{"id":26723,"date":"2021-04-21T13:00:31","date_gmt":"2021-04-21T13:00:31","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26723"},"modified":"2024-05-21T21:44:49","modified_gmt":"2024-05-21T21:44:49","slug":"teorema-de-bayes-probabilidade-composta","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/teorema-de-bayes-probabilidade-composta\/","title":{"rendered":"Teorema de Bayes e a Probabilidade Composta"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Teorema de Bayes e a Probabilidade Composta<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Resumo<\/strong><br \/><em>Nesta aula, foram abordados dois conceitos fundamentais em probabilidade: a probabilidade condicional e a probabilidade composta. Foi enfatizada a diferen\u00e7a entre <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">P(B|A)<\/span>. O Teorema da probabilidade composta estabelece que a probabilidade de um evento <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> pode ser expressa como a soma das probabilidades condicionais <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_i)<\/span> multiplicadas pelas probabilidades dos eventos <span class=\"katex-eq\" data-katex-display=\"false\">B_i<\/span>. Posteriormente, foi apresentado o Teorema de Bayes, que permite calcular a probabilidade condicional <span class=\"katex-eq\" data-katex-display=\"false\">P(B_k|A)<\/span> utilizando a probabilidade condicional <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_k)<\/span>, a probabilidade <span class=\"katex-eq\" data-katex-display=\"false\">P(B_k)<\/span> e a soma das probabilidades condicionais <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_i)<\/span> multiplicadas pelas probabilidades dos eventos <span class=\"katex-eq\" data-katex-display=\"false\">B_i<\/span>. Esses conceitos s\u00e3o fundamentais para compreender e aplicar a probabilidade condicional em diversos contextos, e o Teorema de Bayes fornece uma ferramenta poderosa para atualizar probabilidades a partir de novas informa\u00e7\u00f5es.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJETIVOS DE APRENDIZAGEM:<\/strong><br \/>\nAo finalizar esta aula, o aluno ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Compreender<\/strong> o conceito de probabilidade condicional e diferenciar entre <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">P(B|A)<\/span>.<\/li>\n<li><strong>Calcular<\/strong> a probabilidade de um evento usando probabilidades compostas.<\/li>\n<li><strong>Demonstrar<\/strong> a regra de Bayes.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u><\/strong><br \/>\n<a href=\"#1\">A Probabilidade Composta e a probabilidade condicional<\/a><br \/>\n<a href=\"#2\">O Teorema de Bayes<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/BDUTXmxlsM0\" title=\"YouTube video player\" 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;\">Na <a href=\"https:\/\/toposuranos.com\/probabilidad-condicional-e-independencia-entre-eventos\/\" rel=\"noopener\" target=\"_blank\">aula anterior<\/a>, revisamos o conceito de probabilidade condicional e tamb\u00e9m esclarecemos que nunca se deve confundir uma probabilidade condicional da forma <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span> com <span class=\"katex-eq\" data-katex-display=\"false\">P(B|A).<\/span> Embora na linguagem cotidiana a condicionalidade possa ser confusa, matematicamente s\u00e3o duas coisas muito diferentes que, no entanto, est\u00e3o relacionadas. Esta rela\u00e7\u00e3o \u00e9 descrita pelo Teorema de Bayes, que se baseia na no\u00e7\u00e3o de probabilidade composta para sua formula\u00e7\u00e3o.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>A Probabilidade Composta e a probabilidade condicional<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">TEOREMA:<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=210s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Se <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u00e9 um evento<\/span><\/strong><\/a> e <span class=\"katex-eq\" data-katex-display=\"false\">B_1, B_2, \\cdots, B_n<\/span> formam um conjunto de eventos disjuntos tais que <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega,<\/span> ent\u00e3o se cumpre que:<\/p>\n<p style=\"text-align: center; color: #000000; background-color: #A0FFA0;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{P(A) = \\displaystyle \\sum_{i=1}^n P(A|B_i) P(B_i)}<\/span>\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;\">Esta forma de escrever a probabilidade<\/span> <\/strong><\/a>de <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u00e9 o que chamamos de <strong>Probabilidade Composta de <span class=\"katex-eq\" data-katex-display=\"false\">A.<\/span><\/strong><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">DEMONSTRA\u00c7\u00c3O:<\/span><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span> <\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u00e9 um Evento<\/td>\n<td>; Premissa<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega<\/span><\/td>\n<td>; Premissa<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">B_1, \\cdots, B_n<\/span> s\u00e3o todos disjuntos entre si<\/td>\n<td>; Premissa<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(A\\cap B_i)\\cap(A\\cap B_j) = \\varnothing,<\/span> com <span class=\"katex-eq\" data-katex-display=\"false\">i\\neq j<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">i,j\\in \\{1,2,3,\\cdots n\\}<\/span><\/td>\n<td>; De (1,2,3)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n \\left(A \\cap B_i \\right) = A<\/span><\/td>\n<td>; De (1,2,3)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/td>\n<td><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><\/td>\n<td>; De (4,5)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> P(A|B_i) = \\dfrac{P(A\\cap B_i)}{P(B_i)}<\/span><\/td>\n<td>; Defini\u00e7\u00e3o de Probabilidade Condicional<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> P(A\\cap