{"id":45486,"date":"2023-08-06T16:20:40","date_gmt":"2023-08-06T16:20:40","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45486"},"modified":"2026-03-05T10:21:20","modified_gmt":"2026-03-05T10:21:20","slug":"bayes-formula","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/bayes-formula\/","title":{"rendered":"Bayes&apos; Formula"},"content":{"rendered":"\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 that a company will remain solvent given that it meets the investment criteria?\",\n    \"text\": \"You have developed a set of criteria for assessing potential investments in growth-stage companies. Companies not meeting these criteria are predicted to be insolvent within 24 months. You gathered the following information:\\n\\n- P(insolvency) = 0.50\\n- P(meet criteria) = 0.65\\n- P(meet criteria | solvency) = 0.80\\n\\nThe probability that a company will remain solvent given that it meets the criteria, that is, P(solvency | meet criteria), is closest to:\\n\\nA. 20%\\nB. 50%\\nC. 62%\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"62%.\\n\\nFirst compute P(solvent) = 1 \u2212 P(insolvency) = 0.50.\\n\\nUse the law of total probability to find P(meet criteria | insolvency):\\n\\nP(meet criteria) = P(meet criteria | solvency)P(solvency) + P(meet criteria | insolvency)P(insolvency)\\n\\n0.65 = (0.80 \u00d7 0.50) + P(meet criteria | insolvency)(0.50)\\n\\nSolving gives P(meet criteria | insolvency) = 0.50.\\n\\nApply Bayes\u2019 theorem:\\n\\nP(solvency | meet criteria) = (0.80 \u00d7 0.50) \/ [(0.80 \u00d7 0.50) + (0.50 \u00d7 0.50)] = 0.6153 \u2248 62%.\",\n      \"dateCreated\": \"2026-01-02\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/Ciu1kP6q9d0?si=xppPHXaQWF4JtWGH\" 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>Investors make investment decisions based on their experience and expertise. Their decisions may change in the wake of new knowledge and observations.<\/p>\n\n\n\n<p>Bayes&#8217; formula allows us to update our decisions as we receive new information. In other words, Bayes&#8217; formula is used to calculate an updated or posterior probability given a set of prior probabilities for a given event.<\/p>\n\n\n\n<div style=\"text-align:center; margin:18px 0;\">\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"display:inline-flex; align-items:center; justify-content:center; padding:10px 18px; border:2px solid #1e5bd8; color:#1e5bd8; background:#f5f7fb; border-radius:9999px; text-decoration:none; font-weight:600; white-space:nowrap;\">\nApply Bayes\u2019 formula concepts with a free CFA trial\n<\/a>\n<\/div>\n\n\n<p>Given a set of prior probabilities for an event, if we receive new information, the updated probability is as follows:<\/p>\n<p>$$ \\begin{align*}\\text{Updated probability of an event given the new information}<br \/>= &amp; \\frac {\\text{Probability of the new information given event}}{\\text{Unconditional probability of the new information}} \\\\ &amp; \\times {\\text{Prior probability of event.}}<br \/>\\end{align*} $$<\/p>\n<p>The above equation can be written as:<\/p>\n<p>$$ P(\\text{Event}\\mid \\text{Information})=\\frac { P(\\text{Information}\\mid \\text{Event})}{P(\\text{Information})}\\cdot P(\\text{Event}) $$<\/p>\n<h2>Deriving Bayes&#8217; Formula<\/h2>\n<p>Let \\(B_1, B_2, B_3,\\ldots, B_n\\) be a set of mutually exclusive and exhaustive events.<\/p>\n<p>Using the conditional probability:<\/p>\n<p>$$ P(B_i \\mid A)=\\frac {P(B_i\\cap A)}{P(A)} \\ldots \\ldots (1) $$<\/p>\n<p>And also, the relationship:<\/p>\n<p>$$ P(B_i \\cap A)=P(A \\cap B_i )=P(B_i )\\cdot P(A\\mid B_i ) \\ldots \\ldots (2)$$<\/p>\n<p>Also, using the total probability rule:<\/p>\n<p>$$ P(A)=\\sum_{i=1}^n P(A \\cap B_i) =\\sum_{i=1}^n P(B_i) \\cdot P(A\\mid B_i ) \\ldots \\ldots (3) $$<\/p>\n<p>Substituting equations (2) and (3) in (1), we have:<\/p>\n<p>$$ P(B_i \\mid A)=\\frac {(P(A \\mid B_i )}{\\sum_{i=1}^n P(B_i)\\cdot P(A\\mid B_i)} \\cdot P(B_i) $$<\/p>\n<p>This is the Bayes&#8217; formula, and it allows us to \u2018turnaround\u2019 conditional probabilities, i.e., we can calculate \\(P(B_i \\mid A)\\) if given information only about \\(P(A\\mid B_i)\\).<\/p>\n<p>Note that:<\/p>\n<ol type=\"1\">\n<li>\\(P(B_i)\\) are known as <strong>prior probabilities<\/strong>.<\/li>\n<li>Event A is some event known <strong>to have occurred<\/strong>.<\/li>\n<li>\\(P(B_i \\mid A)\\) is the <strong>posterior probability<\/strong>.<\/li>\n<\/ol>\n<p><strong>Example: Bayes&#8217; Formula<\/strong><\/p>\n<p data-start=\"0\" data-end=\"298\">An investment analyst is studying stock performance using a sample of stocks listed on multiple exchanges. In the sample, 50% of the stocks are listed on the New York Stock Exchange (NYSE), 30% on the London Stock Exchange (LSE), and 20% on the Tokyo Stock Exchange (TSE).<\/p>\n<p data-start=\"300\" data-end=\"459\">The probability that a stock posts a negative return is 40% for NYSE listed stocks, 35% for LSE listed stocks, and 25% for TSE listed stocks.