|
||||
Espérance:
|
||||
|
||||
\vfill
|
||||
|
||||
\paragraph{Exemple - lancé de dé - équipartition}
|
||||
|
||||
$X$ la variable aléatoire décrivant le score obtenu à un lancé de dé.
|
||||
|
||||
$X$ a pour loi
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}[h]{|p{2cm}| *{7}{c|}}
|
||||
\hline
|
||||
Valeur & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline
|
||||
Probabilité& $\frac{1}{6}$ & $\frac{1}{6}$ & $\frac{1}{6}$ & $\frac{1}{6}$ & $\frac{1}{6}$ & $\frac{1}{6}$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
|
||||
\begin{tikzpicture}[scale=0.8]
|
||||
\draw[very thin, gray] (0,0) grid (7,4);
|
||||
|
||||
\draw[line width=2mm,color=red] plot[ycomb] coordinates {(1, 1.666)(2, 1.666)(3, 1.666)(4, 1.666)(5, 1.666)(6, 1.666)};
|
||||
|
||||
\draw[->, very thick] (-0.5,0) -- (7.5,0);
|
||||
\draw[->, very thick] (0,-0.5) -- (0,4.2);
|
||||
|
||||
\draw (0, 4.1) node [above] {Probabilité};
|
||||
\draw (0, 4) node [left] {1};
|
||||
\draw (0, 2) node [left] {0.5};
|
||||
\draw (7, 0) node [above right] {Valeur};
|
||||
\draw (1, 0) node [below] {1};
|
||||
\draw (6, 0) node [below] {6};
|
||||
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
Espérance:
|
||||
\vfill
|
||||
}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\exemple
|
||||
\pagebreak
|
||||
\exemple
|
||||
|
||||
\end{document}
|
||||
BIN
TES/Probabilte_statistiques/Loi_uniforme/1E_revente.pdf
Normal file
BIN
TES/Probabilte_statistiques/Loi_uniforme/1E_revente.pdf
Normal file
Binary file not shown.
39
TES/Probabilte_statistiques/Loi_uniforme/1E_revente.tex
Normal file
39
TES/Probabilte_statistiques/Loi_uniforme/1E_revente.tex
Normal file
@@ -0,0 +1,39 @@
|
||||
\documentclass[10pt,xcolor=table]{classPres}
|
||||
%\usepackage{myXsim}
|
||||
|
||||
\title{}
|
||||
\author{}
|
||||
\date{Septembre 2019}
|
||||
|
||||
\begin{document}
|
||||
\begin{frame}{ Marché noir}
|
||||
À force de confisquer les téléphones portables de ses élèves, un professeur a pu établir le tableau suivant
|
||||
\begin{center}
|
||||
\footnotesize
|
||||
\begin{tabular}[h]{|p{2cm}| *{6}{c|}}
|
||||
\hline
|
||||
Type de portable & Vieux & À clapet & Smartphone & Téléphone satellite & Tablette\\
|
||||
\hline
|
||||
Fréquence (en \%)& 10 & 5 & 50 & 5 & 30\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
Il décide alors de ne plus les rendre en fin de cours mais de les vendre au marché noir. Il se renseigne alors sur les prix de vente:
|
||||
\begin{center}
|
||||
\footnotesize
|
||||
\begin{tabular}[h]{|p{2cm}| *{6}{c|}}
|
||||
\hline
|
||||
Type de portable & Vieux & À clapet & Smartphone & Téléphone satellite & Tablette \\ \hline
|
||||
Prix de revente (en \euro) & 10 & 40 & 150 & 200 & 250 \\ \hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
Combien peut-il espérer gagner en moyenne à chaque fois qu'il confisque un téléphone?
|
||||
\end{frame}
|
||||
|
||||
\end{document}
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "master"
|
||||
%%% End:
|
||||
|
||||
BIN
TES/Probabilte_statistiques/Loi_uniforme/2P_simulation.pdf
Normal file
BIN
TES/Probabilte_statistiques/Loi_uniforme/2P_simulation.pdf
Normal file
Binary file not shown.
116
TES/Probabilte_statistiques/Loi_uniforme/2P_simulation.tex
Normal file
116
TES/Probabilte_statistiques/Loi_uniforme/2P_simulation.tex
Normal file
@@ -0,0 +1,116 @@
|
||||
\documentclass[14pt,xcolor=table]{classPres}
|
||||
%\usepackage{myXsim}
|
||||
|
||||
\title{}
|
||||
\author{}
|
||||
\date{Septembre 2019}
|
||||
|
||||
\begin{document}
|
||||
\begin{frame}{Simulation avec le tableur}
|
||||
\begin{block}{Aléatoire}
|
||||
|
||||
\pause
|
||||
Fonction pour faire de l'aléatoire:
|
||||
\begin{center}
|
||||
\texttt{ALEA()}
|
||||
\end{center}
|
||||
\pause
|
||||
|
||||
Renvoie un nombre aléatoire entre 0 et 1
|
||||
|
||||
\Ovalbox{\texttt{F9}} tirer un nouveau nombre
|
||||
|
||||
\end{block}
|
||||
|
||||
\pause
|
||||
\begin{block}{Conditions}
|
||||
Si ... Alors ... Sinon ...
