{"id":26741,"date":"2021-04-21T13:00:52","date_gmt":"2021-04-21T13:00:52","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26741"},"modified":"2024-05-21T22:41:05","modified_gmt":"2024-05-21T22:41:05","slug":"theoreme-de-bayes-et-probabilite-composee","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/theoreme-de-bayes-et-probabilite-composee\/","title":{"rendered":"Th\u00e9or\u00e8me de Bayes et Probabilit\u00e9 Compos\u00e9e"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Th\u00e9or\u00e8me de Bayes et Probabilit\u00e9 Compos\u00e9e<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>R\u00e9sum\u00e9<\/strong><br \/><em>Dans ce cours, deux concepts fondamentaux en probabilit\u00e9 ont \u00e9t\u00e9 abord\u00e9s : la probabilit\u00e9 conditionnelle et la probabilit\u00e9 compos\u00e9e. La diff\u00e9rence entre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B|A)<\/span><\/span> a \u00e9t\u00e9 soulign\u00e9e. Le th\u00e9or\u00e8me de la probabilit\u00e9 compos\u00e9e stipule que la probabilit\u00e9 d&#8217;un \u00e9v\u00e9nement <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> peut \u00eatre exprim\u00e9e comme la somme des probabilit\u00e9s conditionnelles <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_i)<\/span><\/span> multipli\u00e9es par les probabilit\u00e9s des \u00e9v\u00e9nements <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_i<\/span><\/span>. Par la suite, le th\u00e9or\u00e8me de Bayes a \u00e9t\u00e9 pr\u00e9sent\u00e9, permettant de calculer la probabilit\u00e9 conditionnelle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B_k|A)<\/span><\/span> en utilisant la probabilit\u00e9 conditionnelle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_k)<\/span><\/span>, la probabilit\u00e9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B_k)<\/span><\/span> et la somme des probabilit\u00e9s conditionnelles <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_i)<\/span><\/span> multipli\u00e9es par les probabilit\u00e9s des \u00e9v\u00e9nements <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_i<\/span><\/span>. Ces concepts sont essentiels pour comprendre et appliquer la probabilit\u00e9 conditionnelle dans divers contextes, et le th\u00e9or\u00e8me de Bayes fournit un outil puissant pour mettre \u00e0 jour les probabilit\u00e9s \u00e0 partir de nouvelles informations.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJECTIFS D&#8217;APPRENTISSAGE:<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de:\n<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> le concept de probabilit\u00e9 conditionnelle et diff\u00e9rencier entre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B|A)<\/span><\/span>.<\/li>\n<li><strong>Calculer<\/strong> la probabilit\u00e9 d&#8217;un \u00e9v\u00e9nement en utilisant des probabilit\u00e9s compos\u00e9es.<\/li>\n<li><strong>D\u00e9montrer<\/strong> la r\u00e8gle de Bayes.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>TABLE DES MATI\u00c8RES<\/u><\/strong><br \/>\n<a href=\"#1\">Probabilit\u00e9 Compos\u00e9e et Probabilit\u00e9 Conditionnelle<\/a><br \/>\n<a href=\"#2\">Le Th\u00e9or\u00e8me de Bayes<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/BDUTXmxlsM0\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">Dans le <a href=\"https:\/\/toposuranos.com\/probabilidad-condicional-e-independencia-entre-eventos\/\" rel=\"noopener\" target=\"_blank\">cours pr\u00e9c\u00e9dent<\/a>, nous avons examin\u00e9 le concept de probabilit\u00e9 conditionnelle et avons \u00e9galement clarifi\u00e9 qu&#8217;il ne faut jamais confondre une probabilit\u00e9 conditionnelle de la forme <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span><\/span> avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B|A).<\/span><\/span> Bien que dans le langage courant, la conditionnalit\u00e9 puisse pr\u00eater \u00e0 confusion, math\u00e9matiquement, ce sont deux choses tr\u00e8s diff\u00e9rentes qui, cependant, sont li\u00e9es. Cette relation est d\u00e9crite par le th\u00e9or\u00e8me de Bayes, qui est bas\u00e9 sur la notion de probabilit\u00e9 compos\u00e9e.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Probabilit\u00e9 Compos\u00e9e et Probabilit\u00e9 Conditionnelle<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">TH\u00c9OR\u00c8ME:<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=210s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> est un \u00e9v\u00e9nement<\/span><\/strong><\/a> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_1, B_2, \\cdots, B_n<\/span><\/span> forment un ensemble d&#8217;\u00e9v\u00e9nements disjoints tels que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega,<\/span><\/span> alors il est vrai que:<\/p>\n<p style=\"text-align: center; color: #000000; background-color: #A0FFA0;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{P(A) = \\displaystyle \\sum_{i=1}^n P(A|B_i) P(B_i)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=428s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Cette fa\u00e7on d&#8217;\u00e9crire la probabilit\u00e9<\/span> <\/strong><\/a> de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> est ce que nous appelons <strong>Probabilit\u00e9 Compos\u00e9e de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A.