{"id":1969,"date":"2023-11-22T22:32:03","date_gmt":"2023-11-22T21:32:03","guid":{"rendered":"http:\/\/localhost:8080\/maxblog\/?p=1969"},"modified":"2023-11-22T22:34:26","modified_gmt":"2023-11-22T21:34:26","slug":"modeliser-le-hasard-calculer-des-probabilites","status":"publish","type":"post","link":"https:\/\/www.maxdecours.com\/maxblog\/modeliser-le-hasard-calculer-des-probabilites\/","title":{"rendered":"Mod\u00e9liser le hasard, calculer des probabilit\u00e9s"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"1969\" class=\"elementor elementor-1969\">\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\">Mod\u00e9liser le hasard, calculer des probabilit\u00e9s<\/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_82_2 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\/modeliser-le-hasard-calculer-des-probabilites\/#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\/modeliser-le-hasard-calculer-des-probabilites\/#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\/modeliser-le-hasard-calculer-des-probabilites\/#1_Ensemble_des_issues_et_evenements\" >1. Ensemble des issues et \u00e9v\u00e9nements<\/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\/modeliser-le-hasard-calculer-des-probabilites\/#2_Operations_sur_les_evenements\" >2. Op\u00e9rations sur les \u00e9v\u00e9nements<\/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\/modeliser-le-hasard-calculer-des-probabilites\/#3_Loi_de_probabilite\" >3. Loi de probabilit\u00e9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.maxdecours.com\/maxblog\/modeliser-le-hasard-calculer-des-probabilites\/#4_Relation_de_base_des_probabilites\" >4. Relation de base des probabilit\u00e9s<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.maxdecours.com\/maxblog\/modeliser-le-hasard-calculer-des-probabilites\/#5_Denombrement_a_laide_de_tableaux_et_darbres\" >5. D\u00e9nombrement \u00e0 l&rsquo;aide de tableaux et d\u2019arbres<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.maxdecours.com\/maxblog\/modeliser-le-hasard-calculer-des-probabilites\/#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-9\" href=\"https:\/\/www.maxdecours.com\/maxblog\/modeliser-le-hasard-calculer-des-probabilites\/#1_Utiliser_des_modeles_theoriques_de_reference\" >1. Utiliser des mod\u00e8les th\u00e9oriques de r\u00e9f\u00e9rence<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.maxdecours.com\/maxblog\/modeliser-le-hasard-calculer-des-probabilites\/#2_Construire_un_modele_a_partir_de_frequences_observees\" >2. Construire un mod\u00e8le \u00e0 partir de fr\u00e9quences observ\u00e9es<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/www.maxdecours.com\/maxblog\/modeliser-le-hasard-calculer-des-probabilites\/#3_Calculer_des_probabilites_dans_des_cas_simples\" >3. Calculer des probabilit\u00e9s dans des cas simples<\/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>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">La mod\u00e9lisation du hasard \u00e0 l&rsquo;aide de la th\u00e9orie des\nprobabilit\u00e9s nous permet de quantifier l&rsquo;incertitude et de faire des\npr\u00e9dictions sur les r\u00e9sultats d&rsquo;exp\u00e9riences al\u00e9atoires. En ma\u00eetrisant le\nd\u00e9nombrement et les propri\u00e9t\u00e9s des \u00e9v\u00e9nements, on peut calculer des\nprobabilit\u00e9s dans de nombreux contextes diff\u00e9rents.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h1><span class=\"ez-toc-section\" id=\"Cours\"><\/span>Cours<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"1_Ensemble_des_issues_et_evenements\"><\/span>1. Ensemble des issues et \u00e9v\u00e9nements<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Pour mod\u00e9liser le hasard, nous commen\u00e7ons par d\u00e9finir l&rsquo;<b>ensemble\ndes issues<\/b> <span class=\"katex-eq\" data-katex-display=\"false\"> \\Omega <\/span>, qui est l&rsquo;ensemble de tous les\nr\u00e9sultats possibles d&rsquo;une exp\u00e9rience al\u00e9atoire. Par exemple, lorsqu&rsquo;on lance un\nd\u00e9, <span class=\"katex-eq\" data-katex-display=\"false\"> \\Omega = \\{1, 2, 3, 4, 5, 6\\} <\/span>.