|
||||
|
||||
\begin{center}
|
||||
\texttt{SI(...; ... ; ...)}
|
||||
\end{center}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Marche aléatoire}
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=1.5]
|
||||
\filldraw[very thick, ->] (-2.4,0) -- (2.4,0);
|
||||
\draw[step=1] (-2,-0.1) grid (2,0.1);
|
||||
\draw (0, 0) node [below] {0};
|
||||
\draw (1, 0) node [below] {1};
|
||||
\draw (2, 0) node [below] {2};
|
||||
\draw (-1, 0) node [below] {-1};
|
||||
\draw (-2, 0) node [below] {-2};
|
||||
|
||||
% \filldraw[very thick, ->] (0.1,0.3) -- (0.9,0.3) ;
|
||||
% \filldraw[very thick, ->] (-0.1,0.3) -- (-0.9,0.3) ;
|
||||
\filldraw[very thick, ->] (0.1,0.3) -- node[midway,above] {0,5} (0.9,0.3) ;
|
||||
\filldraw[very thick, ->] (-0.1,0.3) -- node[midway,above] {0,5} (-0.9,0.3) ;
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
P(\mbox{aller à droite ou }+1) = 0.5\\
|
||||
P(\mbox{aller à gauche ou }-1) = 0.5
|
||||
\pause
|
||||
\begin{block}{Où sera-t-on au bout de 100 étapes?}
|
||||
\pause
|
||||
\begin{center}
|
||||
\includegraphics[scale=0.2]{./fig/tabl_marche_alea}
|
||||
\end{center}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}{Marche aléatoire}
|
||||
\framesubtitle{Direction favorite}
|
||||
\begin{center}
|
||||
\begin{tikzpicture}[scale=1.5]
|
||||
\filldraw[very thick, ->] (-2.4,0) -- (2.4,0);
|
||||
\draw[step=1] (-2,-0.1) grid (2,0.1);
|
||||
\draw (0, 0) node [below] {0};
|
||||
\draw (1, 0) node [below] {1};
|
||||
\draw (2, 0) node [below] {2};
|
||||
\draw (-1, 0) node [below] {-1};
|
||||
\draw (-2, 0) node [below] {-2};
|
||||
|
||||
% \filldraw[very thick, ->] (0.1,0.3) -- (0.9,0.3) ;
|
||||
% \filldraw[very thick, ->] (-0.1,0.3) -- (-0.9,0.3) ;
|
||||
\filldraw[very thick, ->] (0.1,0.3) -- node[midway,above] {p} (0.9,0.3) ;
|
||||
\filldraw[very thick, ->] (-0.1,0.3) -- node[midway,above] {...} (-0.9,0.3) ;
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
P(\mbox{aller à droite ou }+1) = p\\
|
||||
P(\mbox{aller à gauche ou }-1) = ...
|
||||
\pause
|
||||
\begin{block}{Où sera-t-on au bout de 100 étapes}
|
||||
\begin{itemize}
|
||||
\item quand $p=0.6$?
|
||||
\item quand $p=0.4$?
|
||||
\item quand $p=0.9$?
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{Marché noir}
|
||||
\framesubtitle{Simulation}
|
||||
|
||||
$X$ variable aléatoire respectant la loi suivante
|
||||
\begin{center}
|
||||
\small
|
||||
\begin{tabular}[h]{|p{2cm}| *{6}{c|}}
|
||||
\hline
|
||||
Valeur & 10 & 40 & 150 & 200 & 250 \\ \hline
|
||||
Probabilité& 0.10 & 0.05 & 0.50 & 0.05 & 0.30\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
Simuler cette variable pour calculer les gains du professeur après 10 confiscations.
|
||||
\end{frame}
|
||||
|
||||
\end{document}
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "master"
|
||||
%%% End:
|
||||
|
||||
BIN
TES/Probabilte_statistiques/Loi_uniforme/3E_banque.pdf
Normal file
BIN
TES/Probabilte_statistiques/Loi_uniforme/3E_banque.pdf
Normal file
Binary file not shown.
52
TES/Probabilte_statistiques/Loi_uniforme/3E_banque.tex
Normal file
52
TES/Probabilte_statistiques/Loi_uniforme/3E_banque.tex
Normal file
@@ -0,0 +1,52 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\title{Loi uniforme - Banque d'exercices}
|
||||
\tribe{Terminale ES}
|
||||
\date{Septembre 2019}
|
||||
|
||||
\pagestyle{empty}
|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{exercise}[subtitle={Technique}]
|
||||
$X$ suit la loi $\mathcal{U}(\intFF{4}{16})$. Calculer les probabilités suivantes
|
||||
\begin{multicols}{3}
|
||||
\begin{enumerate}
|
||||
\item $P(X > 10)$
|
||||
\item $P( X < 15)$
|
||||
\item $P(2 < X < 5)$
|
||||
\item $P(X = 5)$
|
||||
\item $P(X<6 \mbox{ ou } X>14)$
|
||||
\end{enumerate}
|
||||
\end{multicols}
|
||||
\end{exercise}
|
||||
|
||||
\begin{exercise}[subtitle={Heure du déjeuner}]
|
||||
Lorsqu'elle arrive chez sa grand-mère, Fatou arrive pour prendre son petit dejeuner entre 7h et 8h30 de façon aléatoire. Son cousin Moussa le prend toujours à 8h et y consacre 15min.
