{"id":684,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=684"},"modified":"2026-03-27T17:27:09","modified_gmt":"2026-03-27T17:27:09","slug":"bayes-formula-interpretation-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/bayes-formula-interpretation-example\/","title":{"rendered":"Bayes\u2019 Formula"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n\n  \"name\": \"Probability Concepts (2021 Level I CFA\u00ae Exam \u2013 Reading 8)\",\n\n  \"description\": \"This video lesson covers the fundamentals of probability concepts, including random variables, events, and probabilities. It explains key concepts like conditional and unconditional probabilities, multiplication and addition rules, Bayes' formula, tree diagrams, covariance, correlation, and probability distribution tools. Techniques for counting problems, such as combinations, permutations, and factorials, are also detailed for portfolio and investment analysis.\",\n\n  \"uploadDate\": \"2020-10-03T00:00:00+00:00\",\n\n  \"thumbnailUrl\": \"https:\/\/analystprep.com\/path-to-thumbnail\/probability-concepts-thumbnail.jpg\",\n\n  \"contentUrl\": \"https:\/\/youtu.be\/hu47ZbsskEw\",\n\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw\",\n\n  \"duration\": \"PT48M33S\",\n\n  \"publisher\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"logo\": {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/path-to-logo\/logo.jpg\",\n      \"width\": 600,\n      \"height\": 60\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"What is the probability the analyst chose route A given that she arrived on time?\",\n    \"text\": \"A chartered analyst can choose any one of these 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 of the routes and arrive on time, calculate the probability that she chose route A.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"Use Bayes\u2019 formula. With equal route choice, P(A)=P(B)=P(C)=1\/3. Let T be \u201carrives on time.\u201d Then P(A|T) = [P(A)P(T|A)] \/ [P(A)P(T|A)+P(B)P(T|B)+P(C)P(T|C)] = [(1\/3)(0.50)] \/ [(1\/3)(0.50)+(1\/3)(0.52)+(1\/3)(0.60)] = 0.1667\/0.54 = 0.3086 \u2248 30.9%.\",\n      \"dateCreated\": \"2026-02-05\"\n    }\n  }\n}\n<\/script>\n\n\n\n<p><br class=\"clear\" \/><p>\n  <iframe loading=\"lazy\"\n    src=\"\/\/www.youtube.com\/embed\/hu47ZbsskEw\"\n    width=\"611\"\n    height=\"343\"\n    allowfullscreen=\"allowfullscreen\">\n  <\/iframe>\n<\/p>\n<\/p>\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\u2019s a theorem named after Reverend T Bayes and it is widely used in Bayesian methods of statistical inference.<\/p>\n<p><!--more--><\/p>\n<p>This is the logic used to come up with the formula:<\/p>\n<p>Let E<sub>1<\/sub>, E<sub>2, <\/sub>E<sub>3, <\/sub>&#8230;, E<sub>n <\/sub>be a set of mutually exclusive and exhaustive events.<\/p>\n<p>Using the conditional probability:<\/p>\n<p>$$ P(E_i | A) =\\cfrac {P(E_i A)}{P(A)} $$<\/p>\n<p>And also the relationship:<\/p>\n<p>$$ P(E_iA) = P(AE_i) = P(E_i)P(A | E_i) $$<\/p>\n<p>And the total probability rule:<\/p>\n<p>$$ P(A) = \\sum {P(AE_j)} \\quad \\text { for all j} = 1, 2,\u2026,n $$<\/p>\n<p>We can finally substitute for P(E<sub>i<\/sub>A) and P(A) in equation 1<\/p>\n<p>This gives:<\/p>\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<p>This is the Bayes\u2019 formula and it allows us to \u2018turnaround\u2019 conditional probabilities i.e., we can calculate P(E<sub>i <\/sub>| A) if given information only about P(A | E<sub>i<\/sub>).<\/p>\n<p>Note:<\/p>\n<ol>\n<li>P(E<sub>j<\/sub>) are known as <strong>prior probabilities<\/strong>;<\/li>\n<li>the event A is some event known <strong>to have occurred<\/strong>; and<\/li>\n<li>P(E<sub>i<\/sub> | A) is the <strong>posterior probability<\/strong>.