{"id":435,"date":"2019-10-10T13:27:00","date_gmt":"2019-10-10T13:27:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=435"},"modified":"2025-03-16T17:44:04","modified_gmt":"2025-03-16T17:44:04","slug":"probability-rules-examples","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/","title":{"rendered":"Probability Rules"},"content":{"rendered":"<p>[vsw id=&#8221;hu47ZbsskEw&#8221; source=&#8221;youtube&#8221; width=&#8221;611&#8243; height=&#8221;344&#8243; autoplay=&#8221;no&#8221;]<\/p>\n<p>Probability rules are the concepts and facts that must be taken into account while evaluating the probabilities of various events. The CFA curriculum requires candidates to master 3 main rules of probability. These are the multiplication rule, the addition rule, and the law of total probability. We now look at each rule in detail.<\/p>\n<p><!--more--><\/p>\n<h2><strong>Multiplication Rule<\/strong><\/h2>\n<p>We use the multiplication rule to determine the <strong>joint probability<\/strong> of two events, P (AB). A joint probability is the probability of two events happening together. For example, we may be interested in the probability that gas prices and bus fares increase concurrently.<\/p>\n<p><em>Note: If two events are mutually exclusive, then they cannot occur together. Therefore, we say that such events have zero joint probability.<\/em><\/p>\n<p>The multiplication rule states that:<\/p>\n<p>$$ P(AB) = P(A | B)P(B) $$<\/p>\n<h3><strong>Example: Multiplication rule<\/strong><\/h3>\n<p>Let\u2019s assume we have a bag containing 16 blue balls and 14 yellow balls. Suppose we draw two balls at random, one after the other without replacement. Let\u2019s define:<\/p>\n<p>Event B = the first ball is blue<\/p>\n<p>Event A = the second ball is blue<\/p>\n<p>What will be the joint probability of A and B?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>From the wording of the question, we can calculate the conditional probability P(A|B):<\/p>\n<p>The probability that the first ball to be drawn is blue, P(B) = \\(\\cfrac {16}{30}\\)<\/p>\n<p>The probability that the second ball to be drawn is blue given the first one is also blue, P(A | B) = \\(\\cfrac {15}{29}\\)<\/p>\n<p>Thus,<\/p>\n<p>$$ \\begin{align*} P(AB) &amp; = P(A | B)P(B) \\\\ &amp; = \\cfrac {15}{29} * \\cfrac {16}{30} \\\\ &amp; =\\cfrac {240}{870} \\\\ &amp; =\\cfrac {8}{29} \\\\ \\end{align*} $$<\/p>\n<h2><strong>Addition Rule<\/strong><\/h2>\n<p>We use the addition rule to assess the probability that events A or B occur:<\/p>\n<p>$$ \\text P(\\text{A or B}) = \\text P(\\text A) + \\text P(\\text B) \u2013 \\text P(\\text {AB}) $$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1632\" height=\"972\" class=\"aligncenter size-full wp-image-16956\" style=\"max-width: 100%;\" src=\"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/05090015\/page-76.jpg\" alt=\"addition_rule\" \/><\/p>\n<h3><strong>Example: Addition rule<\/strong><\/h3>\n<p>Suppose the probability of relaxed import restrictions is 0.5 and the probability of a price war is 0.2. If the joint probability of relaxed import restrictions and a price war is 0.4, what is the probability of relaxed trade restrictions <strong>or <\/strong>a price war?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\text P(\\text{A or B}) = 0.5 + 0.2 \u2013 0.4 = 0.3 $$<\/p>\n<p><em>Note: If two events are mutually exclusive, then:<\/em><\/p>\n<p>\\( \\text P(\\text{A or B}) = \\text P(\\text A) + \\text P(\\text B)\\) since the joint probability would be (and is always) zero.<\/p>\n<h2><strong>Total Probability Rule<\/strong><\/h2>\n<p>Suppose we have a set of mutually exclusive and exhaustive events B<sub>1<\/sub>, B<sub>2<\/sub>, B<sub>3<\/sub>\u2026B<sub>n<\/sub>.<\/p>\n<p>We can determine the unconditional probability of an event, given the conditional probabilities:<\/p>\n<p>$$ \\text P(\\text A) = \\text P(\\text A | \\text B_1)\\text P(\\text B_1) + \\text P(\\text A | \\text B_3)\\text P(\\text B_3) + \u2026 + \\text P(\\text A | \\text B_n)\\text P(\\text B_n) $$<\/p>\n<p>Perhaps an example will help you understand the concept:<\/p>\n<h3><strong>Example:\u00a0<\/strong><strong>Total Probability Rule<\/strong><\/h3>\n<p>Suppose a local authority subdivides a forest into three regions, B<sub>1<\/sub>, B<sub>2<\/sub>, and B<sub>3<\/sub>. Assume the area covered by each region is 50km<sup>2<\/sup>, 65km<sup>2<\/sup>, and 74km<sup>2, <\/sup>respectively. What is the total forest area, A?<\/p>\n<p>The solution is simple: we just add the respective areas:<\/p>\n<p>$$ 50 + 65 + 74 = 189 \\text{ km}^2 $$<\/p>\n<p>This is the idea behind the law of total probability. First, the total forest area is replaced by P(A). Then, we add the probability of A that falls in each partitions B<sub>1<\/sub>, B<sub>2<\/sub>, and B<sub>3<\/sub>.<\/p>\n<p>\u00a0<\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>[vsw id=&#8221;hu47ZbsskEw&#8221; source=&#8221;youtube&#8221; width=&#8221;611&#8243; height=&#8221;344&#8243; autoplay=&#8221;no&#8221;] Probability rules are the concepts and facts that must be taken into account while evaluating the probabilities of various events. The CFA curriculum requires candidates to master 3 main rules of probability. These are&#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-435","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>Probability Rules with Examples | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn key probability rules, including mutual exclusivity and independence, with practical examples for evaluating event probabilities.\" \/>\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\/probability-rules-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Probability Rules with Examples | CFA Level 1\" \/>\n<meta property=\"og:description\" content=\"Learn key probability rules, including mutual exclusivity and independence, with practical examples for evaluating event probabilities.