{"id":1975,"date":"2023-11-22T22:47:57","date_gmt":"2023-11-22T21:47:57","guid":{"rendered":"http:\/\/localhost:8080\/maxblog\/?p=1975"},"modified":"2023-11-22T22:48:32","modified_gmt":"2023-11-22T21:48:32","slug":"echantillonnage","status":"publish","type":"post","link":"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/","title":{"rendered":"\u00c9chantillonnage"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"1975\" class=\"elementor elementor-1975\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7d71a4a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7d71a4a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-389cd7d\" data-id=\"389cd7d\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a8169f1 elementor-widget elementor-widget-spacer\" data-id=\"a8169f1\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7643655 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7643655\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f297427\" data-id=\"f297427\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-361d80f elementor-widget elementor-widget-heading\" data-id=\"361d80f\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-heading-title elementor-size-xl\">\u00c9chantillonnage<\/div>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-de06178 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"de06178\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-c8b729b\" data-id=\"c8b729b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-1f07f28\" data-id=\"1f07f28\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b7c611b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b7c611b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9a7e2c2\" data-id=\"9a7e2c2\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fffe349 nc-justify-text elementor-widget elementor-widget-text-editor\" data-id=\"fffe349\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/#Introduction\" >Introduction<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/#Cours\" >Cours<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/#1_Echantillon_aleatoire_de_taille_n_pour_une_experience_a_deux_issues\" >1. \u00c9chantillon al\u00e9atoire de taille n pour une exp\u00e9rience \u00e0 deux issues<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/#2_Loi_des_grands_nombres\" >2. Loi des grands nombres<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/#3_Estimation_dune_probabilite_ou_dune_proportion_dans_une_population\" >3. Estimation d'une probabilit\u00e9 ou d'une proportion dans une population<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/#Methodes\" >M\u00e9thodes<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/#1_Lire_et_comprendre_une_fonction_Python_pour_echantillonnage\" >1. Lire et comprendre une fonction Python pour \u00e9chantillonnage<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/#2_Observer_la_loi_des_grands_nombres_via_une_simulation_Python\" >2. Observer la loi des grands nombres via une simulation Python<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.maxdecours.com\/maxblog\/echantillonnage\/#3_Simulation_de_N_echantillons_et_calcul_de_proportion\" >3. Simulation de N \u00e9chantillons et calcul de proportion<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"Introduction\"><\/span>Introduction<span class=\"ez-toc-section-end\"><\/span><\/h1><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">L&rsquo;\u00e9chantillonnage est un outil puissant qui nous permet d&rsquo;estimer des proportions et des probabilit\u00e9s dans une population \u00e0 partir d&rsquo;un sous-ensemble de donn\u00e9es. Il repose sur l&rsquo;id\u00e9e que, en augmentant la taille de l&rsquo;\u00e9chantillon, nos estimations seront de plus en plus proches de la r\u00e9alit\u00e9. Cette approche est largement utilis\u00e9e dans de nombreux domaines tels que la sociologie, la m\u00e9decine, et la science politique pour n&rsquo;en nommer que quelques-uns.