{"id":27375,"date":"2021-08-24T13:26:00","date_gmt":"2021-08-24T13:26:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27375"},"modified":"2026-09-02T20:28:17","modified_gmt":"2026-09-02T20:28:17","slug":"probability-in-terms-of-odds-for-and-against-the-event","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-in-terms-of-odds-for-and-against-the-event\/","title":{"rendered":"Probability in Terms of 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\": \"Suppose you toss a fair coin. What are the odds against obtaining a head?\",\n    \"text\": \"Options:\\nA. 2:2\\nB. 1:2\\nC. 1:1\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. For a fair coin, the probability of obtaining a head is 1\/2. The odds against obtaining a head are calculated as (1 \u2212 P(H)) \u00f7 P(H) = (1 \u2212 1\/2) \u00f7 (1\/2) = 1. Therefore, the odds against obtaining a head are 1:1. The odds for obtaining a head are also 1:1.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw?si=2kGduS-4qydB7vxm\" 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 class=\"wp-block-paragraph\">Odds for and against an event represent a ratio of the desired outcomes versus the field. In other words, the odds for an event are the ratio of the number of ways the event can occur to the number of ways the event does not occur. Thus:<\/p>\n\n\n\n<!--more-->\n\n\n\n<p class=\"wp-block-paragraph\">Given the probability of an event \u2018E\u2019 i.e. P(E),<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$ \\begin{align*}<br>\\text{Odds for E} &amp; =\\cfrac {P(E)}{ \\left\\{ 1 \u2013 P(E) \\right\\} } \\\\<br>\\text{Odds against E} &amp; = \\cfrac { \\left\\{ 1 \u2013 P(E) \\right\\} }{ P(E) } \\\\<br>\\end{align*} $$<\/p>\n\n\n\n<div style=\"margin: 18px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"display: block; text-align: center; padding: 14px 18px; border: 2px solid #1a73e8; border-radius: 18px; color: #ffffff; font-weight: 600; font-size: 16px; text-decoration: none; background-color: #1a73e8;\">\n    Master probability and odds concepts with our Free Trial.\n  <\/a>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Example: Odds for and against an event<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">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\n\n\n<p class=\"wp-block-paragraph\">First, we have to establish the probability of drawing a blue ball:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let P(B) represent the event that a blue ball is drawn from the box. Thus,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$ P(B) = \\cfrac {5}{13} $$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$ \\begin{align*}<br>\\text{The odds for a blue ball} &amp; =\\cfrac {5}{13} \u00f7 \\left(1 \u2013 \\cfrac {5}{13} \\right)\\\\<br>&amp; = \\cfrac {5}{13} \u00f7 \\left( \\cfrac {8}{13} \\right) \\\\<br>&amp; =\\cfrac {5}{13} * \\cfrac {13}{8} \\\\<br>&amp; =\\cfrac {5}{8} \\\\<br>\\end{align*} $$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the odds for a blue ball are 5:8 (pronounced as \u20185 to 8\u2019).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similarly, we can calculate the odds against drawing a blue ball:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$ \\begin{align*}<br>\\text{Odds against a blue ball} &amp;= \\left\\{1 \u2013 \\cfrac{5}{13} \\right\\} \u00f7 \\cfrac{5}{13} \\\\<br>&amp;= \\cfrac{8}{13} \u00f7 \\cfrac{5}{13} \\\\<br>&amp;= \\cfrac{8}{13} * \\cfrac{13}{5} \\\\<br>&amp; = \\cfrac{8}{5}<br>\\end{align*} $$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the odds against are 8:5 (pronounced as \u20188 to 5\u2019).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>You should notice that the odds against an event = reciprocal of odds for the same event.<\/em><\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<h2 class=\"wp-block-heading\"><strong>Question<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Suppose you toss a fair coin. What are the odds against obtaining a head?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. 2:2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">B. 1:2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C. 1:1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Solution<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The correct answer is C.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The probability of obtaining a head, P(H) = 1\/2<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$ \\begin{align*}<br>\\text{Odds against a head} &amp; = \\left\\{ 1 \u2013 P(H) \\right\\} \u00f7 P(H) \\\\<br>&amp; = \\left\\{ 1 \u2013 \\cfrac{1}{2} \\right\\} \u00f7 \\cfrac{1}{2} \\\\<br>&amp; = \\cfrac{1}{2} * \\cfrac{2}{1} \\\\<br>&amp; = \\cfrac{1}{1}<br>\\end{align*} $$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the odds against a head are 1:1, pronounced as \u20181 to 1.\u2019<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Note that in this case, the odds for a head are also 1:1.<\/em><\/p>\n<\/blockquote>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Reading 8 LOS 3c: describe the probability of an event in terms of odds for and against the event;<\/em><\/p>\n\n\n\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\n\n\n<div style=\"text-align: center; margin: 30px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\"\n     target=\"_blank\"\n     rel=\"noopener noreferrer\"\n     style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">\n    Master probability in terms of odds for and against an event, probability calculations, and quantitative analysis with our CFA Level I study materials and practice tools.\n  <\/p>\n<\/div>\n","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 are the ratio of the number of ways the event can occur to the number of&#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-27375","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 v28.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Odds for and Against an Event | AnalystPrep CFA L1 Notes<\/title>\n<meta name=\"description\" content=\"Learn odds for and against an event, including odds in favor and odds against, with key probability concepts for CFA Level 1.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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