{"id":938,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=938"},"modified":"2026-02-26T09:35:44","modified_gmt":"2026-02-26T09:35:44","slug":"calculating-probabilities-given-discrete-uniform-binomial-distribution-functions","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-probabilities-given-discrete-uniform-binomial-distribution-functions\/","title":{"rendered":"Calculating Probabilities Given the Discrete Uniform and the Binomial Distribution Functions"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"A bowl contains blue and orange balls. The probability of drawing a blue ball in any attempt is 0.5. If 5 balls are drawn, what is the probability that exactly 3 of the 5 balls drawn are blue?\",\n    \"text\": \"A bowl contains blue and orange balls. The probability of drawing a blue ball in any attempt is 0.5. Suppose you draw 5 balls from the bowl. What is the probability that 3 of the 5 balls drawn are blue?\\n\\nA. 0.3125\\n\\nB. 0.03125\\n\\nC. 0.25\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A.\\n\\nThis is a binomial probability problem. The probability of exactly 3 blue balls out of 5 draws is:\\n\\nP(X = 3) = 5C3 \u00d7 (0.5)^3 \u00d7 (0.5)^2\\n\\n= 10 \u00d7 0.03125\\n\\n= 0.3125.\"\n    }\n  }\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2021 Level I CFA\u00ae Exam \u2013 Reading 9)\",\n  \"description\": \"This video lesson covers common probability distributions, including discrete and continuous variables, Bernoulli and binomial distributions, and the normal distribution. It also explains confidence intervals, the safety-first criterion, lognormal distributions, Monte Carlo simulations, and how financial analysts apply these concepts in risk management and investment decisions.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/KbEfz3KiJDo\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=KbEfz3KiJDo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo\",\n  \"duration\": \"PT45M31S\"\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo?si=C0M22_3nOVCAf6SP\" 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>We can calculate and interpret probabilities of random variables that assume either the uniform distribution or the binomial distribution.<\/p>\n\n\n\n<!--more-->\n\n\n\n<h2 class=\"wp-block-heading\"><strong>The Discrete Uniform Distribution<\/strong><\/h2>\n\n\n\n<p>As you will recall<strong>, <\/strong>under the uniform distribution, all possible outcomes have equal probabilities. A good example might be the throw of a die, in which case each of the outcomes, 1, 2, 3, 4, 5, and 6, has a 1\/6 probability of occurrence.<\/p>\n\n\n\n<p>Define the random variable X such that X<sub>i<\/sub> = i,&nbsp;&nbsp;&nbsp;&nbsp; where i = 1, 2, 3, \u2026, k)<\/p>\n\n\n\n<p>$$ P(X = x) = \\cfrac {1}{k} $$<\/p>\n\n\n\n<p>$$ \\text{Mean} =\\cfrac {(k + 1)}{2} $$<\/p>\n\n\n\n<p>$$ \\text{Variance} = \\cfrac {(k^2 \u2013 1)}{12} $$<\/p>\n\n\n\n<div style=\"text-align:center; margin:20px 0 24px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-flex; align-items:center; justify-content:center; padding:10px 18px; border:2px solid #1a73e8; border-radius:999px; background:#f5f9ff; color:#1a73e8; text-decoration:none; font-weight:600; font-size:14px;\">\n    Apply probability distribution concepts on CFA-style practice\n  <\/a>\n<\/div>\n\n\n<h3><strong>Example: Uniform Distribution Function<\/strong><\/h3>\n<p>A uniform distribution function has been defined as:<\/p>\n<p>X = {1, 3, 5, 7, 9}<\/p>\n<p>Calculate P(X = 5) and F(5).