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2019-10-10T13:27:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-03-16T17:44:04+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-76.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1632\" \/>\n\t<meta property=\"og:image:height\" content=\"972\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Ngari Joseph\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Ngari Joseph\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\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\/probability-rules-examples\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/\"},\"author\":{\"name\":\"Ngari Joseph\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914\"},\"headline\":\"Probability Rules\",\"datePublished\":\"2019-10-10T13:27:00+00:00\",\"dateModified\":\"2025-03-16T17:44:04+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/\"},\"wordCount\":552,\"image\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/05090015\/page-76.jpg\",\"articleSection\":[\"Quantitative Methods\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/\",\"name\":\"Probability Rules with Examples | CFA Level 1\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/05090015\/page-76.jpg\",\"datePublished\":\"2019-10-10T13:27:00+00:00\",\"dateModified\":\"2025-03-16T17:44:04+00:00\",\"author\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914\"},\"description\":\"Learn key probability rules, including mutual exclusivity and independence, with practical examples for evaluating event probabilities.\",\"breadcrumb\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#primaryimage\",\"url\":\"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/05090015\/page-76.jpg\",\"contentUrl\":\"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/05090015\/page-76.jpg\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Probability Rules\"}]},{\"@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\/ddbf15286b86900dfa9f8ac2603c3914\",\"name\":\"Ngari Joseph\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png\",\"contentUrl\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png\",\"caption\":\"Ngari Joseph\"},\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/ngari-joseph\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Probability Rules with Examples | CFA Level 1","description":"Learn key probability rules, including mutual exclusivity and independence, with practical examples for evaluating event probabilities.","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\/probability-rules-examples\/","og_locale":"en_US","og_type":"article","og_title":"Probability Rules with Examples | CFA Level 1","og_description":"Learn key probability rules, including mutual exclusivity and independence, with practical examples for evaluating event probabilities.","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2019-10-10T13:27:00+00:00","article_modified_time":"2025-03-16T17:44:04+00:00","og_image":[{"width":1632,"height":972,"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-76.jpg","type":"image\/jpeg"}],"author":"Ngari Joseph","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Ngari Joseph","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/"},"author":{"name":"Ngari Joseph","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914"},"headline":"Probability Rules","datePublished":"2019-10-10T13:27:00+00:00","dateModified":"2025-03-16T17:44:04+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/"},"wordCount":552,"image":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#primaryimage"},"thumbnailUrl":"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/05090015\/page-76.jpg","articleSection":["Quantitative Methods"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/","name":"Probability Rules with Examples | CFA Level 1","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"primaryImageOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#primaryimage"},"image":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#primaryimage"},"thumbnailUrl":"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/05090015\/page-76.jpg","datePublished":"2019-10-10T13:27:00+00:00","dateModified":"2025-03-16T17:44:04+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914"},"description":"Learn key probability rules, including mutual exclusivity and independence, with practical examples for evaluating event probabilities.","breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#primaryimage","url":"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/05090015\/page-76.jpg","contentUrl":"https:\/\/cdn.analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/05090015\/page-76.jpg"},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-examples\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"Probability Rules"}]},{"@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\/ddbf15286b86900dfa9f8ac2603c3914","name":"Ngari Joseph","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png","contentUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png","caption":"Ngari Joseph"},"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/ngari-joseph\/"}]}},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/435","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/comments?post=435"}],"version-history":[{"count":67,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/435\/revisions"}],"predecessor-version":[{"id":36640,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/435\/revisions\/36640"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=435"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=435"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=435"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}