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|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\author{Benjamin Bertrand}
|
||||
\title{Binomiale et echantillonnage - Cours}
|
||||
\date{octobre 2020}
|
||||
|
||||
\pagestyle{empty}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
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|
||||
\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\author{Benjamin Bertrand}
|
||||
\title{Binomiale et echantillonnage - Cours}
|
||||
\date{octobre 2020}
|
||||
|
||||
\pagestyle{empty}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
\section{Expérience et loi de Bernoulli}
|
||||
|
||||
\subsection*{Définition}
|
||||
|
||||
Une expérience aléatoire qui a deux issues possibles (que l'on nommera \textbf{succès} et \textbf{échec}) est appelé \textbf{épreuve de Bernoulli}.
|
||||
|
||||
En associant la valeur 1 à un succès et 0 à un échec. On peut modéliser cette expérience avec un variable aléatoire $X$ qui suit un \textbf{loi de Bernoulli} (notée $X \sim \mathcal{B}(p)$) résumée par le tableau suivant:
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{|c|*{2}{C{2cm}|}}
|
||||
\hline
|
||||
Valeurs & 1 & 0 \\
|
||||
\hline
|
||||
Probabilité & p & 1-p \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
où $p$ est la probabilité d'avoir un succès.
|
||||
|
||||
\subsubsection*{Exemple}
|
||||
Un passager qui a 9 chances sur 10 de se présenter à l'embarquement d'un avion.
|
||||
|
||||
\afaire{Préciser ce qu'est le succès, l'échec, déterminer la valeur de $p$ et compléter le tableau}
|
||||
|
||||
\subsection*{Propriétés}
|
||||
Soit $X \sim \mathcal{B}$ alors
|
||||
\begin{itemize}
|
||||
\item L'espérance de $X$ est $E[X] = p$
|
||||
\item L'écart-type de $X$ est $\sigma = \sqrt{p(1-p)}$
|
||||
\end{itemize}
|
||||
|
||||
\subsubsection*{Démonstration}
|
||||
\afaire{Démontrer la formule de l'espérance}
|
||||
|
||||
\section{Loi binomiale}
|
||||
|
||||
On a vu que pour simuler tout un vol, c'est à dire 53 passagers, il fallait répéter 53 fois l'épreuve de Bernoulli vue dans l'exemple précédent. Les répétitions d'épreuve de Bernoulli s'appellent \textbf{schéma de Bernoulli} et sont modéliser avec une loi \textbf{binomiale}.
|
||||
|
||||
\subsection*{Définition}
|
||||
|
||||
La \textbf{loi Binomiale de paramètre $n$ et $p$} notée $\mathcal{B}(n;p)$ est la loi de probabilité qui modélise la somme de répétitions indépendantes et identiques de $n$ situations modélisées par une loi de Bernoulli de paramètre $p$.
|
||||
|
||||
\bigskip
|
||||
|
||||
Ces situations peuvent être représenté par un arbre de probabilité où chaque étage correspond à une répétition.
|
||||
|
||||
\subsubsection*{Exemple}
|
||||
|
||||
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.
|
||||
|
||||
On note $X$ le nombre de fois que Sarah est interrogée.
|
||||
|
||||
\afaire{Quelle loi suit $X$? Représenter la situation avec un arbre de probabilité}
|
||||
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
@ -2,7 +2,7 @@
|
||||
\usepackage{myXsim}
|
||||
|
||||
\author{Benjamin Bertrand}
|
||||
\title{Binomiale et echantillonnage - Cours}
|
||||
\title{Simulation de suréservation}
|
||||
\date{octobre 2020}
|
||||
|
||||
\DeclareExerciseCollection{banque}
|
||||
@ -15,4 +15,4 @@
|
||||
\input{exercises.tex}
|
||||
\printcollection{banque}
|
||||
|
||||
\end{document}
|
||||
\end{document}
|
||||
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\documentclass[a4paper,10pt]{article}
|
||||
\usepackage{myXsim}
|
||||
|
||||
\author{Benjamin Bertrand}
|
||||
\title{Modélisation avec la loi binomiale}
|
||||
\date{octobre 2020}
|
||||
|
||||
\DeclareExerciseCollection{banque}
|
||||
\xsimsetup{
|
||||
step=2,
|
||||
}
|
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|
||||
\begin{document}
|
||||
|
||||
\input{exercises.tex}
|
||||
\printcollection{banque}
|
||||
|
||||
\end{document}
|
||||
@ -1,10 +1,131 @@
|
||||
\collectexercises{banque}
|
||||
\begin{exercise}[subtitle={<++>}, step={1}, origin={<++>}, topics={Binomiale et echantillonnage}, tags={Probabilité, Échantillonnage, Binomiale}]
|
||||
<++>
|
||||
\begin{exercise}[subtitle={Surréservation}, step={1}, origin={Création}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale, Simulation}]
|
||||
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.
|
||||
|
||||
On considère une ligne aérienne entre deux villes pour laquelle:
|
||||
\begin{multicols}{2}
|
||||
\begin{itemize}
|
||||
\item Tous les avions ont 50 places.