<\/span><\/span><\/strong><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">D\u00c9MONSTRATION:<\/span><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span> <\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> est un \u00e9v\u00e9nement<\/td>\n<td>; Pr\u00e9misse<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega<\/span><\/span><\/td>\n<td>; Pr\u00e9misse<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_1, \\cdots, B_n<\/span><\/span> sont tous disjoints entre eux<\/td>\n<td>; Pr\u00e9misse<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(A\\cap B_i)\\cap(A\\cap B_j) = \\varnothing,<\/span><\/span> avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\neq j<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i,j\\in \\{1,2,3,\\cdots n\\}<\/span><\/span><\/td>\n<td>; De (1,2,3)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n \\left(A \\cap B_i \\right) = A<\/span><\/span><\/td>\n<td>; De (1,2,3)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(A) = P\\left( \\bigcup_{i=1}^n \\left(A \\cap B_i \\right) \\right) = \\sum_{i=1}^n P\\left( A \\cap B_i \\right)<\/span><\/span><\/td>\n<td>; De (4,5)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> P(A|B_i) = \\dfrac{P(A\\cap B_i)}{P(B_i)}<\/span><\/span><\/td>\n<td>; D\u00e9finition de Probabilit\u00e9 Conditionnelle<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> P(A\\cap B_i) = P(A|B_i) P(B_i)<\/span><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\displaystyle P(A) = \\sum_{i=1}^n P(A|B_i) P(B_i)}<\/span><\/span><\/td>\n<td>; De (6,7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Le Th\u00e9or\u00e8me de Bayes<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=801s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Dans le m\u00eame contexte que le th\u00e9or\u00e8me pr\u00e9c\u00e9dent<\/span><\/strong><\/a>, le th\u00e9or\u00e8me suivant s&#8217;applique:<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">TH\u00c9OR\u00c8ME:<\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> P(B_k|A) = \\dfrac{P(A|B_k)P(B_k)}{\\displaystyle\\sum_{i=1}^n P(A|B_i)P(B_i)} = \\dfrac{P(A|B_k)P(B_k)}{P(A)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">D\u00c9MONSTRATION:<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=855s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> est un \u00e9v\u00e9nement quelconque<\/span><\/strong><\/a> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_1, B_2, \\cdots, B_n<\/span><\/span> est une collection d&#8217;\u00e9v\u00e9nements disjoints tels que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega,<\/span><\/span> selon le th\u00e9or\u00e8me pr\u00e9c\u00e9dent de la probabilit\u00e9 compos\u00e9e, nous avons:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A) = \\displaystyle \\sum_{i=1}^n P(A|B_i)P(B_i)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Maintenant, en utilisant le fait que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\cap Y) = P(X|Y)P(Y),<\/span><\/span> nous avons que si nous rempla\u00e7ons <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=A<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=B_k,<\/span><\/span> nous arriverons \u00e0<\/p>\n<p style=\"text-align: center; color: #000000; background-color: #b0b0ff;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A) = \\dfrac{P(B_k \\cap A)}{P(B_k|A)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">D&#8217;autre part, nous avons que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_k) = \\dfrac{P(A\\cap B_k)}{P(B_k)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">D&#8217;o\u00f9 il suit que<\/p>\n<p style=\"text-align: center; color: #000000; background-color: #b0ffb0;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B_k \\cap A) = P(A|B_k)P(B_k)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Maintenant, si nous rempla\u00e7ons la partie verte dans la partie bleue, nous aurons<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A) = \\dfrac{P(A|B_k)P(B_k)}{P(B_k|A)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Ce qui est \u00e9quivalent \u00e0 dire<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{P(B_k|A) = \\dfrac{P(A|B_k)P(B_k)}{P(A)}= \\dfrac{P(A|B_k)P(B_k)}{\\displaystyle \\sum_{i=1}^n P(A|B_i) P(B_i)} }<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">C&#8217;est ce que nous voulions d\u00e9montrer.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Th\u00e9or\u00e8me de Bayes et Probabilit\u00e9 Compos\u00e9e R\u00e9sum\u00e9Dans ce cours, deux concepts fondamentaux en probabilit\u00e9 ont \u00e9t\u00e9 abord\u00e9s : la probabilit\u00e9 conditionnelle et la probabilit\u00e9 compos\u00e9e. La diff\u00e9rence entre et a \u00e9t\u00e9 soulign\u00e9e. Le th\u00e9or\u00e8me de la probabilit\u00e9 compos\u00e9e stipule que la probabilit\u00e9 d&#8217;un \u00e9v\u00e9nement peut \u00eatre exprim\u00e9e comme la somme des probabilit\u00e9s conditionnelles multipli\u00e9es par [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26415,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":14,"footnotes":""},"categories":[569,682],"tags":[],"class_list":["post-26741","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-probabilites-et-statistiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Th\u00e9or\u00e8me de Bayes et Probabilit\u00e9 Compos\u00e9e - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Apprenez \u00e0 diff\u00e9rencier entre P(A|B) et P(B|A), et comment 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