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Un <b>\u00e9v\u00e9nement<\/b> est un sous-ensemble de <span class=\"katex-eq\" data-katex-display=\"false\"> \\Omega <\/span>.\nPar exemple, l&rsquo;\u00e9v\u00e9nement \u00ab\u00a0Obtenir un nombre pair\u00a0\u00bb lorsqu&rsquo;on lance un\nd\u00e9 peut \u00eatre repr\u00e9sent\u00e9 par l&rsquo;ensemble <span class=\"katex-eq\" data-katex-display=\"false\"> A = \\{2, 4, 6\\} <\/span>.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"2_Operations_sur_les_evenements\"><\/span>2. Op\u00e9rations sur les \u00e9v\u00e9nements<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">&#8211; <b>R\u00e9union<\/b> : La r\u00e9union de deux \u00e9v\u00e9nements <span class=\"katex-eq\" data-katex-display=\"false\"> A <\/span>\net <span class=\"katex-eq\" data-katex-display=\"false\"> B <\/span> est l&rsquo;ensemble des \u00e9l\u00e9ments qui appartiennent \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\"> A\n<\/span> ou \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\"> B <\/span> (ou les deux). On le note <span class=\"katex-eq\" data-katex-display=\"false\"> A \\cup B <\/span>.<br style=\"mso-special-character:line-break\">\n<br style=\"mso-special-character:line-break\">\n<\/p>\n\n<p class=\"MsoNormal\">&#8211; <b>Intersection<\/b> : L&rsquo;intersection de <span class=\"katex-eq\" data-katex-display=\"false\"> A <\/span>\net <span class=\"katex-eq\" data-katex-display=\"false\"> B <\/span> est l&rsquo;ensemble des \u00e9l\u00e9ments qui appartiennent \u00e0 la fois \u00e0\n<span class=\"katex-eq\" data-katex-display=\"false\"> A <\/span> et \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\"> B <\/span>. On le note <span class=\"katex-eq\" data-katex-display=\"false\"> A \\cap B <\/span>.<\/p>\n\n<p class=\"MsoNormal\"><br>\n&#8211; <b>Compl\u00e9mentaire<\/b> : Le compl\u00e9mentaire d&rsquo;un \u00e9v\u00e9nement <span class=\"katex-eq\" data-katex-display=\"false\"> A <\/span>\nest l&rsquo;ensemble des \u00e9l\u00e9ments de <span class=\"katex-eq\" data-katex-display=\"false\"> \\Omega <\/span> qui ne sont pas dans <span class=\"katex-eq\" data-katex-display=\"false\">\nA <\/span>. On le note <span class=\"katex-eq\" data-katex-display=\"false\"> A^c <\/span>.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"3_Loi_de_probabilite\"><\/span>3. Loi de probabilit\u00e9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">La <b>loi de probabilit\u00e9<\/b> attribue \u00e0 chaque issue de <span class=\"katex-eq\" data-katex-display=\"false\">\n\\Omega <\/span> une probabilit\u00e9. La probabilit\u00e9 d&rsquo;une issue est un nombre entre\n0 et 1, et la somme des probabilit\u00e9s de toutes les issues doit \u00eatre \u00e9gale \u00e0 1.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">La <b>probabilit\u00e9 d&rsquo;un \u00e9v\u00e9nement<\/b> <span class=\"katex-eq\" data-katex-display=\"false\"> A <\/span> est\ndonn\u00e9e par la somme des probabilit\u00e9s des issues qui constituent <span class=\"katex-eq\" data-katex-display=\"false\"> A <\/span>:<\/p>\n\n<p class=\"MsoNormal\"><span lang=\"EN-US\" style=\"mso-ansi-language:EN-US\"><span class=\"katex-eq\" data-katex-display=\"false\">\nP(A) = \\sum_{\\omega \\in A} P(\\{\\omega\\}) <\/span><\/span><\/p>\n\n<p class=\"MsoNormal\"><span lang=\"EN-US\" style=\"mso-ansi-language:EN-US\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\"><b>Exemple : Lancer d&rsquo;un d\u00e9<\/b><\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Consid\u00e9rons un d\u00e9 \u00e9quilibr\u00e9 \u00e0 six faces avec <span class=\"katex-eq\" data-katex-display=\"false\"> \\Omega\n= \\{1, 2, 3, 4, 5, 6\\} <\/span>. Soit <span class=\"katex-eq\" data-katex-display=\"false\"> A = \\{2, 4, 6\\} <\/span>\nl&rsquo;\u00e9v\u00e9nement \u00ab\u00a0Obtenir un nombre pair\u00a0\u00bb. <\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Chaque face a une probabilit\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{1}{6} <\/span>.