{"id":27538,"date":"2021-08-25T07:52:37","date_gmt":"2021-08-25T07:52:37","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27538"},"modified":"2026-02-10T06:59:18","modified_gmt":"2026-02-10T06:59:18","slug":"updating-probability-using-bayes-formula","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/updating-probability-using-bayes-formula\/","title":{"rendered":"Updating Probability Using Bayes\u2019 Formula"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/updating-probability-using-bayes-formula\/#qapage-question-1\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/updating-probability-using-bayes-formula\/#question-1\",\n    \"name\": \"If a person is equally likely to choose routes A, B, or C, what is the probability they chose route A given they arrived on time?\",\n    \"text\": \"A chartered analyst can choose any one of three routes, A, B, or C, to get to work. The probabilities that she arrives on time using routes A, B, and C are 50%, 52%, and 60%, respectively. If she is equally likely to choose any one of the routes and she arrives on time, the probability that she chose route A is closest to:\\nA. 30.9%.\\nB. 16.67%.\\nC. 25%.\",\n    \"answerCount\": 1,\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/updating-probability-using-bayes-formula\/#answer-1\",\n      \"text\": \"A. 30.9%. Using Bayes\u2019 formula: P(A|T) = [P(A)P(T|A)] \/ [P(A)P(T|A) + P(B)P(T|B) + P(C)P(T|C)]. With equal route probabilities (1\/3 each), P(A|T) = (1\/3 \u00d7 0.50) \/ [(1\/3 \u00d7 0.50) + (1\/3 \u00d7 0.52) + (1\/3 \u00d7 0.60)] \u2248 0.30865 \u2248 30.9%.\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/PsZrsg3ZUDE?si=mfrfUPW5lbUg8v0w\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\n<p>Bayes\u2019 formula is used to calculate an updated or posterior probability given a set of prior probabilities for a given event. It is a theorem named after the Reverend T Bayes and is used widely in Bayesian methods of statistical inference.<\/p>\n\n\n\n<!--more-->\n\n\n\n<p>It is the logic used to come up with the formula:<\/p>\n\n\n\n<p>Let \\(E_1, E_2, E_3, &#8230;, E_n\\) be a set of mutually exclusive and exhaustive events.<\/p>\n\n\n\n<p>Using the conditional probability:<\/p>\n\n\n\n<p>$$ P(E_i | A) =\\cfrac {P(E_i A)}{P(A)} $$<\/p>\n\n\n\n<p>And also the relationship:<\/p>\n\n\n\n<p>$$ P(E_iA) = P(AE_i) = P(E_i)P(A | E_i) $$<\/p>\n\n\n\n<p>And the total probability rule:<\/p>\n\n\n\n<p>$$ P(A) = \\sum {P(AE_j)} \\quad \\text { for all j} = 1, 2,\u2026,n $$<\/p>\n\n\n\n<p>We can finally substitute for \\(P(E_iA)\\) and \\(P(A)\\) in equation 1.<\/p>\n\n\n\n<p>This gives:<\/p>\n\n\n\n<p>$$ P(E_{ i }|A)=\\cfrac { P(E_{ i })P(A|E_{ i }) }{ \\sum _{ j=1 }^{ n }{ P(E_{ i })P(A|E_{ i }) } } $$<\/p>\n\n\n\n<p>This is the Bayes\u2019 formula, and it allows us to \u2018turnaround\u2019 conditional probabilities, i.e., we can calculate \\(P(E_i|A)\\) &nbsp;if given information only about \\(P(A|E_i)\\).<\/p>\n\n\n\n<p>Take note of the explanations given below.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\\(P(E_j)\\) are known as <strong>prior probabilities<\/strong>.<\/li>\n\n\n\n<li>Event \\(A\\) is some event known <strong>to have occurred<\/strong>.<\/li>\n\n\n\n<li>\\(P(E_i|A)\\) is the <strong>posterior probability<\/strong>.<\/li>\n<\/ol>\n\n\n\n<!-- TOP CTA \u2013 Full Width Outline Button -->\n<div style=\"margin:24px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:block;\n       width:100%;\n       padding:14px 0;\n       border:2px solid #3b6fd8;\n       border-radius:50px;\n       font-size:18px;\n       font-weight:500;\n       text-align:center;\n       text-decoration:none;\n       color:#3b6fd8;\n       background-color:#f4f6f9;\n       box-sizing:border-box;\n     \">\n     Practice Bayes formula questions with free trial access.\n  <\/a>\n<\/div>\n\n\n\n<h4><strong>Example: Bayes&#8217; Formula<\/strong><\/h4>\n<p>A Civil Engineer wishes to investigate the punctuality of electric trains by considering the number of train journeys. In the sample, 50% of trains were destined for New York, 30% for Vegas, and 20% for Washington, DC. The probabilities of a train arriving late in New York, Vegas, and Washington, DC, are 40%, 35%, and 25%, respectively. If the Engineer picks a train at random from this group, what is the probability that it would be one destined for New York?