|
||||
\begin{enumerate}
|
||||
\item Quelle est la loi suivie par $X$ la variable aléatoire décrivant l'heure d'arrivée de Fatou?
|
||||
\item Calculer la probabilité que Fatou arrive avant Moussa.
|
||||
\item Quelle est la probabilité qu'ils se croisent?
|
||||
\item Sachant que Fatou est arrivée après 8h, quelle est la probabilité qu'elle passe un moment à table avec Moussa?
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\begin{exercise}[subtitle={Injustice}]
|
||||
Lors de la restitution des notes d'une intérrogation notées sur 20, le professeur annonce que suite à de nombreuses tricheries, il a décidé de noter chaque élèves aléatoirement entre 5 et 15.
|
||||
\begin{enumerate}
|
||||
\item Quelle est la probabilité qu'un élèves ait une note supérieure à 12?
|
||||
\item Quelle devrait être la moyenne de la classe?
|
||||
\item Un élève qui n'a pas triché est éffondré à l'annonce de ce système de notation. Il espérait avoir une note supérieur à 14. Le professeur lui indique que sa note est supérieur à 12. Quelle est alors la probabilité qu'il ait une note supérieure à 14?
|
||||
\item Le professeur souhaiterai que 75\% des élèves aient plus de 10. Proposer une autre façon de noter aléatoirement.
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\vfill
|
||||
|
||||
\printexercise{exercise}{1}
|
||||
\printexercise{exercise}{2}
|
||||
\printexercise{exercise}{3}
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 45 KiB |
75
TES/Probabilte_statistiques/Loi_uniforme/index.rst
Normal file
75
TES/Probabilte_statistiques/Loi_uniforme/index.rst
Normal file
@@ -0,0 +1,75 @@
|
||||
Probabilité discrète et loi uniforme pour l'année 2019-2020 en terminale ES
|
||||
###########################################################################
|
||||
|
||||
:date: 2019-10-01
|
||||
:modified: 2019-10-01
|
||||
:authors: Bertrand Benjamin
|
||||
:category: TESL
|
||||
:tags: Probabilité, Variable aléatoire
|
||||
:summary: Probabilité discrète vers la loi uniforme pour l'année 2019-2020 en terminale ESL
|
||||
|
||||
Un des objectifs de cette séquence est de consolider les acquis au sujet des probabilités discrètes (calcul, espérance et représentation graphique) pour amener à élargir le concept de probabilité sur des ensembles continus.
|
||||
|
||||
Étape 1: Variable aléatoire et espérance
|
||||
========================================
|
||||
|
||||
Exercice de la confiscation de téléphone portable. Chaque cours un élève sort son téléphone en classe et se le fait confisquer. Le prof en profite pour le revendre au marché noir. Le lendemain l'élève revient avec le même téléphone.
|
||||
|
||||
On donne:
|
||||
|
||||
- la répartition des téléphones par élèves
|
||||
- le prix de revente
|
||||
|
||||
On va chercher combien en moyenne se fait le prof par cours puis sur un mois. Pour les élèves allant trop vite, on peut complexifier en demandant ce qui se passe si le prof confisque 2 téléphones.
|
||||
|
||||
On invitera les élèves à tracer des tableaux et faire des graphiques pour illustrer leurs propos.
|
||||
|
||||
Cahier de bord: Calcul d'une probabilité discrète, variable aléatoire
|
||||
|
||||
Exercices techniques: Répétition d'expériences, utilisation de loi de probabilité, calcul d'espérance. On illustrera les concepts avec `des exemples <./1B_marche_noir.pdf>`_.
|
||||
|
||||
Étape 2: Simulation avec le tableur
|
||||
===================================
|
||||
|
||||
.. image:: 2P_simulation.pdf
|
||||
:height: 200px
|
||||
:alt: Simulation avec le tableur
|
||||
|
||||
Certainement un peu ambitieuse cette étape. Dans l'idée j'aimerai trouver une expérience aléatoire qu'on ne sait pas analyser mais que l'on pourrait simuler.
|
||||
|
||||
Idée 1
|
||||
------
|
||||
On va vouloir simuler une marche aléatoire en une dimension avec le tableur.
|
||||
|
||||
On monte de 1 avec une probabilité de *p* et on descend de 1 autrement. Au départ, *p* sera fixé à 0.5 puis on ce posera la question de ce qui se passe si on modifie cette valeur.
|
||||
|
||||
Pour faire de l'aléatoire avec le tableur, on ne peut utiliser que la fonction `ALEA` qui donne un nombre aléatoire entre 0 et 1. Il va falloir trouver comment adapter ce comportement pour choisir aléatoirement entre monter et descendre. C'est le but de ce TP. Ce questionnement devrait nous permettre d'approcher la loi uniforme.