<\/p>\n<p data-start=\"461\" data-end=\"614\">If the analyst selects a stock at random from the sample, what is the probability that it is listed on the NYSE, given that it has a negative return?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>We are looking for P(NYSE | Negative Return).<\/p>\n<p>Let\u2019s define the following events:<\/p>\n<p>NYSE is the event \u201cA stock chosen at random is listed on the NYSE.\u201d<\/p>\n<p>LSE is the event \u201cA stock chosen at random is listed on the LSE.\u201d<\/p>\n<p>TSE is the event \u201cA stock chosen at random is listed on the TSE.\u201d<\/p>\n<p>Finally, let NR be the event \u201cA randomly chosen stock posts a negative return.\u201d<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\begin{align*}<br \/>P\\left(NYSE\\middle|NR\\right)&amp; =\\frac{P\\left(NYSE\\right)P(NR|NYSE)}{P\\left(NYSE\\right)P\\left(NR\\middle|NYSE\\right)+P\\left(LSE\\right)P\\left(NR\\middle|LSE\\right)+P\\left(TSE\\right)P(NR|TSE)}\\\\<br \/>&amp; =\\frac {0.5\\times 0.4}{0.5\\times 0.4+0.3 \\times 0.35+0.2 \\times 0.25} \\\\<br \/>&amp;=\\frac {0.2}{0.355} \\\\<br \/>&amp; =0.5634\\approx 56.3\\%<br \/>\\end{align*} $$<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>You have developed a set of criteria for assessing potential investments in growth-stage companies. Companies not meeting these criteria are predicted to be insolvent within 24 months. You gathered the following information when validating your criteria:<\/p>\n<ul>\n<li>Fifty percent of the companies that have been assessed will become insolvent within 24 months: \\(P(\\text{insolvency}) = 0.50\\).<\/li>\n<li>Sixty-five percent of the companies assessed meet the criteria: \\(P(\\text{meet criteria}) = 0.65\\).<\/li>\n<li>The probability that a company will meet the criteria given that it remains solvent for 24 months is 0.80: \\(P(\\text{meet criteria} \\mid \\text{solvency}) = 0.80\\).<\/li>\n<\/ul>\n<p>The probability that a company will remain solvent, given that it meets the criteria, that is, \\(P(\\text{solvency} \\mid \\text{meet criteria})\\), is <em>closest to<\/em>:<\/p>\n<ol type=\"A\">\n<li>20%.<\/li>\n<li>50%.<\/li>\n<li>62%.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>Using Bayes&#8217; formula, we have:<\/p>\n<p>$$ \\begin{align*}<br \/>&amp; P(\\text{solvency} \\mid \\text{meet criteria})\\\\ &amp; =\\frac {P(\\text{meet criteria}\\mid \\text{solvency})P(\\text{solvency})}{{ [P(\\text{meet criteria}\\mid \\text{solvency})P(\\text{solvency})} \\\\ {+P(\\text{meet criteria}\\mid \\text{insolvency})P(\\text{insolvency}) ] }} \\\\<br \/>&amp;=\\frac {0.80 \\times 0.50}{0.80 \\times 0.50+P(\\text{meet criteria}\\mid \\text{insolvency})\\times 0.50} \\\\<br \/>\\end{align*} $$<\/p>\n<p>Clearly, we need to calculate the \\(P(\\text{meet criteria}\\mid \\text{insolvency})\\). Using the total probability:<\/p>\n<p>$$ \\begin{align*}<br \/>P(\\text{meet criteria}) &amp; = P(\\text{meet criteria} \\mid \\text{solvency})P(\\text{solvency}) \\\\ &amp; + P(\\text{meet criteria} \\mid \\text{insolvency}) P(\\text{insolvency}) \\\\<br \/>\\Rightarrow 0.65 &amp; = 0.80 \\times 0.50 + P(\\text{meet criteria} \\mid \\text{insolvency})\\times 0.50<br \/>\\end{align*} $$<\/p>\n<p>$$ \\therefore P(\\text{meet criteria} \\mid \\text{insolvency})=\\frac {0.65-0.80 \\times 0.50}{0.50}=0.50 $$<\/p>\n<p>As such,<\/p>\n<p>$$ P(\\text{solvency} \\mid \\text{meet criteria})=\\frac {0.80 \\times 0.50}{0.80 \\times 0.50+0.50 \\times 0.50}=\\bf{0.6153}\\approx 62\\% $$<\/p>\n<\/blockquote>\n\n\n<div style=\"text-align:center; margin:32px 0 10px;\">\n\n<a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"display:inline-flex; align-items:center; justify-content:center; padding:12px 24px; border-radius:9999px; background:#1e5bd8; color:#ffffff; font-weight:700; text-decoration:none;\">\nStart Free Trial \u2192\n<\/a>\n\n<p style=\"margin-top:12px; font-size:16px; line-height:1.5;\">\nPrepare for CFA Level I by calculating posterior probabilities and applying Bayes\u2019 formula in financial decision making.\n<\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Investors make investment decisions based on their experience and expertise. Their decisions may change in the wake of new knowledge and observations. Bayes&#8217; formula allows us to update our decisions as we receive new information. In other words, Bayes&#8217; formula&#8230;<\/p>\n","protected":false},"author":7,"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-45486","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 Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how Bayes&#039; Formula updates probabilities based on new data and its applications in investment analysis and decision-making.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/bayes-formula\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bayes&#039; 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