<\/p><p class=\"MsoNormal\">\u00a0<\/p><h1><span class=\"ez-toc-section\" id=\"Cours\"><\/span>Cours<span class=\"ez-toc-section-end\"><\/span><\/h1><p class=\"MsoNormal\">\u00a0<\/p><h2><span class=\"ez-toc-section\" id=\"1_Echantillon_aleatoire_de_taille_n_pour_une_experience_a_deux_issues\"><\/span>1. \u00c9chantillon al\u00e9atoire de taille n pour une exp\u00e9rience \u00e0 deux issues<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Une exp\u00e9rience \u00e0 deux issues est une exp\u00e9rience al\u00e9atoire qui n&rsquo;a que deux r\u00e9sultats possibles. Par exemple, le lancer d&rsquo;une pi\u00e8ce de monnaie a deux issues possibles : pile ou face.<\/p><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Un <b>\u00e9chantillon al\u00e9atoire<\/b> de taille <span class=\"katex-eq\" data-katex-display=\"false\"> n <\/span> pour une telle exp\u00e9rience est un ensemble de <span class=\"katex-eq\" data-katex-display=\"false\"> n <\/span> r\u00e9sultats ind\u00e9pendants obtenus en r\u00e9p\u00e9tant l&rsquo;exp\u00e9rience <span class=\"katex-eq\" data-katex-display=\"false\"> n <\/span> fois. Prenons l&rsquo;exemple du lancer d&rsquo;une pi\u00e8ce de monnaie :<\/p><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">&#8211; Si nous lan\u00e7ons la pi\u00e8ce 10 fois (donc <span class=\"katex-eq\" data-katex-display=\"false\"> n = 10 <\/span>) et obtenons les r\u00e9sultats suivants : P, F, P, P, F, F, P, P, F, P, alors cet ensemble de r\u00e9sultats est un \u00e9chantillon al\u00e9atoire de taille 10.<\/p><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">L&rsquo;ind\u00e9pendance des r\u00e9sultats est importante : le r\u00e9sultat d&rsquo;un lancer ne doit pas influencer les r\u00e9sultats des autres lancers.<\/p><p class=\"MsoNormal\">\u00a0<\/p><h2><span class=\"ez-toc-section\" id=\"2_Loi_des_grands_nombres\"><\/span>2. Loi des grands nombres<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">La <b>loi des grands nombres<\/b> est un concept cl\u00e9 en statistiques et probabilit\u00e9s. Elle peut \u00eatre expliqu\u00e9e simplement ainsi :<\/p><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">\u00ab Lorsque <span class=\"katex-eq\" data-katex-display=\"false\"> n <\/span> est grand, la fr\u00e9quence observ\u00e9e tend \u00e0 se rapprocher de la probabilit\u00e9. \u00bb<\/p><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Prenons \u00e0 nouveau l&rsquo;exemple du lancer de pi\u00e8ce de monnaie. Si la pi\u00e8ce est \u00e9quilibr\u00e9e, la probabilit\u00e9 d&rsquo;obtenir pile est de 0,5. Si nous lan\u00e7ons la pi\u00e8ce un petit nombre de fois, la fr\u00e9quence observ\u00e9e d&rsquo;obtenir pile peut varier consid\u00e9rablement. Mais si nous lan\u00e7ons la pi\u00e8ce un tr\u00e8s grand nombre de fois, disons 10 000 fois, la fr\u00e9quence observ\u00e9e de pile se rapprochera de la probabilit\u00e9 r\u00e9elle, soit 0,5.<\/p><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Cela signifie que plus nous avons de donn\u00e9es, plus nos observations sont susceptibles de refl\u00e9ter la r\u00e9alit\u00e9.<\/p><p class=\"MsoNormal\">\u00a0<\/p><h2>3. Estimation d&rsquo;une probabilit\u00e9 ou d&rsquo;une proportion dans une population<\/h2><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Le principe d&rsquo;estimation d&rsquo;une probabilit\u00e9 ou d&rsquo;une proportion dans une population par une fr\u00e9quence observ\u00e9e sur un \u00e9chantillon est fondamental en statistiques.<\/p><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Supposons que nous voulons conna\u00eetre la proportion de personnes qui pr\u00e9f\u00e8rent le chocolat plut\u00f4t que la vanille dans une ville. Il serait co\u00fbteux et long de poser la question \u00e0 chaque habitant. \u00c0 la place, nous pouvons s\u00e9lectionner un \u00e9chantillon al\u00e9atoire de personnes et leur poser la question.<\/p><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Si, par exemple, dans notre \u00e9chantillon de 100 personnes, 70 pr\u00e9f\u00e8rent le chocolat, nous pourrions estimer que 70% de la population de la ville pr\u00e9f\u00e8re le chocolat. Notre \u00e9chantillon nous donne une <b>estimation<\/b> de la proportion r\u00e9elle dans la population enti\u00e8re.