{"id":27456,"date":"2021-08-25T04:23:58","date_gmt":"2021-08-25T04:23:58","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27456"},"modified":"2026-01-15T15:21:26","modified_gmt":"2026-01-15T15:21:26","slug":"dependent-and-independent-events","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/","title":{"rendered":"Dependent and Independent 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\": \"Probability of Two Heads in Two Coin Tosses\",\n    \"text\": \"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\\nA. 1\/2\\nB. 1\/4\\nC. 2\/3\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"1\/4\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"1\/2\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"2\/3\"\n      }\n    ]\n  }\n}\n<\/script>\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/PsZrsg3ZUDE\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Two or more events are independent if the occurrence of one event does not influence the occurrence of the other event(s). Let us put this in annotations:<\/p>\n<p><!--more--><\/p>\n<p>$$ \\text P ( \\text A\u00a0 |\u00a0 \\text B)\u00a0 =\u00a0 \\text P (\\text A) $$<\/p>\n<p>or<\/p>\n<p>$$ \\text P (\\text B\u00a0 |\u00a0 \\text A)\u00a0 =\u00a0 \\text P (\\text B) $$<\/p>\n<p>The events are said to be dependent if this condition is not satisfied.<\/p>\n<h4><strong>Example 1: Independent Events<\/strong><\/h4>\n<p>Suppose you rolled a die and flipped a coin. The probability of getting any number face on the die does not influence the probability of getting a head or a tail on the coin. The two events are independent.<\/p>\n<h4><strong>Example 2: Dependent Events<\/strong><\/h4>\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 on your second attempt is greatly changed because you drew a blue ball the first time. Such events are said to be dependent.<\/p>\n<p>If two events are independent:<\/p>\n<p>$$ \\text P (\\text {AB})\u00a0 = \\text P (\\text A)\u00a0 \u00d7 \\text P (\\text B) $$<\/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. \\(\\frac{1}{2}\\).<\/p>\n<p>B. \\(\\frac{1}{4}\\).<\/p>\n<p>C. \\(\\frac{2}{3}\\).<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>The coin has no memory. As such, the first outcome has no influence on the second outcome. Therefore:<\/p>\n<p>$$ \\begin {align*}<br \/>\n\\text P (\\text {Head and Head})\u00a0 &amp;\u00a0 =\u00a0 \\text P (\\text H) \u00d7 \\text P (\\text H) \\\\<br \/>\n&amp;\u00a0 = \\cfrac\u00a0 {1} {2} \u00d7 \\cfrac\u00a0 {1} {2} \\\\<br \/>\n&amp;\u00a0 =\u00a0 \\cfrac\u00a0 {1} {4}\u00a0 \\\\<br \/>\n\\end {align*} $$<\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Two or more events are independent if the occurrence of one event does not influence the occurrence of the other event(s). Let us put this in annotations:<\/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-27456","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>Dependent and Independent Events | CFA Level I<\/title>\n<meta name=\"description\" content=\"Explains dependent and independent events in probability, including definitions, examples, and how event relationships affect probability calculations.\" \/>\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\/dependent-and-independent-events\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dependent and Independent Events | CFA Level I\" \/>\n<meta property=\"og:description\" content=\"Explains dependent and independent events in probability, including definitions, examples, and how event relationships affect probability calculations.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-25T04:23:58+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-01-15T15:21:26+00:00\" \/>\n<meta name=\"author\" content=\"Kosikos Tuitoek\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Kosikos Tuitoek\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/\"},\"author\":{\"name\":\"Kosikos Tuitoek\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\"},\"headline\":\"Dependent and Independent Events\",\"datePublished\":\"2021-08-25T04:23:58+00:00\",\"dateModified\":\"2026-01-15T15:21:26+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/\"},\"wordCount\":258,\"articleSection\":[\"Quantitative Methods\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/\",\"name\":\"Dependent and Independent Events | CFA Level I\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\"},\"datePublished\":\"2021-08-25T04:23:58+00:00\",\"dateModified\":\"2026-01-15T15:21:26+00:00\",\"author\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\"},\"description\":\"Explains dependent and independent events in probability, including definitions, examples, and how event relationships affect probability calculations.\",\"breadcrumb\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Dependent and Independent Events\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\",\"name\":\"AnalystPrep | CFA\u00ae Exam Study Notes\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20\",\"name\":\"Kosikos Tuitoek\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g\",\"caption\":\"Kosikos Tuitoek\"},\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/kosikos-tuitoek-enockanalystprep-com\/\"}]}<\/script>\n<meta property=\"og:video\" content=\"https:\/\/www.youtube.com\/embed\/PsZrsg3ZUDE\" \/>\n<meta property=\"og:video:type\" content=\"text\/html\" \/>\n<meta property=\"og:video:duration\" content=\"3674\" \/>\n<meta property=\"og:video:width\" content=\"480\" \/>\n<meta property=\"og:video:height\" content=\"270\" \/>\n<meta property=\"ya:ovs:adult\" content=\"false\" \/>\n<meta property=\"ya:ovs:upload_date\" content=\"2021-08-25T04:23:58+00:00\" \/>\n<meta property=\"ya:ovs:allow_embed\" content=\"true\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Dependent and Independent Events | CFA Level I","description":"Explains dependent and independent events in probability, including definitions, examples, and how event relationships affect probability calculations.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/","og_locale":"en_US","og_type":"article","og_title":"Dependent and Independent Events | CFA Level I","og_description":"Explains dependent and independent events in probability, including definitions, examples, and how event relationships affect probability calculations.","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2021-08-25T04:23:58+00:00","article_modified_time":"2026-01-15T15:21:26+00:00","author":"Kosikos Tuitoek","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Kosikos Tuitoek","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/"},"author":{"name":"Kosikos Tuitoek","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"headline":"Dependent and Independent Events","datePublished":"2021-08-25T04:23:58+00:00","dateModified":"2026-01-15T15:21:26+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/"},"wordCount":258,"articleSection":["Quantitative Methods"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/","name":"Dependent and Independent Events | CFA Level I","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"datePublished":"2021-08-25T04:23:58+00:00","dateModified":"2026-01-15T15:21:26+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20"},"description":"Explains dependent and independent events in probability, including definitions, examples, and how event relationships affect probability calculations.","breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/dependent-and-independent-events\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"Dependent and Independent Events"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/","name":"AnalystPrep | CFA\u00ae Exam Study Notes","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/73df713e3b6e82ee139e1eff20cebe20","name":"Kosikos Tuitoek","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8260edaa3f7ba04cf6b536b3f7fd769007ecb789b3289ac0cc4c3ab8b3f7f061?s=96&d=mm&r=g","caption":"Kosikos Tuitoek"},"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/kosikos-tuitoek-enockanalystprep-com\/"}]},"og_video":"https:\/\/www.youtube.com\/embed\/PsZrsg3ZUDE","og_video_type":"text\/html","og_video_duration":"3674","og_video_width":"480","og_video_height":"270","ya_ovs_adult":"false","ya_ovs_upload_date":"2021-08-25T04:23:58+00:00","ya_ovs_allow_embed":"true"},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/27456","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/users\/15"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/comments?post=27456"}],"version-history":[{"count":29,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/27456\/revisions"}],"predecessor-version":[{"id":58096,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/27456\/revisions\/58096"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=27456"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=27456"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=27456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}