\nLa probabilit\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\"> A <\/span> est donc :<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><span class=\"katex-eq\" data-katex-display=\"false\"> P(A) = P(\\{2\\}) + P(\\{4\\}) + P(\\{6\\}) = \\frac{1}{6}\n+ \\frac{1}{6} + \\frac{1}{6} = \\frac{1}{2} <\/span>.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"4_Relation_de_base_des_probabilites\"><\/span>4. Relation de base des probabilit\u00e9s<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">La relation suivante lie les probabilit\u00e9s des \u00e9v\u00e9nements <span class=\"katex-eq\" data-katex-display=\"false\">\nA <\/span>, <span class=\"katex-eq\" data-katex-display=\"false\"> B <\/span>, <span class=\"katex-eq\" data-katex-display=\"false\"> A \\cup B <\/span> (la r\u00e9union de <span class=\"katex-eq\" data-katex-display=\"false\">\nA <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> B <\/span>) et <span class=\"katex-eq\" data-katex-display=\"false\"> A \\cap B <\/span> (l&rsquo;intersection\nde <span class=\"katex-eq\" data-katex-display=\"false\"> A <\/span> et <span class=\"katex-eq\" data-katex-display=\"false\"> B <\/span>):<\/p>\n\n<p class=\"MsoNormal\"><span lang=\"EN-US\" style=\"mso-ansi-language:EN-US\"><span class=\"katex-eq\" data-katex-display=\"false\"> P(A\n\\cup B) + P(A \\cap B) = P(A) + P(B) <\/span><\/span><\/p>\n\n<p class=\"MsoNormal\"><span lang=\"EN-US\" style=\"mso-ansi-language:EN-US\">&nbsp;<\/span><\/p>\n\n<p class=\"MsoNormal\">Cette relation est importante car elle permet de calculer la\nprobabilit\u00e9 de la r\u00e9union ou de l&rsquo;intersection de deux \u00e9v\u00e9nements \u00e0 partir des\nprobabilit\u00e9s des \u00e9v\u00e9nements individuels.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"5_Denombrement_a_laide_de_tableaux_et_darbres\"><\/span>5. D\u00e9nombrement \u00e0 l&rsquo;aide de tableaux et d\u2019arbres<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Le d\u00e9nombrement des issues possibles peut \u00eatre facilit\u00e9 \u00e0\nl&rsquo;aide de <b>tableaux<\/b> ou d&rsquo;<b>arbres de probabilit\u00e9<\/b>.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">&#8211; <b>Tableaux<\/b> : On peut utiliser un tableau pour\norganiser les donn\u00e9es et d\u00e9nombrer facilement les r\u00e9sultats possibles.<br style=\"mso-special-character:line-break\">\n<br style=\"mso-special-character:line-break\">\n<\/p>\n\n<p class=\"MsoNormal\">&#8211; <b>Arbres de probabilit\u00e9 <\/b>: Les arbres de probabilit\u00e9\nrepr\u00e9sentent graphiquement toutes les issues possibles d&rsquo;une exp\u00e9rience. Chaque\nbranche de l&rsquo;arbre repr\u00e9sente une issue possible et est \u00e9tiquet\u00e9e avec la\nprobabilit\u00e9 de cette issue.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>Exemple :<\/b><\/p>\n\n<p class=\"MsoNormal\">Consid\u00e9rons le lancer de deux pi\u00e8ces de monnaie. L&rsquo;ensemble\ndes issues est <span class=\"katex-eq\" data-katex-display=\"false\"> \\Omega = \\{ (P, P), (P, F), (F, P), (F, F) \\} <\/span>\no\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\"> P <\/span> repr\u00e9sente pile et <span class=\"katex-eq\" data-katex-display=\"false\"> F <\/span> face.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>a. Tableau de d\u00e9nombrement&nbsp;:<\/b><\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Chaque cellule du tableau repr\u00e9sente une issue possible.