{"id":26720,"date":"2021-04-21T13:00:54","date_gmt":"2021-04-21T13:00:54","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26720"},"modified":"2024-05-21T21:39:47","modified_gmt":"2024-05-21T21:39:47","slug":"bayes-theorem-and-compound-probability","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/bayes-theorem-and-compound-probability\/","title":{"rendered":"Bayes&#8217; Theorem and Compound Probability"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Bayes&#8217; Theorem and Compound Probability<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Abstract<\/strong><br \/><em>In this class, two fundamental concepts in probability were discussed: conditional probability and compound probability. The difference between <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">P(B|A)<\/span> was emphasized. The Compound Probability Theorem states that the probability of an event <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> can be expressed as the sum of the conditional probabilities <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_i)<\/span> multiplied by the probabilities of the events <span class=\"katex-eq\" data-katex-display=\"false\">B_i<\/span>. Subsequently, Bayes&#8217; Theorem was introduced, which allows the calculation of the conditional probability <span class=\"katex-eq\" data-katex-display=\"false\">P(B_k|A)<\/span> using the conditional probability <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_k)<\/span>, the probability <span class=\"katex-eq\" data-katex-display=\"false\">P(B_k)<\/span>, and the sum of the conditional probabilities <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B_i)<\/span> multiplied by the probabilities of the events <span class=\"katex-eq\" data-katex-display=\"false\">B_i<\/span>. These concepts are essential for understanding and applying conditional probability in various contexts, and Bayes&#8217; Theorem provides a powerful tool for updating probabilities based on new information.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nBy the end of this class, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Understand<\/strong> the concept of conditional probability and differentiate between <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">P(B|A)<\/span>.<\/li>\n<li><strong>Calculate<\/strong> the probability of an event using compound probabilities.<\/li>\n<li><strong>Demonstrate<\/strong> Bayes&#8217; rule.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>TABLE OF CONTENTS<\/u><\/strong><br \/>\n<a href=\"#1\">Compound Probability and Conditional Probability<\/a><br \/>\n<a href=\"#2\">Bayes&#8217; Theorem<\/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;\">In the <a href=\"https:\/\/toposuranos.com\/probabilidad-condicional-e-independencia-entre-eventos\/\" rel=\"noopener\" target=\"_blank\">previous class<\/a>, we reviewed the concept of conditional probability and also clarified that one should never confuse a conditional probability of the form <span class=\"katex-eq\" data-katex-display=\"false\">P(A|B)<\/span> with <span class=\"katex-eq\" data-katex-display=\"false\">P(B|A).<\/span> Although in everyday language conditionality can be confusing, mathematically they are two very different things that, however, are related. This relationship is described by Bayes&#8217; Theorem, which is based on the notion of compound probability for its formulation.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Compound Probability and Conditional Probability<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">THEOREM:<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=210s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">If <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> is an event<\/span><\/strong><\/a> and <span class=\"katex-eq\" data-katex-display=\"false\">B_1, B_2, \\cdots, B_n<\/span> form a set of disjoint events such that <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega,<\/span> then the following holds:<\/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;\">This way of writing the probability<\/span> <\/strong><\/a>of <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> is what we call the <strong>Compound Probability of <span class=\"katex-eq\" data-katex-display=\"false\">A.<\/span><\/strong><\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">PROOF:<\/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> is an Event<\/td>\n<td>; Premise<\/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>; Premise<\/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> are all mutually exclusive<\/td>\n<td>; Premise<\/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> with <span class=\"katex-eq\" data-katex-display=\"false\">i\\neq j<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">i,j\\in \\{1,2,3,\\cdots n\\}<\/span><\/td>\n<td>; From (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>; From (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>; From (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>; Definition of Conditional Probability<\/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>; From (6,7)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Bayes&#8217; Theorem<\/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;\">In the same context as the previous theorem<\/span><\/strong><\/a>, the following theorem holds:<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #800000;\">THEOREM:<\/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;\">PROOF:<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=BDUTXmxlsM0&amp;t=855s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">If <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> is any event<\/span><\/strong><\/a> and <span class=\"katex-eq\" data-katex-display=\"false\">B_1, B_2, \\cdots, B_n<\/span> is a collection of disjoint events such that <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{i=1}^n B_i = \\Omega,<\/span> by the previous Compound Probability Theorem, we have that:<\/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;\">Now, using the fact that <span class=\"katex-eq\" data-katex-display=\"false\">P(X\\cap Y) = P(X|Y)P(Y),<\/span> we have that if we replace <span class=\"katex-eq\" data-katex-display=\"false\">Y=A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">X=B_k,<\/span> we will arrive at<\/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;\">On the other hand, we have that<\/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;\">From which it follows that<\/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;\">Now, if we replace the <span style=\"background-color: #b0ffb0;\">green<\/span> within the <span style=\"background-color: #b0b0ff;\">blue<\/span>, we will have<\/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;\">Which is equivalent to saying<\/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;\">This is what we wanted to prove.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bayes&#8217; Theorem and Compound Probability AbstractIn this class, two fundamental concepts in probability were discussed: conditional probability and compound probability. The difference between and was emphasized. The Compound Probability Theorem states that the probability of an event can be expressed as the sum of the conditional probabilities multiplied by the probabilities of the events . [&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":17,"footnotes":""},"categories":[567,670],"tags":[],"class_list":["post-26720","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-probabilities-and-statistics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Bayes&#039; Theorem and Compound Probability - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Learn to differentiate between P(A|B) and P(B|A), and how to use Bayes&#039; Theorem to update and calculate compound probability.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/bayes-theorem-and-compound-probability\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bayes&#039; 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