<\/p><p class=\"MsoNormal\">\u00a0<\/p><h1><span class=\"ez-toc-section\" id=\"Methodes\"><\/span>M\u00e9thodes<span class=\"ez-toc-section-end\"><\/span><\/h1><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Ces m\u00e9thodes sont en liaison avec la partie \u00ab\u00a0Algorithmique et programmation\u00a0\u00bb, qu\u2019il est recommand\u00e9 de lire en premier pour comprendre ce qui suit.<\/p><p class=\"MsoNormal\">Vous pouvez lancer les codes sur un interpr\u00e9teur Python, afin d\u2019observer exp\u00e9rimentalement les notions abord\u00e9es.<\/p><p class=\"MsoNormal\">\u00a0<\/p><h2><span class=\"ez-toc-section\" id=\"1_Lire_et_comprendre_une_fonction_Python_pour_echantillonnage\"><\/span>1. Lire et comprendre une fonction Python pour \u00e9chantillonnage<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Voici une fonction Python qui simule le lancer d&rsquo;une pi\u00e8ce de monnaie <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> fois et renvoie le nombre de succ\u00e8s (par exemple, obtenir \u00ab\u00a0pile\u00a0\u00bb) :<\/p><p class=\"MsoNormal\">\u00a0<\/p><pre style=\"line-height: 125%;\"><i><span style=\"color: #408080;\"># Tout texte qui suit le symbole \u2018#\u2019 sur une ligne est un commentaire qui ne modifie pas le r\u00e9sultat du programme.<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><i><span style=\"color: #408080;\"># La biblioth\u00e8que 'random' nous permet de g\u00e9n\u00e9rer des nombres al\u00e9atoires.<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><i><span style=\"color: #408080;\"># On l\u2019importe afin de pouvoir l\u2019utiliser\u00a0:<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><b><span lang=\"EN-US\" style=\"color: green; mso-ansi-language: EN-US;\">import<\/span><\/b><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"> <b><span style=\"color: blue;\">random<\/span><\/b><\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\">\u00a0<\/span><\/pre><pre style=\"line-height: 125%;\"><b><span lang=\"EN-US\" style=\"color: green; mso-ansi-language: EN-US;\">def<\/span><\/b><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"> <span style=\"color: blue;\">lancer_piece<\/span>(n):<\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><\/span>nb_succes <span style=\"color: #666666;\">=<\/span> <span style=\"color: #666666;\">0<\/span><span style=\"mso-spacerun: yes;\">\u00a0 <\/span><i><span style=\"color: #408080;\"># Initialisation du compteur de succ\u00e8s<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><b><span lang=\"EN-US\" style=\"color: green; mso-ansi-language: EN-US;\">for<\/span><\/b><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"> i <b><span style=\"color: #aa22ff;\">in<\/span><\/b> <span style=\"color: green;\">range<\/span>(n):<span style=\"mso-spacerun: yes;\">\u00a0 <\/span><i><span style=\"color: #408080;\"># R\u00e9p\u00e8te n fois<\/span><\/i><\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><i><span style=\"color: #408080;\"># Choix al\u00e9atoire entre 'Pile' et 'Face'<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><b><span lang=\"EN-US\" style=\"color: green; mso-ansi-language: EN-US;\">if<\/span><\/b><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"> random<span style=\"color: #666666;\">.<\/span>choice([<span style=\"color: #ba2121;\">'Pile'<\/span>, <span style=\"color: #ba2121;\">'Face'<\/span>]) <span style=\"color: #666666;\">==<\/span> <span style=\"color: #ba2121;\">'Pile'<\/span>:<span style=\"mso-spacerun: yes;\">\u00a0 <\/span><\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/span>nb_succes <span style=\"color: #666666;\">+=<\/span> <span style=\"color: #666666;\">1<\/span><span style=\"mso-spacerun: yes;\">\u00a0 <\/span><i><span style=\"color: #408080;\"># Augmente le compteur si 'Pile' est obtenu<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><b><span style=\"color: green;\">return<\/span><\/b> nb_succes<span style=\"mso-spacerun: yes;\">\u00a0 <\/span><i><span style=\"color: #408080;\"># Retourne le nombre total de succ\u00e8s<\/span><\/i><\/pre><pre style=\"line-height: 125%;\">\u00a0<\/pre><pre style=\"line-height: 125%;\"><i><span style=\"color: #408080;\"># Exemple d\u2019utilisation avec affichage du r\u00e9sultat<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><b><span style=\"color: green;\">print<\/span><\/b>(lancer_piece(<span style=\"color: #666666;\">10<\/span>))<\/pre><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Cette fonction utilise `random.choice()` pour simuler le lancer d&rsquo;une pi\u00e8ce et compte le nombre de fois o\u00f9 &lsquo;Pile&rsquo; est obtenu.