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{|c|c|c|} \\hline\u00a0 &amp;\n\\text{Pi\u00e8ce 1: Pile (P)} &amp; \\text{Pi\u00e8ce 1: Face (F)} \\\\ \\hline \\text{Pi\u00e8ce\n2: Pile (P)} &amp; (P, P) &amp; (F, P) \\\\ \\hline \\text{Pi\u00e8ce 2: Face (F)} &amp;\n(P, F) &amp; (F, F) \\\\ \\hline \\end{array} <\/span><span style=\"font-family:\n&quot;Times New Roman&quot;,serif;mso-fareast-font-family:&quot;Times New Roman&quot;;mso-fareast-language:\nFR\"><\/span><\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Ce tableau montre toutes les issues possibles du lancer de\ndeux pi\u00e8ces de monnaie.<\/p>\n\n<p class=\"MsoNormal\">Dans cet exemple, l&rsquo;ensemble des issues est <span class=\"katex-eq\" data-katex-display=\"false\"> \\Omega =\n\\{ (P, P), (P, F), (F, P), (F, F) \\} <\/span>. Chaque issue est \u00e9quiprobable,\ndonc chaque cellule a une probabilit\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{1}{4} <\/span>.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>b. Arbre de probabilit\u00e9 :<\/b><\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Un arbre de probabilit\u00e9 commence par un n\u0153ud racine et se\nramifie pour chaque \u00e9tape de l&rsquo;exp\u00e9rience.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><span style=\"font-family:Consolas\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/span><span style=\"mso-spacerun:yes\">&nbsp;<\/span><\/span><span style=\"font-family:Consolas;mso-fareast-font-family:&quot;MS Mincho&quot;\">\u250c<\/span><span style=\"font-family:Consolas\">\u2500\u2500\u2500(1\/2)\u2500\u2500\u2500&gt; P<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"font-family:Consolas\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/span><span style=\"font-family:Consolas;\nmso-fareast-font-family:&quot;MS Mincho&quot;\">\u2502<\/span><span style=\"font-family:Consolas\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/span><span style=\"font-family:\nConsolas;mso-fareast-font-family:&quot;MS Mincho&quot;\">\u250c<\/span><span style=\"font-family:\nConsolas\">\u2500\u2500\u2500(1\/2)\u2500\u2500\u2500&gt; P : (P, P)<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"font-family:Consolas\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp;&nbsp; <\/span>(d\u00e9but)<span style=\"mso-spacerun:yes\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/span><span style=\"font-family:Consolas;\nmso-fareast-font-family:&quot;MS Mincho&quot;\">\u2514<\/span><span style=\"font-family:Consolas\">\u2500\u2500\u2500(1\/2)\u2500\u2500\u2500&gt;\nF : (P, F)<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"font-family:Consolas\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/span><span style=\"font-family:Consolas;\nmso-fareast-font-family:&quot;MS Mincho&quot;\">\u2502<\/span><span style=\"font-family:Consolas\"><\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"font-family:Consolas\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/span><span style=\"font-family:Consolas;\nmso-fareast-font-family:&quot;MS Mincho&quot;\">\u2514<\/span><span style=\"font-family:Consolas\">\u2500\u2500\u2500(1\/2)\u2500\u2500\u2500&gt;\nF<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"font-family:Consolas\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/span><span style=\"font-family:Consolas;mso-fareast-font-family:&quot;MS Mincho&quot;\">\u250c<\/span><span style=\"font-family:Consolas\">\u2500\u2500\u2500(1\/2)\u2500\u2500\u2500&gt; P : (F, P)<\/span><\/p>\n\n<p class=\"MsoNormal\"><span style=\"font-family:Consolas\"><span style=\"mso-spacerun:yes\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/span><span style=\"font-family:Consolas;mso-fareast-font-family:&quot;MS Mincho&quot;\">\u2514<\/span><span style=\"font-family:Consolas\">\u2500\u2500\u2500(1\/2)\u2500\u2500\u2500&gt; F : (F, F)<\/span><\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Dans cet arbre, chaque branche est \u00e9tiquet\u00e9e avec la\nprobabilit\u00e9 de cette issue. Par exemple, le chemin \u00ab\u00a0P \u2192 P\u00a0\u00bb correspond\n\u00e0 l&rsquo;issue \u00ab\u00a0(P, P)\u00a0\u00bb et a une probabilit\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{1}{2} \\times\n\\frac{1}{2} = \\frac{1}{4} <\/span>. Les autres issues sont calcul\u00e9es de mani\u00e8re\nsimilaire.