|
||||
|
||||
Ensuite les questions associée à cette marche pourront être le nombre de fois que l'on repasse par 0 en 1000 étapes. Ou quelles sont les valeurs maximales atteintes.
|
||||
|
||||
Idée 2
|
||||
------
|
||||
On va simuler l'exercice de l'étape 1. Le but étant de calculer le gain moyen de notre prof.
|
||||
|
||||
Pour faire cette simulation, on aura besoin de `ALEA`. Il faudra alors trouver une astuce pour répartir les gains en fonctions des résultats données par cette fonction.
|
||||
|
||||
Cahier de bord: Simulation d'une variable discrète avec `ALEA` (définie comme générant une variable aléatoire uniforme sur [0, 1]).
|
||||
|
||||
|
||||
Étape 3: Calcul d'une probabilité continue
|
||||
==========================================
|
||||
|
||||
On va se donner une variable aléatoire uniforme sur un intervalle et on cherche à calculer la probabilité que le nombre sorti soit entre 2 bornes. On pourra commencer avec une loi uniforme sur [0, 1] pour faire le parallèle avec l'étape 2. Puis on pourra généraliser aux lois uniforme sur n'importe quel intervalle.
|
||||
|
||||
Exercices techniques: Calculs de probabilité.
|
||||
|
||||
Étape 4: Simulation pour calculer l'espérance d'une loi uniforme
|
||||
================================================================
|
||||
|
||||
Avec le tableur, on va chercher à calculer l'espérance d'une loi uniforme sur [0, 1]. Aux élèves de trouver comment faire en s'inspirant de l'étape 2 idée 2!
|
||||
|
||||
Puis on généralise à n'importe quel loi uniforme.
|
||||
|
||||
Cahier de bord: Espérance de la loi uniforme.
|
||||
|
||||
On ne peut pas à ce moment de l'année parler du calcule de l'aire sous la courbe. Si un élève souhaite avoir une démonstration du résultat, on lui demandera de patienter mais elle qu'elle arrivera en cours d'année.
|
||||
Binary file not shown.
@@ -0,0 +1,83 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
|
||||
\title{Arbre de probabilité - Bilan}
|
||||
\date{Novembre 2019}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\section*{Probabilité d'un évènement sous conditions}
|
||||
|
||||
Dans d'activité sur l'enquête des fumeurs, on a été amené à vérifier la véracité de la phrase suivante
|
||||
|
||||
\begin{quote}
|
||||
Sachant qu'aucun de ses parents ne soit fumeur, la probabilité qu'il ne soit pas aussi est de plus de 50\%.
|
||||
\end{quote}
|
||||
|
||||
La probabilité à calculer est soumise a une condition, on appelle cela une \textbf{probabilité conditionnelle}. En notant
|
||||
\[
|
||||
F = \left\{\mbox{Le jeune est fumeur}\right\} \qquad
|
||||
P = \left\{\mbox{Un de ses parents est fumeur}\right\}
|
||||
\]
|
||||
|
||||
On a alors calculer la probabilité de la manière suivante
|
||||
\[
|
||||
P_{\overline{P}} (\overline{F}) = \frac{\mbox{Nombre de non fumeur avec des parents non fumeur}}{\mbox{Nombre de non fumeur}} = \frac{700}{1000} = 0.7 = 70\%
|
||||
\]
|
||||
|
||||
\begin{definition}
|
||||
Soient $A$ et $B$ deux évènements avec $P(A) \neq 0$
|
||||
|
||||
On note $P_A(B)$ la probabilité de l'évènement \textbf{B sachant que A est réalisé} et on a
|
||||
\[
|
||||
P_A(B) = \frac{\mbox{Nombre d'éléments dans } A\cap B}{\mbox{Nombre d'éléments dans } A}
|
||||
\]
|
||||
\end{definition}
|
||||
|
||||
\section*{Arbre de probabilité}
|
||||
|
||||
On a représenté la situation de l'activité avec un arbre de probabilité. Deux arbres sont possibles suivant qu'on l'on s'intéresse d'abord à la condition que le jeune soit fumeur ou d'abord à ses parents.
|
||||
|
||||
\bigskip
|
||||
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{center}
|
||||
Condition sur le jeune
|
||||
|
||||
\begin{tikzpicture}
|
||||
\node {$\bullet$}
|
||||
child {node {$F$}
|
||||
child {node {$P$}}
|
||||
child {node {$\overline{P}$}}
|
||||
}
|
||||
child[missing] {}
|
||||
child { node {$\overline{F}$}
|
||||
child {node {$P$}}
|
||||
child {node {$\overline{P}$}}
|
||||
} ;
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
\begin{center}
|
||||
Condition sur le jeune
|
||||
|
||||
\begin{tikzpicture}
|
||||
\node {$\bullet$}
|
||||
child {node {$P$}
|
||||
child {node {$F$}}
|
||||
child {node {$\overline{F}$}}
|
||||
}
|
||||
child[missing] {}
|
||||
child { node {$\overline{P}$}
|
||||
child {node {$F$}}
|
||||
child {node {$\overline{F}$}}
|
||||
} ;
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\afaire{Compléter les arbres avec les probabilités calculées en classe}
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
@@ -0,0 +1,50 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\title{Probabilité conditionnelle - tableau croisé}
|
||||
\tribe{Terminale ES}
|
||||
\date{Novembre 2019}
|
||||
|
||||
\geometry{left=10mm,right=10mm, top=10mm}
|
||||
|
||||
\pagestyle{empty}
|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{exercise}[subtitle={Fumeur ou pas}]
|
||||
On interroge un échantillon de \np{1500} jeunes ayant entre 14 et 18ans pour savoir s'ils fument et si au moins l'un de leurs parents fume.