<\/p><p class=\"MsoNormal\">\u00a0<\/p><h2><span class=\"ez-toc-section\" id=\"2_Observer_la_loi_des_grands_nombres_via_une_simulation_Python\"><\/span>2. Observer la loi des grands nombres via une simulation Python<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Nous pouvons observer la loi des grands nombres en action en tra\u00e7ant la fr\u00e9quence cumul\u00e9e de succ\u00e8s \u00e0 mesure que <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> augmente.<\/p><p class=\"MsoNormal\">\u00a0<\/p><pre style=\"line-height: 125%;\"><i><span style=\"color: #408080;\"># 'matplotlib.pyplot' est une biblioth\u00e8que pour cr\u00e9er des graphiques.<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><b><span style=\"color: green;\">import<\/span><\/b> <b><span style=\"color: blue;\">matplotlib.pyplot<\/span><\/b> <b><span style=\"color: green;\">as<\/span><\/b> <b><span style=\"color: blue;\">plt<\/span><\/b><\/pre><pre style=\"line-height: 125%;\"><b><span style=\"color: green;\">import<\/span><\/b> <b><span style=\"color: blue;\">random<\/span><\/b><\/pre><pre style=\"line-height: 125%;\">\u00a0<\/pre><pre style=\"line-height: 125%;\"><b><span style=\"color: green;\">def<\/span><\/b> <span style=\"color: blue;\">loi_grands_nombres<\/span>(N):<\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\">nb_succes_cumule <span style=\"color: #666666;\">=<\/span> <span style=\"color: #666666;\">0<\/span><\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span>frequences <span style=\"color: #666666;\">=<\/span> []<\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><b><span style=\"color: green;\">for<\/span><\/b> n <b><span style=\"color: #aa22ff;\">in<\/span><\/b> <span style=\"color: green;\">range<\/span>(<span style=\"color: #666666;\">1<\/span>, N<span style=\"color: #666666;\">+1<\/span>):<\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><b><span style=\"color: green;\">if<\/span><\/b> random<span style=\"color: #666666;\">.<\/span>choice([<span style=\"color: #ba2121;\">'Pile'<\/span>, <span style=\"color: #ba2121;\">'Face'<\/span>]) <span style=\"color: #666666;\">==<\/span> <span style=\"color: #ba2121;\">'Pile'<\/span>:<\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span>nb_succes_cumule <span style=\"color: #666666;\">+=<\/span> <span style=\"color: #666666;\">1<\/span><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><i><span style=\"color: #408080;\"># Calcul de la fr\u00e9quence cumul\u00e9e et ajout \u00e0 la liste des fr\u00e9quences<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>frequences<span style=\"color: #666666;\">.<\/span>append(nb_succes_cumule <span style=\"color: #666666;\">\/<\/span> n)<\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0<\/span><i><span style=\"color: #408080;\"># Cr\u00e9ation du graphique<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span>plt<span style=\"color: #666666;\">.<\/span>plot(<span style=\"color: green;\">range<\/span>(<span style=\"color: #666666;\">1<\/span>, N<span style=\"color: #666666;\">+1<\/span>), frequences, label<span style=\"color: #666666;\">=<\/span><span style=\"color: #ba2121;\">\"Fr\u00e9quence cumul\u00e9e de 'Pile'\"<\/span>)<\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span>plt<span style=\"color: #666666;\">.<\/span>axhline(y<span style=\"color: #666666;\">=0.5<\/span>, color<span style=\"color: #666666;\">=<\/span><span style=\"color: #ba2121;\">'r'<\/span>, linestyle<span style=\"color: #666666;\">=<\/span><span style=\"color: #ba2121;\">'-'<\/span>, label<span style=\"color: #666666;\">=<\/span><span style=\"color: #ba2121;\">\"Probabilit\u00e9 r\u00e9elle\"<\/span>)<\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span>plt<span style=\"color: #666666;\">.<\/span>xlabel(<span style=\"color: #ba2121;\">'Nombre de lancers'<\/span>)<\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\">plt<span style=\"color: #666666;\">.