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>c. Analyse des repr\u00e9sentations<\/b><\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Ces deux repr\u00e9sentations facilitent le calcul des\nprobabilit\u00e9s d&rsquo;\u00e9v\u00e9nements. Par exemple, la probabilit\u00e9 d&rsquo;obtenir au moins un\n\u00ab\u00a0Pile\u00a0\u00bb (\u00e9v\u00e9nement <span class=\"katex-eq\" data-katex-display=\"false\"> A = \\{ (P, P), (P, F), (F, P) \\} <\/span>)\npeut \u00eatre trouv\u00e9e en additionnant les probabilit\u00e9s des issues correspondantes:<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><span class=\"katex-eq\" data-katex-display=\"false\"> P(A) = P((P, P)) + P((P, F)) + P((F, P)) =\n\\frac{1}{4} + \\frac{1}{4} + \\frac{1}{4} = \\frac{3}{4} <\/span><\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h1><span class=\"ez-toc-section\" id=\"Methodes\"><\/span>M\u00e9thodes<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"1_Utiliser_des_modeles_theoriques_de_reference\"><\/span>1. Utiliser des mod\u00e8les th\u00e9oriques de r\u00e9f\u00e9rence<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>D\u00e9<\/b> : Un d\u00e9 \u00e9quilibr\u00e9 est un exemple classique. Pour\nun d\u00e9 \u00e0 6 faces, chaque face a une chance \u00e9gale d&rsquo;appara\u00eetre, soit <span class=\"katex-eq\" data-katex-display=\"false\">\n\\frac{1}{6} <\/span>. Ainsi, la probabilit\u00e9 de chaque \u00e9v\u00e9nement simple (comme\nobtenir un \u00ab\u00a03\u00a0\u00bb) est d\u00e9finie a priori comme <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{1}{6} <\/span>.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>Pi\u00e8ce \u00e9quilibr\u00e9e<\/b> : De m\u00eame, lorsqu&rsquo;on lance une pi\u00e8ce\n\u00e9quilibr\u00e9e, les deux issues possibles (Pile et Face) sont \u00e9quiprobables, avec\nune probabilit\u00e9 de <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{1}{2} <\/span> chacune.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>Tirage au sort<\/b> : Supposons qu&rsquo;on tire une carte au\nhasard d&rsquo;un jeu de 52 cartes. Chaque carte a une probabilit\u00e9 a priori \u00e9gale de <span class=\"katex-eq\" data-katex-display=\"false\">\n\\frac{1}{52} <\/span> d&rsquo;\u00eatre tir\u00e9e.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Dans ces cas, les probabilit\u00e9s sont d\u00e9finies avant toute\nexp\u00e9rience en supposant que chaque issue est \u00e9galement probable.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"2_Construire_un_modele_a_partir_de_frequences_observees\"><\/span>2. Construire un mod\u00e8le \u00e0 partir de fr\u00e9quences observ\u00e9es<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Parfois, les probabilit\u00e9s ne sont pas connues a priori et\ndoivent \u00eatre estim\u00e9es \u00e0 partir de donn\u00e9es observ\u00e9es.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>Exemple<\/b> : Supposons qu&rsquo;on souhaite conna\u00eetre la\nprobabilit\u00e9 qu&rsquo;il pleuve un jour donn\u00e9. On peut collecter des donn\u00e9es sur le\nnombre de jours de pluie sur une p\u00e9riode donn\u00e9e (disons, 100 jours) et\nconstruire un mod\u00e8le probabiliste.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Si, sur ces 100 jours, il a plu 20 jours, la fr\u00e9quence\nobserv\u00e9e de jours de pluie est <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{20}{100} = 0.2 <\/span>. On peut\nutiliser cette fr\u00e9quence comme estimation de la probabilit\u00e9 qu&rsquo;il pleuve un\njour au hasard.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>Attention<\/b> : Il faut comprendre que cette probabilit\u00e9\nest une estimation bas\u00e9e sur des observations pass\u00e9es et qu&rsquo;elle ne pr\u00e9dit pas\nparfaitement la r\u00e9alit\u00e9.