B_i) = P(A|B_i) P(B_i)<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\displaystyle P(A) = \\sum_{i=1}^n P(A|B_i) P(B_i)}<\/span><\/td>\n<td>; De (6,7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/p>\n<h2>O Teorema 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;\">No mesmo contexto que o teorema anterior<\/span><\/strong><\/a>, se cumpre o seguinte teorema:<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">TEOREMA:<\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><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>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">DEMONSTRA\u00c7\u00c3O:<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=855s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Se <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u00e9 um evento qualquer<\/span><\/strong><\/a> e <span class=\"katex-eq\" data-katex-display=\"false\">B_1, B_2, \\cdots, B_n<\/span> \u00e9 uma cole\u00e7\u00e3o de eventos disjuntos tais que <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega,<\/span> pelo teorema anterior da probabilidade composta, temos que:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A) = \\displaystyle \\sum_{i=1}^n P(A|B_i)P(B_i)<\/span>\n<p style=\"text-align: justify; color: #000000;\">Agora, usando o fato de que <span class=\"katex-eq\" data-katex-display=\"false\">P(X\\cap Y) = P(X|Y)P(Y),<\/span> temos que se substituirmos <span class=\"katex-eq\" data-katex-display=\"false\">Y=A<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">X=B_k,<\/span> chegaremos a que<\/p>\n<p style=\"text-align: center; color: #000000; background-color: #b0b0ff;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A) = \\dfrac{P(B_k \\cap A)}{P(B_k|A)}<\/span>\n<p style=\"text-align: justify; color: #000000;\">Por outro lado, temos que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_k) = \\dfrac{P(A\\cap B_k)}{P(B_k)}<\/span>\n<p style=\"text-align: justify; color: #000000;\">De onde se infere que<\/p>\n<p style=\"text-align: center; color: #000000; background-color: #b0ffb0;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B_k \\cap A) = P(A|B_k)P(B_k)<\/span>\n<p style=\"text-align: justify; color: #000000;\">Agora, se substituirmos o <span style=\"background-color: #b0ffb0;\">verde<\/span> dentro do <span style=\"background-color: #b0b0ff;\">azul<\/span>, teremos que<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A) = \\dfrac{P(A|B_k)P(B_k)}{P(B_k|A)}<\/span>\n<p style=\"text-align: justify; color: #000000;\">O que \u00e9 equivalente a dizer<\/p>\n<p style=\"text-align: center; color: #000000;\"><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>\n<p style=\"text-align: justify; color: #000000;\">Isto \u00e9 o que se queria demonstrar.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Teorema de Bayes e a Probabilidade Composta ResumoNesta aula, foram abordados dois conceitos fundamentais em probabilidade: a probabilidade condicional e a probabilidade composta. Foi enfatizada a diferen\u00e7a entre e . O Teorema da probabilidade composta estabelece que a probabilidade de um evento pode ser expressa como a soma das probabilidades condicionais multiplicadas pelas probabilidades dos [&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":2,"footnotes":""},"categories":[571,672],"tags":[],"class_list":["post-26723","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematica-pt","category-probabilidades-e-estatistica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Teorema de Bayes e a Probabilidade Composta - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprenda a diferenciar entre P(A|B) e P(B|A), e como usar o Teorema de Bayes para atualizar e calcular a probabilidade composta.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/pt\/teorema-de-bayes-probabilidade-composta\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Teorema de Bayes e a Probabilidade Composta\" \/>\n<meta property=\"og:description\" content=\"Aprenda a diferenciar entre P(A|B) e P(B|A), e como usar o Teorema de Bayes para atualizar e calcular a probabilidade composta.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/pt\/teorema-de-bayes-probabilidade-composta\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-04-21T13:00:31+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-21T21:44:49+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/04\/bayes-e1712957676811-1024x285.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Teorema de Bayes e a Probabilidade Composta\" \/>\n<meta name=\"twitter:description\" content=\"Aprenda a diferenciar entre P(A|B) e P(B|A), e como usar o Teorema de Bayes para atualizar e calcular a probabilidade composta.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/04\/bayes-e1712957676811.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/teorema-de-bayes-probabilidade-composta\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/teorema-de-bayes-probabilidade-composta\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Teorema de Bayes e a Probabilidade Composta\",\"datePublished\":\"2021-04-21T13:00:31+00:00\",\"dateModified\":\"2024-05-21T21:44:49+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/teorema-de-bayes-probabilidade-composta\\\/\"},\"wordCount\":831,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/teorema-de-bayes-probabilidade-composta\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/04\\\/bayes-e1712957676811.jpg\",\"articleSection\":[\"Matem\u00e1tica\",\"Probabilidades e Estat\u00edstica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/teorema-de-bayes-probabilidade-composta\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/teorema-de-bayes-probabilidade-composta\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/teorema-de-bayes-probabilidade-composta\\\/\",\"name\":\"Teorema de Bayes e a Probabilidade Composta - 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