<\/span>ylabel(<span style=\"color: #ba2121;\">'Fr\u00e9quence'<\/span>)<\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span>plt<span style=\"color: #666666;\">.<\/span>legend()<\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><\/span>plt<span style=\"color: #666666;\">.<\/span>show()<\/pre><pre style=\"line-height: 125%;\">\u00a0<\/pre><pre style=\"line-height: 125%;\"><i><span style=\"color: #408080;\"># Exemple d\u2019utilisation<\/span><\/i><\/pre><pre style=\"line-height: 125%;\">loi_grands_nombres(<span style=\"color: #666666;\">1000<\/span>)<\/pre><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Ce code cr\u00e9e un graphique montrant comment la fr\u00e9quence de succ\u00e8s se rapproche de la probabilit\u00e9 r\u00e9elle \u00e0 mesure que le nombre d&rsquo;exp\u00e9riences augmente, illustrant la loi des grands nombres.<\/p><p class=\"MsoNormal\">\u00a0<\/p><h2><span class=\"ez-toc-section\" id=\"3_Simulation_de_N_echantillons_et_calcul_de_proportion\"><\/span>3. Simulation de N \u00e9chantillons et calcul de proportion<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Enfin, voici un exemple de code qui simule <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> \u00e9chantillons de taille <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> et calcule la proportion des cas o\u00f9 l&rsquo;\u00e9cart entre <span class=\"katex-eq\" data-katex-display=\"false\">p<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\u0192<\/span> est inf\u00e9rieur ou \u00e9gal \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\">1\/\\sqrt{n}<\/span>:<\/p><p class=\"MsoNormal\">\u00a0<\/p><pre style=\"line-height: 125%;\"><i><span style=\"color: #408080;\"># La biblioth\u00e8que 'math' contient des fonctions math\u00e9matiques de base.<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><b><span style=\"color: green;\">import<\/span><\/b> <b><span style=\"color: blue;\">math<\/span><\/b><\/pre><pre style=\"line-height: 125%;\"><b><span style=\"color: green;\">import<\/span><\/b> <b><span style=\"color: blue;\">random<\/span><\/b><\/pre><pre style=\"line-height: 125%;\">\u00a0<\/pre><pre style=\"line-height: 125%;\"><b><span style=\"color: green;\">def<\/span><\/b> <span style=\"color: blue;\">proportion_ecart<\/span>(N, n, p):<\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span>compteur_cas_favorables <span style=\"color: #666666;\">=<\/span> <span style=\"color: #666666;\">0<\/span><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><b><span lang=\"EN-US\" style=\"color: green; mso-ansi-language: EN-US;\">for<\/span><\/b><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"> _ <b><span style=\"color: #aa22ff;\">in<\/span><\/b> <span style=\"color: green;\">range<\/span>(N):<\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>nb_succes <span style=\"color: #666666;\">=<\/span> <span style=\"color: #666666;\">0<\/span><\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><i><span style=\"color: #408080;\"># Simulation de n exp\u00e9riences et comptage des succ\u00e8s<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><b><span lang=\"EN-US\" style=\"color: green; mso-ansi-language: EN-US;\">for<\/span><\/b><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"> i <b><span style=\"color: #aa22ff;\">in<\/span><\/b> <span style=\"color: green;\">range<\/span>(n):<\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/span><b><span style=\"color: green;\">if<\/span><\/b> random<span style=\"color: #666666;\">.