{"id":34136,"date":"2021-04-21T13:00:44","date_gmt":"2021-04-21T13:00:44","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34136"},"modified":"2025-08-16T10:17:25","modified_gmt":"2025-08-16T10:17:25","slug":"theorema-bayesii-et-probabilitas-composita","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/theorema-bayesii-et-probabilitas-composita\/","title":{"rendered":"Theorema Bayesii et Probabilitas Composita"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Theorema Bayesii et Probabilitas Composita<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Summarium<\/strong><br \/><em>In hac lectione duo conceptus fundamentales in probabilitate tractati sunt: probabilitas conditionalis et probabilitas composita. Emphasis facta est differentiae inter <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>. Theorema probabilitatis compositae statuit probabilitatem eventus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> exprimi posse ut summam probabilitatum conditionalium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_i)<\/span><\/span> multiplicatarum per probabilitates eventuum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_i<\/span><\/span>. Deinde, propositum est Theorema Bayesii, quod sinit computare probabilitatem conditionalem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B_k|A)<\/span><\/span> utens probabilitate conditionali <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_k)<\/span><\/span>, probabilitate <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B_k)<\/span><\/span> et summa probabilitatum conditionalium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_i)<\/span><\/span> multiplicatarum per probabilitates eventuum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_i<\/span><\/span>. Hi conceptus fundamentales sunt ad intellegendam et adhibendam probabilitatem conditionalem in variis contextibus, et Theorema Bayesii praebet instrumentum validum ad probabilitates renovandas ex nova informatione.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>PROPOSITA DISCENDI:<\/strong><br \/>\nAd finem huius lectionis, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Intelligere<\/strong> notionem probabilitatis conditionalis et distinguere inter <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>Computare<\/strong> probabilitatem eventus utens probabilitatibus compositis.<\/li>\n<li><strong>Demonstrrare<\/strong> regulam Bayesii<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>INDEX CONTENTORUM<\/u><\/strong><br \/>\n<a href=\"#1\">Probabilitas Composita et probabilitas conditionalis<\/a><br \/>\n<a href=\"#2\">Theorema Bayesii<\/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;\">In <a href=\"https:\/\/toposuranos.com\/probabilidad-condicional-e-independencia-entre-eventos\/\" rel=\"noopener\" target=\"_blank\">praecedenti lectione<\/a>, tractavimus notionem probabilitatis conditionalis atque etiam declaravimus numquam confundendam esse probabilitatem conditionalem formae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B|A).<\/span><\/span> Quamquam in sermone cotidiano conditionalitas potest esse confusa, mathematice sunt duo diversa quae tamen inter se coniunguntur. Haec relatio describitur a Theorema Bayesii, quod fundatur in notione probabilitatis compositae ad suam formulationem.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Probabilitas Composita et probabilitas conditionalis<\/h2>\n<p style=\"text-align: justify;\"><span style=\"color: #800000;\">THEOREMA:<\/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 eventus<\/span><\/strong><\/a> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_1, B_2, \\cdots, B_n<\/span><\/span> constituunt collectionem eventuum disiunctam talium ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega,<\/span><\/span> tunc valet quod:<\/p>\n<p style=\"text-align: center; 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;\"><a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=428s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Haec ratio scribendi probabilitatem<\/span> <\/strong><\/a><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> est quod vocamus <strong>Probabilitatem Compositam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A.<\/span><\/span><\/strong><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #800000;\">DEMONSTRATIO:<\/span><\/p>\n<table style=\"text-align: justify;\">\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 Eventus<\/td>\n<td>; Praemissa<\/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>; Praemissa<\/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> sunt omnes disiuncti inter se<\/td>\n<td>; Praemissa<\/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> cum <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>; Ex (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>; Ex (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>; Ex (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>; Definitio Probabilitatis Conditionalis<\/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>; Ex (6,7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Theorema Bayesii<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=801s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">In eodem contextu ac theorema praecedens<\/span><\/strong><\/a>, valet sequens theorema:<\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #800000;\">THEOREMA:<\/span><\/p>\n<p style=\"text-align: center;\"><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;\"><span style=\"color: #800000;\">DEMONSTRATIO:<\/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 eventus quilibet<\/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 collectio eventuum disiuncta talium ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega,<\/span><\/span> per theorema praecedens probabilitatis compositae habemus:<\/p>\n<p style=\"text-align: center;\"><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;\">Nunc, utens eo quod <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\cap Y) = P(X|Y)P(Y),<\/span><\/span> habemus quod si substituimus <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> perveniemus ad<\/p>\n<p style=\"text-align: center; 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;\">Altera ex parte, habemus<\/p>\n<p style=\"text-align: center;\"><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;\">Unde colligitur<\/p>\n<p style=\"text-align: center; 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;\">Nunc, si substituimus id quod est <span style=\"background-color: #b0ffb0;\">viride<\/span> intra id quod est <span style=\"background-color: #b0b0ff;\">caeruleum<\/span>, habebimus<\/p>\n<p style=\"text-align: center;\"><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;\">Quod est aequivalens dicere<\/p>\n<p style=\"text-align: center;\"><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;\">Hoc est quod demonstrandum erat.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Theorema Bayesii et Probabilitas Composita SummariumIn hac lectione duo conceptus fundamentales in probabilitate tractati sunt: probabilitas conditionalis et probabilitas composita. Emphasis facta est differentiae inter et . Theorema probabilitatis compositae statuit probabilitatem eventus exprimi posse ut summam probabilitatum conditionalium multiplicatarum per probabilitates eventuum . Deinde, propositum est Theorema Bayesii, quod sinit computare probabilitatem conditionalem utens [&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":6,"footnotes":""},"categories":[1298,1364],"tags":[],"class_list":["post-34136","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematica","category-probabilitates-et-statistica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Theorema Bayesii et Probabilitas Composita - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce distinguere inter P(A|B) et P(B|A), atque quomodo uti Theorema Bayesii ad probabilitatem compositam renovandam et computandam.\" \/>\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\/la\/theorema-bayesii-et-probabilitas-composita\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Theorema Bayesii et Probabilitas Composita\" \/>\n<meta property=\"og:description\" content=\"Disce distinguere inter P(A|B) et P(B|A), atque quomodo uti Theorema Bayesii ad probabilitatem compositam renovandam et computandam.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/theorema-bayesii-et-probabilitas-composita\/\" \/>\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=\"2021-04-21T13:00:44+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-08-16T10:17:25+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=\"Theorema Bayesii et Probabilitas Composita\" \/>\n<meta name=\"twitter:description\" content=\"Disce distinguere inter P(A|B) et P(B|A), atque quomodo uti Theorema Bayesii ad probabilitatem compositam renovandam et computandam.\" \/>\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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/theorema-bayesii-et-probabilitas-composita\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/theorema-bayesii-et-probabilitas-composita\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Theorema Bayesii et Probabilitas Composita\",\"datePublished\":\"2021-04-21T13:00:44+00:00\",\"dateModified\":\"2025-08-16T10:17:25+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/theorema-bayesii-et-probabilitas-composita\\\/\"},\"wordCount\":733,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/theorema-bayesii-et-probabilitas-composita\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/04\\\/bayes-e1712957676811.jpg\",\"articleSection\":[\"Mathematica\",\"Probabilitates et Statistica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/theorema-bayesii-et-probabilitas-composita\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/theorema-bayesii-et-probabilitas-composita\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/theorema-bayesii-et-probabilitas-composita\\\/\",\"name\":\"Theorema Bayesii et Probabilitas Composita - 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