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<h2><span class=\"ez-toc-section\" id=\"3_Calculer_des_probabilites_dans_des_cas_simples\"><\/span>3. Calculer des probabilit\u00e9s dans des cas simples<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>Exp\u00e9rience al\u00e9atoire \u00e0 deux \u00e9preuves :<\/b><\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>Exemple<\/b> : Consid\u00e9rons le lancer de deux pi\u00e8ces\n\u00e9quilibr\u00e9es. On peut construire un espace des \u00e9chantillons <span class=\"katex-eq\" data-katex-display=\"false\"> \\Omega =\n\\{(P,P), (P,F), (F,P), (F,F)\\} <\/span>, o\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\"> P <\/span> est Pile et <span class=\"katex-eq\" data-katex-display=\"false\">\nF <\/span> est Face.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Pour calculer la probabilit\u00e9 de l&rsquo;\u00e9v\u00e9nement \u00ab\u00a0Au moins\nune pi\u00e8ce montre Face\u00a0\u00bb, on compte le nombre d&rsquo;issues favorables (3: <span class=\"katex-eq\" data-katex-display=\"false\">\n(P,F), (F,P), (F,F) <\/span>) et on divise par le nombre total d&rsquo;issues (4).\nAinsi, <span class=\"katex-eq\" data-katex-display=\"false\"> P(\\text{&quot;Au moins un F&quot;}) = \\frac{3}{4} <\/span>.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>Exp\u00e9rience al\u00e9atoire \u00e0 trois \u00e9preuves :<\/b><\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\"><b>Exemple<\/b> : Supposons le lancer de trois d\u00e9s\n\u00e9quilibr\u00e9s. On souhaite trouver la probabilit\u00e9 que la somme des num\u00e9ros obtenus\nsoit 7. <\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n<p class=\"MsoNormal\">Pour ce faire, on liste toutes les triplets possibles qui\nsomment \u00e0 7, par exemple <span class=\"katex-eq\" data-katex-display=\"false\"> (1,2,4), (2,2,3) <\/span>, etc. On compte le\nnombre d&rsquo;issues favorables et on divise par le nombre total d&rsquo;issues (<span class=\"katex-eq\" data-katex-display=\"false\">6\n\\times 6 \\times 6 = 216<\/span>) pour trouver la probabilit\u00e9 recherch\u00e9e.<\/p>\n\n<p class=\"MsoNormal\">&nbsp;<\/p>\n\n\n\n\n\n<style>@font-face\n\t{font-family:\"MS Mincho\";\n\tpanose-1:2 2 6 9 4 2 5 8 3 4;\n\tmso-font-alt:\"\uff2d\uff33 \u660e\u671d\";\n\tmso-font-charset:128;\n\tmso-generic-font-family:modern;\n\tmso-font-pitch:fixed;\n\tmso-font-signature:-536870145 1791491579 134217746 0 131231 0;}@font-face\n\t{font-family:\"Cambria Math\";\n\tpanose-1:2 4 5 3 5 4 6 3 2 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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>Mod\u00e9liser le hasard, calculer des probabilit\u00e9s Introduction &nbsp; La mod\u00e9lisation du hasard \u00e0 l&rsquo;aide de la th\u00e9orie des probabilit\u00e9s nous permet de quantifier l&rsquo;incertitude et de faire des pr\u00e9dictions sur les r\u00e9sultats d&rsquo;exp\u00e9riences al\u00e9atoires. En ma\u00eetrisant le d\u00e9nombrement et les propri\u00e9t\u00e9s des \u00e9v\u00e9nements, on peut calculer des probabilit\u00e9s dans de nombreux contextes diff\u00e9rents. &nbsp; Cours [&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\/1969"}],"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=1969"}],"version-history":[{"count":5,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts\/1969\/revisions"}],"predecessor-version":[{"id":1974,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/posts\/1969\/revisions\/1974"}],"wp:attachment":[{"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/media?parent=1969"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/categories?post=1969"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.maxdecours.com\/maxblog\/wp-json\/wp\/v2\/tags?post=1969"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}