{"id":410,"date":"2019-08-17T13:28:00","date_gmt":"2019-08-17T13:28:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=410"},"modified":"2026-03-10T20:16:23","modified_gmt":"2026-03-10T20:16:23","slug":"probability-odds-for-and-against","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-odds-for-and-against\/","title":{"rendered":"Stating the Probability of an Event in Terms of the Odds for and Against the Event"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"If a fair coin is tossed, what are the odds against obtaining a head?\",\n    \"text\": \"Suppose you toss a fair coin. What are the odds against obtaining a head?\\n\\nA. 2:2\\nB. 1:2\\nC. 1:1\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. The probability of obtaining a head is P(H) = 1\/2. The odds against a head are calculated as (1 \u2212 P(H)) \u00f7 P(H) = (1 \u2212 1\/2) \u00f7 (1\/2) = (1\/2) \u00f7 (1\/2) = 1:1.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Probability Concepts (2021 Level I CFA\u00ae Exam \u2013 Reading 8)\",\n  \"description\": \"This video lesson covers probability concepts in quantitative methods, including definitions of random variables, events, and outcomes. It explores probability rules, odds, conditional probabilities, multiplication rules, and Bayes\u2019 formula. The lesson also discusses portfolio analysis, covariance, correlation, and practical applications using tree diagrams and counting problems.\",\n  \"uploadDate\": \"2020-10-01T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/hu47ZbsskEw\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=hu47ZbsskEw\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw\",\n  \"duration\": \"PT48M33S\"\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw?si=g8c9qAYF6otgDj50\" 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<p>Odds for and against an event represent a ratio of the desired outcomes versus the field. In other words, the odds for an event refer to the ratio of the number of ways the event can occur to the number of ways the event does not occur. Therefore:<\/p>\n<p><!--more--><\/p>\n<p>given the probability of an event \u2018E\u2019 i.e. P(E),<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{Odds for E} &amp; =\\cfrac {P(E)}{ \\left\\{ 1 \u2013 P(E) \\right\\} } \\\\<br \/>\\text{Odds against E} &amp; = \\cfrac { \\left\\{ 1 \u2013 P(E) \\right\\} }{ P(E) } \\\\<br \/>\\end{align*} $$<\/p>\n<h3><strong>Example: Odds for and Against an Event<\/strong><\/h3>\n<p>A box contains five blue balls, two green balls, and six yellow balls. What are the odds of drawing a blue ball from the box?<\/p>\n<p>First, we have to establish the probability of drawing a blue ball:<\/p>\n<p>Let P(B) represent the event that a blue ball is drawn from the box. Therefore,<\/p>\n<p>$$ P(B) = \\cfrac {5}{13} $$<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{The odds for a blue ball} &amp; =\\cfrac {5}{13} \u00f7 \\left(1 \u2013 \\cfrac {5}{13} \\right)\\\\<br \/>&amp; = \\cfrac {5}{13} \u00f7 \\left( \\cfrac {8}{13} \\right) \\\\<br \/>&amp; =\\cfrac {5}{13} * \\cfrac {13}{8} \\\\<br \/>&amp; =\\cfrac {5}{8} \\\\<br \/>\\end{align*} $$<\/p>\n<p>Therefore, the odds for a blue ball are 5:8 (pronounced as \u20185 to 8\u2019).<\/p>\n<p>Similarly, we can calculate the odds against drawing a blue ball:<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{Odds against a blue ball} &amp;= \\left\\{1 \u2013 \\cfrac{5}{13} \\right\\} \u00f7 \\cfrac{5}{13} \\\\<br \/>&amp;= \\cfrac{8}{13} \u00f7 \\cfrac{5}{13} \\\\<br \/>&amp;= \\cfrac{8}{13} * \\cfrac{13}{5} \\\\<br \/>&amp; = \\cfrac{8}{5}<br \/>\\end{align*} $$<\/p>\n<p>Therefore, the odds against are 8:5 (pronounced as \u20188 to 5\u2019).<\/p>\n<p><em>You should notice that the odds against an event = reciprocal of odds for the same event.<\/em><\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Suppose you toss a fair coin, what are the odds against obtaining a head?<\/p>\n<p>A. 2:2<\/p>\n<p>B. 1:2<\/p>\n<p>C. 1:1<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is C.<\/p>\n<p>The probability of obtaining a head, P(H) = 1\/2<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\begin{align*}<br \/>\\text{Odds against a head} &amp; = \\left\\{ 1 \u2013 P(H) \\right\\} \u00f7 P(H) \\\\<br \/>&amp; = \\left\\{ 1 \u2013 \\cfrac{1}{2} \\right\\} \u00f7 \\cfrac{1}{2} \\\\<br \/>&amp; = \\cfrac{1}{2} * \\cfrac{2}{1} \\\\<br \/>&amp; = \\cfrac{1}{1}<br \/>\\end{align*} $$<\/p>\n<p>Therefore, the odds against a head are 1:1, pronounced as \u20181 to 1.\u2019<\/p>\n<p><em>Note that in this case, the odds for a head are also 1:1.<\/em><\/p>\n<\/blockquote>\n<p><em>Reading 8 LOS 8c: state the probability of an event in terms of odds for and against the event;<\/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>","protected":false},"excerpt":{"rendered":"<p>Odds for and against an event represent a ratio of the desired outcomes versus the field. In other words, the odds for an event refer to the ratio of the number of ways the event can occur to the number&#8230;<\/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-410","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Odds and Probability Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to calculate odds for and against an event, including key probability concepts and real-world applications in finance.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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