{"id":445,"date":"2019-10-10T13:27:00","date_gmt":"2019-10-10T13:27:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=445"},"modified":"2026-09-10T09:40:49","modified_gmt":"2026-09-10T09:40:49","slug":"independent-dependent-events-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/independent-dependent-events-example\/","title":{"rendered":"Independent vs. Dependent Events"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Mike Jamerson tosses a coin twice. What is the probability of getting a head on the first toss and a head on the second toss?\",\n    \"text\": \"Options:\\nA. 1\/2\\nB. 1\/4\\nC. 2\/3\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. The outcomes of the two coin tosses are independent, meaning the result of the first toss does not affect the second toss. The probability of getting a head on each toss is 1\/2. Therefore, the probability of getting heads on both tosses is P(H and H) = P(H) \u00d7 P(H) = 1\/2 \u00d7 1\/2 = 1\/4.\"\n    }\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" \n  width=\"611\" \n  height=\"344\" \n  src=\"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  allowfullscreen>\n<\/iframe><\/p>\n<p>Two or more events are independent if the occurrence of one event has no influence on the occurrence of the other event(s). Let us put this in annotations:<\/p>\n<p><!--more--><\/p>\n<p>$$ \\text P( \\text A | \\text B) = \\text P(\\text A) $$<\/p>\n<p>or<\/p>\n<p>$$ \\text P(\\text B | \\text A) = \\text P(\\text B) $$<\/p>\n<p>If this condition is not satisfied, the events are said to be dependent.<\/p>\n<div style=\"margin:28px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"display:block;width:100%;padding:12px 24px;border-radius:999px;background:#1a73e8;color:#ffffff;font-size:15px;font-weight:500;text-align:center;text-decoration:none;box-sizing:border-box;\">\n    Practice Independent Events Questions with Our Free Trial.\n  <\/a>\n<\/div>\n<h2><strong>Example 1: Independent Events<\/strong><\/h2>\n<p>Suppose you rolled a die and flipped a coin, the probability of getting any number face on the die would not influence the probability of getting a head or a tail on the coin. The two events are independent.<\/p>\n<h2><strong>Example 2: Dependent Events<\/strong><\/h2>\n<p>Suppose you have a bag containing 8 green balls and 5 blue balls. If you draw a blue ball without replacing it, the probability of drawing another blue ball in your second attempt is greatly changed because you drew a blue ball the first time. Such events are said to be dependent.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Mike Jamerson tosses a coin twice. What is the probability of getting a head on the first toss and a head on the second toss?<\/p>\n<p>A. 1\/2<\/p>\n<p>B. 1\/4<\/p>\n<p>C. 2\/3<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>The coin has no memory. Therefore, the first outcome has no influence on the second outcome. Thus:<\/p>\n<p>$$ \\begin{align*}<br \/>\\text P(\\text{head and head}) &amp; = \\text P(\\text H) * \\text P(\\text H) \\\\<br \/>&amp; =\\cfrac {1}{2} * \\cfrac {1}{2} \\\\<br \/>&amp; = \\cfrac {1}{4} \\\\<br \/>\\end{align*} $$<\/p>\n<p><em>Note: If two events are independent:<\/em><\/p>\n<p>$$ \\text P(\\text{AB}) =\\text P( \\text A) * \\text P(\\text B) $$<\/p>\n<\/blockquote>\n<p><em>Reading 8 LOS 8g<\/em><\/p>\n<p><em>Distinguish between dependent and independent events<\/em><\/p>\n<div class=\"notes_inv\">\n<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<div style=\"text-align:center;margin:35px 0;\">\n  <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"display:inline-block;padding:15px 35px;border-radius:999px;background:#1a73e8;color:#ffffff;font-size:17px;font-weight:600;text-decoration:none;\">\n    Start Free Trial \u2192\n  <\/a>\n\n  <p style=\"margin:25px 0 0;font-size:16px;line-height:1.6;text-align:center;\">\n    Master CFA Level I Quantitative Methods concepts, including independent and dependent events, probability rules, conditional probability, and exam-style practice questions with study notes, mock exams, and video lessons.\n  <\/p>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Two or more events are independent if the occurrence of one event has no influence on the occurrence of the other event(s). Let us put this in annotations:<\/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-445","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>Independent VS Dependent Events Example | CFA Level I - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Two or more events are said to be independent if the occurrence of one event has no influence on the occurrence of the other event(s)\" \/>\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\/independent-dependent-events-example\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Independent VS Dependent Events Example | CFA Level I - 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