|
||||
\item 53 réservations sont vendues pour chaque vol (on supposera qu'elles sont toutes vendues)
|
||||
\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).
|
||||
\item Un billet vendu rapporte 100\euro. Un billet remboursé coûte 250\euro à la compagnie.
|
||||
\item Chaque personne ayant réservé une place se présente au non à l'embarquement indépendamment des autres personnes ayant réservé sur le même vol.
|
||||
\end{itemize}
|
||||
\end{multicols}
|
||||
|
||||
\begin{center}
|
||||
\textbf{La compagnie prend elle un risque en vendant 53 places au lieu des 50 disponibles?}
|
||||
\end{center}
|
||||
|
||||
Pour évaluer les risques liés à une surréservation, nous allons \textbf{simuler} avec le tableur plusieurs vols sur cette ligne.
|
||||
|
||||
\begin{enumerate}
|
||||
\item On commence par simuler un vol où 53 places ont été vendues.
|
||||
|
||||
\begin{minipage}{0.6\textwidth}
|
||||
Pour savoir si une personne se présente ou non à l'embarquement, nous utiliserons la commande \calc{=SI(ALEA()>0.9;0;1)}. Cette commande renvoie:
|
||||
\begin{itemize}
|
||||
\item 0 si le passager ne s'est pas présenté
|
||||
\item 1 s'il s'est présenté.
|
||||
\end{itemize}
|
||||
\begin{enumerate}
|
||||
\item Réaliser la simulation pour le premier vol
|
||||
\item Combien de personnes se sont-elles présentées à l'embarquement?
|
||||
\item Quelle formule peut-on rentrer en \texttt{B56} pour calculer ce nombre?
|
||||
\end{enumerate}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
\includegraphics[scale=0.27]{./fig/vol1}
|
||||
\end{minipage}
|
||||
|
||||
\item
|
||||
\begin{enumerate}
|
||||
\item Réaliser cette simulation pour 100 vols de cette ligne.
|
||||
|
||||
\hspace{-2cm}
|
||||
\includegraphics[scale=0.17]{./fig/vol100}
|
||||
|
||||
\item Quelle formule doit-on entrer en \texttt{CX56} pour calculer la moyenne du nombre de passager?
|
||||
\item Pensez vous que le risque de surréservation est grand?
|
||||
\end{enumerate}
|
||||
\item On veut maintenant évaluer le risque de surréservation. Pour savoir si un vol est en surréservation, on utilise la commande \calc{=SI(nbr_passagers > 50;1;0)} (avec \lstinline|nbr_passagers| à remplacer le nom de la case)
|
||||
\begin{enumerate}
|
||||
\item Compléter le tableau pour connaître les vols en surréservation.
|
||||
|
||||
\hspace{-2cm}
|
||||
\includegraphics[scale=0.15]{./fig/vol100_overbooking}
|
||||
\item Quelle formule doit-on rentrer en \texttt{CX59} pour compter le nombre de vols en surréservation?
|
||||
\item Quelle formule doit-on entrer dans \texttt{CX61} pour calculer la fréquence des vols en surréservation?
|
||||
\end{enumerate}
|
||||
\item Répondre à la question initiale en vous basant sur les résultats de votre simulation.
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\begin{solution}
|
||||
<++>
|
||||
\end{solution}
|
||||
\begin{exercise}[subtitle={Modélisation}, step={2}, origin={Création}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale}]
|
||||
Dans chacune des situations suivantes, dessiner l'arbre de probabilité qui décrit la situation puis expliquer si oui ou non elle peut être modélisé par une loi binomiale en précisant les paramètres.
|
||||
\begin{enumerate}
|
||||
\item Dans mon jardin j'ai planté 3 fraisiers suffisamment éloignés pour qu'ils ne se gênent pas. D'expérience, ils donnent des fruits dans 90\% des cas. Je m'intéresse au nombre de fraisier qui donneront des fruits.
|
||||
\item Bob mange à la cantine 3 fois par semaine. À chaque fois, il se demande s'il prend un dessert plutôt qu'un fromage ce qu'il fait 2 fois sur 3. On s'intéresse au nombre de fois où il a mangé du dessert en une semaine.
|
||||
\item Dans un sachet, il reste 6 bonbons: 2 à la fraise et 4 au réglisse. J'en choisi 4 au hasard et je les mange. Je m'intéresse au nombre de bonbon à la fraise que j'ai mangé.
|
||||
\item Je joue avec un dé à 6 faces. J'ai le droit à un maximum de 4 lancers. J'arrête de lancer dès que j'ai obtenu un 6. Je compte le nombre de lancer que je fais.