|
||||
|
||||
Les résultats de l'enquête sont consignés dans le tableau suivant.
|
||||
\begin{center}
|
||||
\begin{tabular}{|>{\columncolor{highlightbg}}c|*{2}{p{4cm}|}|p{4cm}|}
|
||||
\hline
|
||||
\rowcolor{highlightbg}
|
||||
& Fumeur & Non fumeur & Total\\
|
||||
\hline
|
||||
Au moins un parent fumeur & 300 & 300 & 600\\
|
||||
\hline
|
||||
Aucun parent fumeur & 200 & 700 & 900\\
|
||||
\hline
|
||||
Total &500 & \np{1000} & \np{1500}\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
On choisit au hasard un jeune parmi ceux interrogés.
|
||||
|
||||
Pour chacune des phrases suivantes, justifier si elles sont vraies ou fausses.
|
||||
\begin{enumerate}
|
||||
\item La probabilité qu'il soit fumeur est de plus de 30\%.
|
||||
\item La probabilité qu'il soit fumeur et qu'aucun parent ne soit fumeur est de moins de 10\%.
|
||||
\item La probabilité qu'au moins un de ses parents soit fumeur et qu'il ne le soit pas est de 20\%.
|
||||
\item La probabilité qu'il soit fumeur ou qu'un de ses parents le soit est de plus de 70\%.
|
||||
\item Sachant qu'il est fumeur, la probabilité que ses parents le soit aussi est de $\frac{3}{5}$
|
||||
\item Sachant qu'aucun de ses parents ne soit fumeur, la probabilité qu'il ne soit pas aussi est de plus de 50\%.
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
\vfill
|
||||
\printexercise{exercise}{1}
|
||||
\vfill
|
||||
\printexercise{exercise}{1}
|
||||
\vfill
|
||||
\end{document}
|
||||
Binary file not shown.
@@ -0,0 +1,45 @@
|
||||
\documentclass[10pt,xcolor=table]{classPres}
|
||||
%\usepackage{myXsim}
|
||||
|
||||
\title{}
|
||||
\author{}
|
||||
\date{Novembre 2019}
|
||||
|
||||
\begin{document}
|
||||
\begin{frame}{Fumeur ou pas}
|
||||
\begin{center}
|
||||
\begin{tabular}{|>{\columncolor{highlightbg}}p{3cm}|*{2}{p{2cm}|}|p{2cm}|}
|
||||
\hline
|
||||
\rowcolor{highlightbg}
|
||||
& Fumeur & Non fumeur & Total\\
|
||||
\hline
|
||||
Au moins un parent fumeur & 300 & 300 & 600\\
|
||||
\hline
|
||||
Aucun parent fumeur & 200 & 700 & 900\\
|
||||
\hline
|
||||
Total &500 & \np{1000} & \np{1500}\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\pause
|
||||
\vfill
|
||||
Nom des évènements
|
||||
\[
|
||||
F = \left\{\mbox{Le jeune est fumeur}\right\}
|
||||
\]
|
||||
\[
|
||||
P = \left\{\mbox{Un de ses parents est fumeur}\right\}
|
||||
\]
|
||||
\vfill
|
||||
\pause
|
||||
Représenter ce tirage sous forme d'un arbre.
|
||||
\vfill
|
||||
\end{frame}
|
||||
|
||||
\end{document}
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "master"
|
||||
%%% End:
|
||||
|
||||
Binary file not shown.