<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\begin{align*} P(X = x) &amp; = \\cfrac {1}{k} \\\\ K &amp; = 5\\\\ \\end{align*} $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ P(X = 5) =\\cfrac {1}{5} = 0.2 $$<\/p>\n<p>F(5) is the cumulative probability of an outcome less than or equal to 5. Thus,<\/p>\n<p>$$ \\begin{align*} F(5) &amp; = P(X = 1) + P(X = 3) + P(X = 5) \\\\ &amp; = 0.2 + 0.2 + 0.2 \\\\ &amp; = 0.6 \\\\ \\end{align*} $$<\/p>\n<h2><strong>The Binomial Distribution <\/strong><\/h2>\n<p>The binomial distribution is a sequence of n Bernoulli trials where the outcome for every trial can be a success or a failure.<\/p>\n<p>Suppose the probability of a success is \u03b8:<\/p>\n<p>$$ P\\left( X=x \\right) =\\left( \\begin{matrix} n \\\\ x \\end{matrix} \\right) { \\theta }^{ x }\\left( 1-\\theta \\right) ^{ n-x },x=0,1,2,&#8230;,n;0&lt;\\theta &lt;1 $$<\/p>\n<p>Where<\/p>\n<p>$$ \\left( \\begin{matrix} n \\\\ x \\end{matrix} \\right) ={ \\quad }^{ n }{ C }_{ x }=\\frac { n! }{ \\left( n-x \\right) !x! } $$<\/p>\n<p>$$ \\text{Mean of X} = n \\theta $$<\/p>\n<p>$$ \\text{Variance of X} = n \\theta (1 &#8211; \\theta )$$<\/p>\n<p>Note: sometimes the probability of success can be denoted by \u201cp\u201d and (1-p) to denote the probability of failure.<\/p>\n<h3><strong>Example: Binomial Distribution<\/strong><\/h3>\n<p>The probability of surviving an attack by a certain disease is 60%. What is the probability that at least 11 out of a group of 12 people who catch the disease will survive?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The number of survivors is distributed binomially with parameters n = 12 and \u03b8 = 0.6:<\/p>\n<p>$$ \\begin{align*} P(X \\ge 11) &amp; = P(X = 11 \\text{ or } 12) \\\\ &amp;={ _{ 12 }{ C }_{ 11 }{ \\ast 0.6 }^{ 11 }*{ 0.4 }^{ 1 }+_{ 12 }{ C }_{ 12 }{ \\ast 0.6 }^{ 12 } } \\\\ &amp;= \\cfrac {12!}{(1!11!)} * 0.6^{11} * 0.4^1 + \\cfrac {12!}{(0!12!)} *0.6^{12} \\\\ &amp;= 12 * 0.003628 * 0.4 + 0.6^{12} \\\\ &amp;= 0.01959 \\end{align*} $$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>A bowl contains blue and orange balls. The probability of drawing a blue ball in any attempt is 0.5. Suppose you draw 5 balls from the bag, what is the probability that 3 of the 5 balls drawn are blue?<\/p>\n<p>A. 0.3125<\/p>\n<p>B. 0.03125<\/p>\n<p>C. 0.25<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is A.<\/p>\n<p>P(X = 3) = <sub>5<\/sub>C<sub>3 <\/sub>* 0.5<sup>3<\/sup> * 0.5<sup>2<\/sup><\/p>\n<p>= 10 * 0.03125<\/p>\n<p>= 0.3125<\/p>\n<\/blockquote>\n<div class=\"notes_inv\">\u00a0<\/div>\n\n\n<div style=\"text-align:center; margin:40px 0 20px;\">\n\n <div style=\"text-align:center; margin:28px 0 12px;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n     style=\"display:inline-flex; align-items:center; justify-content:center; padding:14px 26px; border-radius:12px; background:#1a73e8; color:#ffffff; text-decoration:none; font-weight:700; font-size:16px;\">\n    Start Free Trial \u2192\n  <\/a>\n<\/div>\n\n  <p style=\"\n        margin-top:14px;\n        font-size:15px;\n        line-height:1.6;\n        max-width:620px;\n        margin-left:auto;\n        margin-right:auto;\n        color:#333333;\n     \">\n     Strengthen your CFA Level I Quantitative Methods preparation with realistic exam-style practice. Build confidence solving binomial and uniform distribution problems before test day.\n  <\/p>\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>We can calculate and interpret probabilities of random variables that assume either the uniform distribution or the binomial distribution.<\/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-938","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>Binomial Distribution Probability | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to calculate probabilities using binomial and uniform 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