From daef72df19b7343304c8a4b29d99e62334e45f2f Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Wed, 28 Oct 2020 12:04:44 +0100 Subject: [PATCH] =?UTF-8?q?Feat:=20exercices=20de=20mod=C3=A9lisation=20po?= =?UTF-8?q?ur=20les=20math=20complementaires?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../2E_modelisation.pdf | Bin 0 -> 49471 bytes .../2E_modelisation.tex | 18 +++++ .../exercises.tex | 69 +++++++++++++++++- .../01_Binomiale_et_echantillonnage/index.rst | 4 + 4 files changed, 90 insertions(+), 1 deletion(-) create mode 100644 Complementaire/01_Binomiale_et_echantillonnage/2E_modelisation.pdf create mode 100644 Complementaire/01_Binomiale_et_echantillonnage/2E_modelisation.tex diff --git a/Complementaire/01_Binomiale_et_echantillonnage/2E_modelisation.pdf b/Complementaire/01_Binomiale_et_echantillonnage/2E_modelisation.pdf new file mode 100644 index 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) z{W}i3W65;h>tM&7^{P=F@yNhJvGT-*s*N6kaS8Qi@zN~&E$oL&<6;kN@lfac_*_A= zF;@ihLS;=67}@38Ma;qvrzh>~gn?gvR-di3&On87+JN}G4BosJKQM2LH6T>HC2TiR z)Zo};!Y+Fy;qoQFY|nrtoJHCM6crNczM!%~c<##vCak?7i)LI!`=Z@MX?q1W?HEmQ zn-6iES}uQoSF*Y;!EDUIdmLd|&TXvZh_#E?9Q^dfq}>QYA3TFyAb}s+5=jk$G9Lq~iGCY?;+^ zeoD35CDnkT1+IY(psc}^ox`>a%#>jvz2o?(Zkd5|e)hJ{!MnZ|ITzsP=}VV(4%pVP z25H=BB3PqAP;e}yPW4jAvtsEbtrq&hXeol-+DW72#_!NXuc^mbp@EKnkl3DHXH@33 z+t!C1`}DQ9_jj|TCSzwtVOcB9+oC{VXBe>#YCOqbOqu7kyWY@rER&~#LZE<&sjLX@?sgx1Ckux zn6P+k{RZu;MD6_f(}v@}@qa>3NZ@u^Wgp8MSkn~FGFiQ1}3`yf?|08cg9Pc8F3i_e&pai zN8DShJ$$lo0)rh_y`4OR9U`f%!0WzNuo0QGRCLx{NjjT8Z8rDsmA;gpLJK)^avU&7 zw7MW;MIs8%nu*7&jmhjTe-PL0pu~Sg;Od8ll8Ikj+uE7? z&?l|E#MO3DSexq`HJ1ftyn>&^y8ru&b#&5qaB}+v_K-|$?5r%1#Kgj~B9Q+BIoSG} literal 0 HcmV?d00001 diff --git a/Complementaire/01_Binomiale_et_echantillonnage/2E_modelisation.tex b/Complementaire/01_Binomiale_et_echantillonnage/2E_modelisation.tex new file mode 100644 index 0000000..d5260aa --- /dev/null +++ b/Complementaire/01_Binomiale_et_echantillonnage/2E_modelisation.tex @@ -0,0 +1,18 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Modélisation avec la loi binomiale} +\date{octobre 2020} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + step=2, +} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} + +\end{document} diff --git a/Complementaire/01_Binomiale_et_echantillonnage/exercises.tex b/Complementaire/01_Binomiale_et_echantillonnage/exercises.tex index eb78fa2..9a1dfc3 100644 --- a/Complementaire/01_Binomiale_et_echantillonnage/exercises.tex +++ b/Complementaire/01_Binomiale_et_echantillonnage/exercises.tex @@ -1,5 +1,5 @@ \collectexercises{banque} -\begin{exercise}[subtitle={Surréservation}, step={1}, origin={Création}, 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: @@ -61,4 +61,71 @@ \end{enumerate} \end{exercise} +\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} + +\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} diff --git a/Complementaire/01_Binomiale_et_echantillonnage/index.rst b/Complementaire/01_Binomiale_et_echantillonnage/index.rst index fbdfbae..d55c6e8 100644 --- a/Complementaire/01_Binomiale_et_echantillonnage/index.rst +++ b/Complementaire/01_Binomiale_et_echantillonnage/index.rst @@ -28,6 +28,10 @@ Bilan: définitions de loi de Bernoulli et de la loi binomiale (caractères pour 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