@@ -0,0 +1,70 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
|
||||
\title{Arbre de probabilité - Bilan 2}
|
||||
\date{Novembre 2019}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\section*{Arbre de probabilité (suite)}
|
||||
|
||||
\subsection*{Propriétés}
|
||||
\begin{minipage}{0.5\textwidth}
|
||||
Soit $A$ et $B$ deux évènements de $\Omega$ avec $P(A) \neq 0$. Alors on peut considérer l'arbre de probabilité ci-contre et on obtient les propriétés suivantes:
|
||||
|
||||
\begin{itemize}
|
||||
\item La somme des probabilités des branches issues d'un même noeud est égale à 1. On a alors
|
||||
\[
|
||||
P(A) + P(\overline{ A }) = 1 \mbox{ ou encore } P_A(B) + P_A(\overline{ B }) = 1
|
||||
\]
|
||||
\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
|
||||
\[
|
||||
P(B) = P(A\cap B) + P(\overline{A} \cap B)
|
||||
\]
|
||||
\end{itemize}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
\begin{tikzpicture}[grow=right, sloped, scale=1.5]
|
||||
\node {.}
|
||||
child [red] {node {$A$}
|
||||
child {node {$B$}
|
||||
edge from parent
|
||||
node[above] {$P_A(B)$}
|
||||
}
|
||||
child [black] {node {$\overline{B}$}
|
||||
edge from parent
|
||||
node[above] {$P_A(\overline{B})$}
|
||||
}
|
||||
edge from parent
|
||||
node[above] {$P(A)$}
|
||||
}
|
||||
child[missing] {}
|
||||
child { node {$\overline{A}$}
|
||||
child {node {$B$}
|
||||
edge from parent
|
||||
node[above] {$P_{\overline{A}}(B)$}
|
||||
}
|
||||
child {node {$\overline{B}$}
|
||||
edge from parent
|
||||
node[above] {$P_{\overline{A}}(\overline{B})$}
|
||||
}
|
||||
edge from parent
|
||||
node[above] {$P(\overline{A})$}
|
||||
}%
|
||||
;
|
||||
\end{tikzpicture}
|
||||
|
||||
\end{minipage}
|
||||
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
@@ -0,0 +1,116 @@
|
||||
\documentclass[a4paper,10pt, twocolumn, landscape]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\title{Probabilité conditionnelle - Arbres}
|
||||
\tribe{Terminale ES}
|
||||
\date{Novembre 2019}
|
||||
|
||||
\geometry{left=10mm,right=10mm, top=10mm}
|
||||
|
||||
\pagestyle{empty}
|
||||
\setlength{\columnseprule}{0pt}
|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{exercise}[subtitle={Neuf ou occasion}]
|
||||
Un concessionnaire automobile vend chaque année 65\% de véhicules neufs. Une étude montre que parmi les acheteurs de véhicules neufs, 40\% adhèrent à un contrat d'assurance. Par ailleurs, 7\% des acheteurs ont acquis un véhicule d'occasion et adhéré à un contrat de maintenance.
|
||||
|
||||
On choisit un client au hasard parmi les clients de ce concessionnaire et on considère les évènements suivants:
|
||||
\begin{itemize}
|
||||
\item $N = \left\{ \mbox{ Le client achète un véhicule neuf } \right\}$
|
||||
\item $M = \left\{ \mbox{ Le client souscrit à un contrat de maintenance } \right\}$
|
||||
\end{itemize}
|
||||
\begin{enumerate}
|
||||
\item Traduire les données de l'énoncé en terme de probabilité en utilisant les évènements $N$ et $M$.
|
||||
\item À partir des données de l'énoncé, construire un arbre de probabilité traduisant la situation.
|
||||
\item Traduire en français les probabilités suivantes, les calculer puis les placer sur l'arbre.
|
||||
\begin{multicols}{3}
|
||||
\begin{enumerate}
|
||||
\item $P(\overline{N})$
|
||||
\item $P_N(\overline{M})$
|
||||
\item $P(M \cap N)$
|
||||
\item $P(M \cap \overline{N})$
|
||||
\item $P_{\overline{N}}(M)$
|
||||
\item $P_{\overline{N}}(\overline{M})$
|
||||
\end{enumerate}
|
||||
\end{multicols}
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
\vfill
|
||||
\begin{exercise}[subtitle={Technique}]
|
||||
On considère 2 évènements $A$ et $B$.
|
||||
|
||||
On donne les probabilités suivantes
|
||||
\begin{multicols}{3}
|
||||
\begin{itemize}
|
||||
\item $P(A) = 0.4$
|
||||
\item $P(\overline{B}) = 0.5$
|
||||
\item $P_A(B) = 0.2$
|
||||
\item $P_B(A) = 0.16$
|
||||
\item $P_{\overline{A}}(\overline{B}) = 0.3$
|
||||
\item $P_{\overline{B}}(A) = 0.64$
|
||||
\end{itemize}
|
||||
\end{multicols}
|
||||
\begin{enumerate}
|
||||
\item Construire et compléter les deux arbres de probabilité possibles.
|
||||
\item Calculer de deux façons différentes $P(A\cap B)$.
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\begin{exercise}[subtitle={Lecture sur arbre}]
|
||||
\begin{minipage}{0.2\textwidth}
|
||||
\begin{tikzpicture}[sloped]
|
||||
\node {.}
|
||||
child {node {$F$}
|
||||
child {node {$E$}
|
||||
edge from parent
|
||||
node[above] {0.8}
|
||||
}
|
||||
child {node {$\overline{E}$}
|
||||
edge from parent
|
||||
node[above] {...}
|
||||
}
|
||||
edge from parent
|
||||
node[above] {0.3}
|
||||
}
|
||||
child[missing] {}
|
||||
child { node {$\overline{F}$}
|
||||
child {node {$E$}
|
||||
edge from parent
|
||||
node[above] {...}
|
||||
}
|
||||
child {node {$\overline{E}$}
|
||||
edge from parent
|
||||
node[above] {0.1}
|
||||
}
|
||||
edge from parent
|
||||
node[above] {...}
|
||||
} ;
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.25\textwidth}
|
||||
On considère 2 évènements $F$ et $E$. Lire ou calculer les probabilités suivantes
|
||||
\begin{multicols}{2}
|
||||
\begin{enumerate}
|
||||
\item $P(\overline{F})$
|
||||
\item $P_F(\overline{E})$
|
||||
\item $P_{\overline{F}}(E)$
|
||||
\item $P_{\overline{F}}(\overline{E})$
|
||||
\item $P(E \cap F)$
|
||||
\item $P(E \cap \overline{F})$
|
||||
\end{enumerate}
|
||||
\end{multicols}
|
||||
\end{minipage}
|
||||
|
||||
\end{exercise}
|
||||
|
||||
\pagebreak
|
||||
|
||||
\printexercise{exercise}{1}
|
||||
\vfill
|
||||
\printexercise{exercise}{2}
|
||||
\vfill
|
||||
\printexercise{exercise}{3}
|
||||
\vfill
|
||||
\end{document}
|
||||
Binary file not shown.