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\collectexercisesstop{banque}
|
||||
\begin{exercise}[subtitle={Création}, step={2}, origin={Création}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale}]
|
||||
Proposer une expérience aléatoire qui pourrait être modélisée avec une loi binomiale. Vous détaillerez ensuite les paramètres et justifierez la modélisation.
|
||||
\end{exercise}
|
||||
|
||||
\begin{exercise}[subtitle={Jeux}, step={2}, origin={Création}, topics={Binomiale et echantillonnage}, tags={Probabilité, Binomiale}]
|
||||
Bob joue à un jeu où il estime qu'il a 70\% de chance de gagner une partie. Entre 2 parties, il prend le temps de se reposer pour que la précédente partie n'influence pas la suivante.
|
||||
|
||||
On note $V$ l'évènement "Bob gagne la partie".
|
||||
|
||||
Bob fait 2 parties et on note $X$ la variable aléatoire qui compte le nombre de victoire.
|
||||
\begin{enumerate}
|
||||
\item Faire un arbre qui modélise la situation.
|
||||
\item Déterminer la probabilité que Bob gagne une seule partie.
|
||||
\item Avec quelle loi peut-on modéliser la variable aléatoire $X$? Préciser les paramètres.
|
||||
\item Démontrer que l'espérance de $X$ est de 1,4.
|
||||
\item Si Bob joue tous les jours deux parties, combien en moyenne peut-il espérer en gagner quotidiennement?
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\begin{exercise}[subtitle={Repas}]
|
||||
Bob adore manger des légumes. Chaque jour, il choisit au hasard un fruit dans une panière quotidiennement remplie par ses parents contenant 7 bananes, 5 pommes et 2 kiwi.
|
||||
|
||||
Ses parents veulent essayer de prévoir la consommation en banane de Bob sur 3 jours.
|
||||
|
||||
On note donc $X$ le nombre bananes mangées par Bob sur 3 jours et $B$ l'évènement "Bob mange une banane".
|
||||
\begin{enumerate}
|
||||
\item Faire un arbre qui modélise la situation.
|
||||
\item Déterminer la probabilité que Bob gagne deux bananes.
|
||||
\item Avec quelle loi peut-on modéliser la variable aléatoire $X$? Préciser les paramètres.
|
||||
\item Calculer l'espérance de $X$. Interpréter le résultat.
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\begin{exercise}[subtitle={Auto-école}]
|
||||
Dans une auto-école, à chaque session 75\% des candidats réussissent à avoir leur code.
|
||||
|
||||
\begin{enumerate}
|
||||
\item On interroge au hasard 4 candidats d'une session pour savoir s'ils ont eu leur code. On note $X$ variable aléatoire qui compte le nombre de réponse positive.
|
||||
\begin{enumerate}
|
||||
\item Avec quelle loi peut-on modéliser la variable aléatoire $X$? Préciser les paramètres.
|
||||
\item Calculer les probabilités suivantes
|
||||
\[
|
||||
P(X = 1) \qquad \qquad
|
||||
P(X = 4) \qquad \qquad
|
||||
P(X \leq 1)
|
||||
\]
|
||||
\item Quelle est la probabilité qu'au moins un candidat ait répondu positivement.
|
||||
\item En moyenne combien de réponses positives peut-on espérer avoir?
|
||||
\end{enumerate}
|
||||
\item Cette fois-ci, on choisit un candidat et on note $Y$ le nombre de sessions qu'il a du passer avant d'avoir code.
|
||||
\begin{enumerate}
|
||||
\item Faire un arbre pour représenter la situtation.
|
||||
\item Peut-on modéliser $Y$ avec une loi binomiale? Si oui, préciser les paramètres.
|
||||
\end{enumerate}
|
||||
\end{enumerate}
|
||||
\end{exercise}
|
||||
|
||||
\collectexercisesstop{banque}
|
||||
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"
|
||||
id="image5322"
|
||||
x="4134.0459"
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y="1710.1553" />
|
||||
</g>
|
||||
</svg>
|
||||
|
After Width: | Height: | Size: 57 KiB |
@ -13,13 +13,25 @@ Binomiale et echantillonnage
|
||||
|
||||
Activité avec le tableur où l'on essaie de simuler une situation de suréservation d'un avion.
|
||||
|
||||
.. image:: ./1E_surreservation.pdf
|
||||
:height: 200px
|
||||
:alt: Simulation avec le tableur de la surréservation d'avions.
|
||||
|
||||
Bilan: définitions de loi de Bernoulli et de la loi binomiale (caractères pour modéliser et représentation par un arbre).
|
||||
|
||||
.. image:: ./1B_bernoulli_binomiale.pdf
|
||||
:height: 200px
|
||||
:alt: Cours sur la loi de bernoulli et la loi binomiale.
|
||||
|
||||
Étape 2: Étude de situations aléatoires et répétées
|
||||
===================================================
|
||||
|
||||
Plusieurs situations pouvant être modélisées ou pas par une loi binomiale où l'on demande sur des petits arbres de calculer des probabilités.
|
||||
|
||||
.. image:: ./2E_modelisation.pdf
|
||||
:height: 200px
|
||||
:alt: Éxercices de modélisation avec la loi binomiale.
|
||||
|
||||
Cours: formule de calcul de probabilité pour la loi binomiale et graphique pour les représenter.
|
||||
|
||||
Étape 3: Augmenter le nombre de répétitions
|
||||
|
||||
Loading…
Reference in New Issue
Block a user