<\/li>\n<\/ol>\n<div style=\"text-align: center; margin: 24px 0;\">\n  <div style=\"max-width: 680px; margin: 0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display: flex; align-items: center; justify-content: center;\n       width: 100%; padding: 8px 16px;\n       border: 2px solid #2f6fed; border-radius: 999px;\n       color: #2f6fed; text-decoration: none;\n       font-size: 15px; font-weight: 500;\n       line-height: 1.2; white-space: nowrap;\">\n      Apply Bayes\u2019 formula in probability analysis with our free trial.\n    <\/a>\n  <\/div>\n<\/div>\n<h2><strong>Example: Bayes&#8217; Formula<\/strong><\/h2>\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% Vegas and 20% 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 the destination of the train would be New York?<\/p>\n<p><strong>Solution<\/strong>:<\/p>\n<p>We are looking for P(New York | Late).<\/p>\n<p>Let\u2019s define the following events:<\/p>\n<p>First, N is the event \u201cA train chosen at random whose destination is New York.\u201d<\/p>\n<p>Secondly, V is the event \u201cA train chosen at random whose destination is Vegas.\u201d<\/p>\n<p>And W is the event \u201cA train chosen at random whose destination is Washington DC.\u201d<\/p>\n<p>Finally, let L be the event \u201cA randomly chosen train arrives late.\u201d<\/p>\n<p>$$ \\begin{align*} P(N|L) &amp; =\\cfrac { P(N)P(L|N) }{ P(N)P(L|N)+P(V)P(L|V)+P(W)P(L|W) } \\\\ &amp; =\\cfrac { 0.5\\ast 0.4 }{ 0.5\\ast 0.4+0.3\\ast 0.35+0.2\\ast 0.25 } \\\\ &amp; =\\cfrac { 0.2 }{ 0.355 } \\\\ &amp; =0.5634 \\\\ &amp; =56.3\\% \\\\ \\end{align*} $$<\/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 these 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 arrive on time, calculate the probability that she chose route A.<\/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 A.<\/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>B the event, \u201cChooses route B\u201d; and<\/p>\n<p>C 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) =\u00a0 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\u00a0 given that she arrives on time.<\/p>\n<p>Hence, applying Bayes\u2019 formula:<\/p>\n<p>$$ \\begin{align*} P(A|T) &amp; =\\cfrac { P(A)P(T|A) }{ P(A)P(T|A)+P(B)P(T|B)+P(C)P(T|C) } \\\\ &amp; =\\cfrac { \\frac {1}{3} \\ast 0.5 }{ \\frac {1}{3} \\ast 0.5+\\frac {1}{3} \\ast 0.52+\\frac {1}{3} \\ast 0.6 } \\\\ &amp; =\\cfrac { 0.16667 }{ 0.54 } \\\\ &amp; =0.30865 \\\\ &amp; =30.9\\% \\\\ \\end{align*} $$<\/p>\n<\/blockquote>\n<p>Tip: good understanding of Bayes\u2019 theorem can only be preceded by a good understanding of the <a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/\">rules of probability<\/a>. The focus should be on the calculations.<\/p>\n<p>\u00a0<\/p>\n<p><em>Reading 8 LOS 8n<\/em><\/p>\n<p><em>Calculate and interpret an updated probability using Bayes&#8217; formula.<\/em><\/p>\n<div class=\"notes_inv\"><hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>\n<p>\u00a0<\/p>\n<div style=\"text-align: center; margin: 40px 0 10px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display: inline-flex; align-items: center; justify-content: center;\n     padding: 10px 24px; border-radius: 999px;\n     font-size: 15px; font-weight: 600;\n     background-color: #4a72c9; color: #ffffff;\n     text-decoration: none; line-height: 1.2; white-space: nowrap;\">\n    Start Free Trial \u2192\n  <\/a>\n\n  <p style=\"margin: 12px auto 0; max-width: 560px; font-size: 15px; line-height: 1.6; color: #333333;\">\n    Practice Bayes\u2019 theorem and conditional probability with structured CFA Level I questions.\n  <\/p>\n<\/div>\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\u2019s a theorem named after Reverend T Bayes and it is widely used in Bayesian methods of statistical inference.<\/p>\n","protected":false},"author":2,"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-684","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&#039; Formula Example | CFA Level I<\/title>\n<meta name=\"description\" content=\"Learn how Bayes\u2019 formula is applied with a worked example to update probabilities using prior information and conditional probabilities.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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