@@ -0,0 +1,149 @@
|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\title{Probabilité conditionnelle - Annales}
|
||||
\tribe{Terminale ES}
|
||||
\date{Novembre 2019}
|
||||
|
||||
\geometry{left=10mm,right=10mm, top=10mm}
|
||||
|
||||
\pagestyle{empty}
|
||||
\setlength{\columnseprule}{0pt}
|
||||
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{exercise}[subtitle={Aéroport}]
|
||||
Dans un aéroport, les portiques de sécurité servent à détecter les objets métalliques que peuvent emporter les voyageurs.
|
||||
|
||||
On choisit au hasard un voyageur franchissant un portique.
|
||||
|
||||
\begin{itemize}
|
||||
\item $S$ l'événement \og le voyageur fait sonner le portique \fg{};
|
||||
\item $M$ l'événement \og le voyageur porte un objet métallique \fg{}.
|
||||
\end{itemize}
|
||||
|
||||
On considère qu'un voyageur sur 500 porte sur lui un objet métallique. Et on note que
|
||||
|
||||
\begin{itemize}
|
||||
\item Lorsqu'un voyageur franchit le portique avec un objet métallique, la probabilité que le portique sonne est égale à $0,98$;
|
||||
\item Lorsqu'un voyageur franchit le portique sans objet métallique, la probabilité que le portique ne sonne pas est aussi égale à $0,98$.
|
||||
\end{itemize}
|
||||
|
||||
\begin{minipage}{0.6\textwidth}
|
||||
\begin{enumerate}
|
||||
\item À l'aide des données de l'énoncé, préciser les valeurs de $P(M)$, $P_{M}(S)$ et $P_{\overline{M}}(\overline{S})$.
|
||||
\item Recopier et compléter l'arbre pondéré ci-contre illustrant cette situation.
|
||||
\item Montrer que: $P(S)=\np{0,02192}$.
|
||||
\item En déduire la probabilité qu'un voyageur porte un objet métallique sachant qu'il a fait sonner le portique. (On arrondira le résultat à $10^{-3}$.)
|
||||
\end{enumerate}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
\begin{minipage}{0.3\textwidth}
|
||||
\begin{tikzpicture}[sloped]
|
||||
\node {.}
|
||||
child {node {$M$}
|
||||
child {node {$S$}
|
||||
edge from parent
|
||||
node[above] {...}
|
||||
}
|
||||
child {node {$\overline{S}$}
|
||||
edge from parent
|
||||
node[above] {...}
|
||||
}
|
||||
edge from parent
|
||||
node[above] {...}
|
||||
}
|
||||
child[missing] {}
|
||||
child { node {$\overline{M}$}
|
||||
child {node {$S$}
|
||||
edge from parent
|
||||
node[above] {...}
|
||||
}
|
||||
child {node {$\overline{S}$}
|
||||
edge from parent
|
||||
node[above] {...}
|
||||
}
|
||||
edge from parent
|
||||
node[above] {...}
|
||||
} ;
|
||||
\end{tikzpicture}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
\end{exercise}
|
||||
\begin{exercise}[subtitle={Sponsort}]
|
||||
Un navigateur s'entraîne régulièrement dans le but de battre le record du monde de
|
||||
traversée de l'Atlantique à la voile.
|
||||
|
||||
\emph{Dans cet exercice, les résultats seront arrondis au millième si nécessaire.}
|
||||
|
||||
Une entreprise nommée \og Régate \fg, s'intéresse aux résultats de ce navigateur.
|
||||
|
||||
La probabilité qu'il réalise la traversée en moins de 6 jours est de 0,16.
|
||||
|
||||
Si le navigateur réalise la traversée en moins de 6 jours, l'entreprise le sponsorise avec une probabilité de 0,95.
|
||||
|
||||
Sinon, l'entreprise hésite et le sponsorise avec une probabilité de 0,50.
|
||||
|
||||
On note $M$ l'évènement \og la traversée est réalisée par le navigateur en moins de 6 jours \fg et $F$ l'évènement \og l'entreprise sponsorise le navigateur \fg.
|
||||
|
||||
\begin{enumerate}
|
||||
\item Représenter cette situation à l'aide d'un arbre pondéré.
|
||||
\item Montrer que la probabilité que l'entreprise ne sponsorise pas le navigateur à la
|
||||
prochaine course est $0,428$.
|
||||
\item L'entreprise a finalement choisi de ne pas financer le navigateur.
|
||||
|
||||
Calculer la probabilité que le navigateur ait tout de même réalisé la traversée en moins
|
||||
de $6$ jours.