<\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>We are looking for \\(P\\text{(New York | Late)}\\).<\/p>\n<p>Let us define the events that are critical in our calculation.<\/p>\n<p>First, \\(N\\) is the event \u201cA train chosen at random would be one destined for New York.\u201d<\/p>\n<p>Secondly, \\(V\\) is the event \u201cA train chosen at random would be destined for Vegas.\u201d<\/p>\n<p>And \\(W\\) is the event \u201cA train chosen at random would be destined for Washington DC.\u201d<\/p>\n<p>Finally, let \\(L\\) be the event \u201cA randomly chosen would arrive late.\u201d<\/p>\n<p>$$ \\begin{align*} P(N|L) &amp; =\\cfrac { P(N)P(L|N) }{ P(N)P(L|N)+P(V)P(L|V)+P(W)P(L|W) } \\\\ &amp; =\\cfrac { 0.5\u00d70.4 }{ 0.5\u00d7 0.4+0.3\u00d70.35+0.2\u00d7 0.25 } \\\\ &amp; =\\cfrac { 0.2 }{ 0.355 } \\\\ &amp; =0.5634 \\\\ &amp; =56.3\\% \\\\ \\end{align*} $$<\/p>\n<p>We have computed \\(P(N | L)\\) given only \\(P(L | N)\\), hence the phrase \u2018turnaround conditional probability\u2019.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>A chartered analyst can choose any one of three routes, A, B, or C, to get to work. The probabilities that she arrives on time using routes A, B, and C are 50%, 52%, and 60%, in that order. If she is equally likely to choose any one of the routes and arrive on time, the probability that she chose route A is <em>closest<\/em> to:<\/p>\n<p>A. 30.9%.<\/p>\n<p>B. 16.67%.<\/p>\n<p>C. 25%.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>First, you should define the relevant events.<\/p>\n<p>Let \\(A\\) be the event \u201cChooses route A.\u201d<\/p>\n<p>Let \\(B\\) be the event \u201cChooses route B.\u201d<\/p>\n<p>And let \\(C\\) be the event \u201cChooses route C.\u201d<\/p>\n<p>Lastly, define event T as \u201cArrives to work on time.\u201d<\/p>\n<p>Now, what we have is \\(P(T | A)\\), i.e., the probability that the analyst arrives on time, given that she chooses route A.<\/p>\n<p>However, we want to find the turnaround probability \\(P(A | T)\\), i.e., the probability that the analyst chooses route A, given that she arrives on time.<\/p>\n<p>This is what calls for the application of Bayes\u2019 formula:<\/p>\n<p>$$ \\begin{align*} P(A|T) &amp; =\\cfrac { P(A)P(T|A) }{ P(A)P(T|A)+P(B)P(T|B)+P(C)P(T|C) } \\\\ &amp; =\\cfrac { \\frac {1}{3} \u00d70.5 }{ \\frac {1}{3} \u00d70.5+\\frac {1}{3} \u00d7 0.52+\\frac {1}{3} \u00d7 0.6 } \\\\ &amp; =\\cfrac { 0.16667 }{ 0.54 } \\\\ &amp; =0.30865 \\\\ &amp; =30.9\\% \\\\ \\end{align*} $$<\/p>\n<\/blockquote>\n\n\n<!-- BOTTOM CTA \u2013 Refined Version -->\n<div style=\"text-align:center; background-color:#f4f6f9; padding:35px 20px; border-radius:12px; margin-top:40px;\">\n\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"\n       display:inline-block;\n       padding:14px 34px;\n       background-color:#3b6fd8;\n       color:#ffffff;\n       border-radius:50px;\n       font-size:16px;\n       font-weight:600;\n       text-decoration:none;\n       margin-bottom:18px;\n     \">\n     Start Free Trial\n  <\/a>\n\n  <p style=\"max-width:700px; margin:0 auto; font-size:16px; line-height:1.6; color:#333;\">\n    Strengthen your CFA Level I probability skills with exam-style Bayes formula problems, clear step by step solutions, and timed practice designed to improve accuracy under exam conditions.\n  <\/p>\n\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Bayes\u2019 formula is used to calculate an updated or posterior probability given a set of prior probabilities for a given event. It is a theorem named after the Reverend T Bayes and is used widely in Bayesian methods of statistical&#8230;<\/p>\n","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-27538","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Bayes\u2019 Formula for Updating Probability | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to update probabilities using Bayes&#039; formula, incorporating prior knowledge to refine probability estimates.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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