<\/span>choice([<span style=\"color: green;\">True<\/span>, <span style=\"color: green;\">False<\/span>]):<span style=\"mso-spacerun: yes;\">\u00a0 <\/span><i><span style=\"color: #408080;\"># Choix al\u00e9atoire entre succ\u00e8s (True) et \u00e9chec (False)<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>nb_succes <span style=\"color: #666666;\">+=<\/span> <span style=\"color: #666666;\">1<\/span><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>f <span style=\"color: #666666;\">=<\/span> nb_succes <span style=\"color: #666666;\">\/<\/span> n<span style=\"mso-spacerun: yes;\">\u00a0 <\/span><i><span style=\"color: #408080;\"># Calcul de la fr\u00e9quence des succ\u00e8s<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><i><span style=\"color: #408080;\"># V\u00e9rification de la condition sur l'\u00e9cart<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><i><span style=\"color: #408080;\"># math.sqrt(n) calcule la racine carr\u00e9e de n<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><b><span lang=\"EN-US\" style=\"color: green; mso-ansi-language: EN-US;\">if<\/span><\/b><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"> <span style=\"color: green;\">abs<\/span>(f <span style=\"color: #666666;\">-<\/span> p) <span style=\"color: #666666;\">&lt;=<\/span> <span style=\"color: #666666;\">1\/<\/span>math<span style=\"color: #666666;\">.<\/span>sqrt(n):<span style=\"mso-spacerun: yes;\">\u00a0 <\/span><\/span><\/pre><pre style=\"line-height: 125%;\"><span lang=\"EN-US\" style=\"mso-ansi-language: EN-US;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/span>compteur_cas_favorables <span style=\"color: #666666;\">+=<\/span> <span style=\"color: #666666;\">1<\/span><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><i><span style=\"color: #408080;\"># Retourne la proportion des cas favorables<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><span style=\"mso-spacerun: yes;\">\u00a0\u00a0\u00a0 <\/span><b><span style=\"color: green;\">return<\/span><\/b> compteur_cas_favorables <span style=\"color: #666666;\">\/<\/span> N<span style=\"mso-spacerun: yes;\">\u00a0 <\/span><\/pre><pre style=\"line-height: 125%;\">\u00a0<\/pre><pre style=\"line-height: 125%;\"><i><span style=\"color: #408080;\"># Exemple d\u2019utilisation<\/span><\/i><\/pre><pre style=\"line-height: 125%;\"><b><span style=\"color: green;\">print<\/span><\/b>(proportion_ecart(<span style=\"color: #666666;\">1000<\/span>, <span style=\"color: #666666;\">100<\/span>, <span style=\"color: #666666;\">0.5<\/span>))<\/pre><p class=\"MsoNormal\">\u00a0<\/p><p class=\"MsoNormal\">Ce code utilise \u2019random.choice([True, False])\u2019 pour simuler des succ\u00e8s (True) et des \u00e9checs (False) et compte le nombre de cas o\u00f9 la condition sur l&rsquo;\u00e9cart est respect\u00e9e.<\/p><p class=\"MsoNormal\">\u00a0<\/p><p><style>@font-face\n\t{font-family:\"Cambria Math\";\n\tpanose-1:2 4 5 3 5 4 6 3 2 4;\n\tmso-font-charset:0;\n\tmso-generic-font-family:roman;\n\tmso-font-pitch:variable;\n\tmso-font-signature:-536870145 1107305727 0 0 415 0;}@font-face\n\t{font-family:Calibri;\n\tpanose-1:2 15 5 2 2 2 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elementor-top-column elementor-element elementor-element-a39de31\" data-id=\"a39de31\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-342de24 elementor-widget elementor-widget-spacer\" data-id=\"342de24\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>\u00c9chantillonnage Introduction \u00a0 L&rsquo;\u00e9chantillonnage est un outil puissant qui nous permet d&rsquo;estimer des proportions et des probabilit\u00e9s dans une population \u00e0 partir d&rsquo;un sous-ensemble de donn\u00e9es. Il repose sur l&rsquo;id\u00e9e que, en augmentant la taille de l&rsquo;\u00e9chantillon, nos estimations seront de plus en plus proches de la r\u00e9alit\u00e9. Cette approche est largement utilis\u00e9e dans de [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_canvas","format":"standard","meta":{"footnotes":""},"categories":[25,26,5],"tags":[],"_links":{"self":[{"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts\/1975"}],"collection":[{"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/comments?post=1975"}],"version-history":[{"count":4,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts\/1975\/revisions"}],"predecessor-version":[{"id":1979,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts\/1975\/revisions\/1979"}],"wp:attachment":[{"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/media?parent=1975"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/categories?post=1975"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/tags?post=1975"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}