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\begin{exercise}[subtitle={Paiements}]
|
||||
Un commerçant dispose dans sa boutique d'un terminal qui permet à ses clients, s'ils souhaitent
|
||||
régler leurs achats par carte bancaire, d'utiliser celle-ci en mode sans contact (quand le montant de
|
||||
la transaction est inférieur ou égal à 30~\euro) ou bien en mode code secret (quel que soit le montant
|
||||
de la transaction).
|
||||
|
||||
Il remarque que 80\,\% de ses clients règlent des sommes inférieures ou égales à 30~\euro. Parmi eux :
|
||||
\begin{itemize}
|
||||
\item 40\,\% paient en espèces;
|
||||
\item 40\,\% paient avec une carte bancaire en mode sans contact ;
|
||||
\item les autres paient avec une carte bancaire en mode code secret.
|
||||
\end{itemize}
|
||||
Et que 20\,\% de ses clients règlent des sommes strictement supérieures à 30~\euro. Parmi eux :
|
||||
\begin{itemize}
|
||||
\item 70\,\% paient avec une carte bancaire en mode code secret ;
|
||||
\item les autres paient en espèces.
|
||||
\end{itemize}
|
||||
On interroge au hasard un client qui vient de régler un achat dans la boutique.
|
||||
|
||||
On considère les évènements suivants :
|
||||
|
||||
$V$ : \og le client a réglé un montant inférieur ou égal à 30~\euro \fg ;
|
||||
|
||||
$E$ : \og le client a réglé en espèces\fg ;
|
||||
|
||||
$C$ : \og le client a réglé avec sa carte bancaire en mode code secret\fg ;
|
||||
|
||||
$S$ : \og le client a réglé avec sa carte bancaire en mode sans contact \fg.
|
||||
|
||||
\medskip
|
||||
|
||||
\begin{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item Donner la probabilité de l'évènement $V$, notée $P(V)$, ainsi que la probabilité de $S$ sachant
|
||||
$V$ notée $P_V(S)$.
|
||||
\item Traduire la situation de l'énoncé à l'aide d'un arbre pondéré.
|
||||
\end{enumerate}
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item Calculer la probabilité que pour son achat, le client ait réglé un montant inférieur ou égal à $30$~\euro{} et qu'il ait utilisé sa carte bancaire en mode sans contact.
|
||||
\item Montrer que la probabilité de l'évènement: \og pour son achat, le client a réglé avec sa carte
|
||||
bancaire en utilisant l'un des deux modes\fg{} est égale à $0,62$.
|
||||
\end{enumerate}
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\end{document}
|
||||
@@ -0,0 +1,47 @@
|
||||
Probabilités conditionnelles pour l'année 2019-2020 en terminale ES
|
||||
###################################################################
|
||||
|
||||
:date: 2019-11-23
|
||||
:modified: 2019-11-23
|
||||
:authors: Bertrand Benjamin
|
||||
:category: TESL
|
||||
:tags: Probabilité, Variable aléatoire, Conditionnement
|
||||
:summary: Construction d'arbre, formule de Bayes et probabilité totale pour l'année 2019-2020 en terminale ESL
|
||||
|
||||
Étape 1: Arbre de probabilité
|
||||
=============================
|
||||
|
||||
.. image:: 1E_tableau_croisee.pdf
|
||||
:height: 200px
|
||||
:alt: Tableau croisée et vrai/faux
|
||||
|
||||
.. image:: 1P_tableau_fumeurs.pdf
|
||||
:height: 200px
|
||||
:alt: Version projetable
|
||||
|
||||
On propose d'étudier un tableau croisé. Pour cela, on propose des phrases et les élèves doivent dire si elles sont vraies ou fausse. On cherchera petit à petit à traduire les phrases en notation ensembliste et probabiliste.
|
||||
|
||||
.. image:: 1B_arbre.pdf
|
||||
:height: 200px
|
||||
:alt: Bilan de la première activité
|
||||
|
||||
Étape 2: Formule de Bayes
|
||||
=========================
|
||||
|
||||
.. image:: 2E_arbres.pdf
|
||||
:height: 200px
|
||||
:alt: Exercices de contruction et lecture d'arbres
|
||||
|
||||
Retrouver la formule de Bayes à l'aide d'un arbre de probabilité. On peut ajouter une question bonus au dernier exercice pour demander de calculer P(E) et ainsi découvrir la formule de probabilité totale.
|
||||
|
||||
.. image:: 2B_bayes_intersection.pdf
|
||||
:height: 200px
|
||||
:alt: Bilan sur les calculs de probabilités sur les arbres
|
||||
|
||||
Étape 3: Annales de BAC
|
||||
=======================
|
||||
|
||||
.. image:: 3E_annales.pdf
|
||||
:height: 200px
|
||||
:alt: Annales 2018 Liban - Asie - Pondichery
|
||||
|
||||
Reference in New Issue
Block a user