{"id":2939,"date":"2019-06-26T05:38:42","date_gmt":"2019-06-26T05:38:42","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=2939"},"modified":"2023-06-13T14:50:19","modified_gmt":"2023-06-13T14:50:19","slug":"calculate-probabilities-of-mutually-exclusive-events","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/general-probability\/calculate-probabilities-of-mutually-exclusive-events\/","title":{"rendered":"Calculate probabilities of mutually exclusive events"},"content":{"rendered":"<h2><strong>Mutually Exclusive Events<\/strong><\/h2>\n<p>\\(A\\) and \\(B\\) are <strong>mutually exclusive<\/strong> events if \\(A\\) and \\(B\\) cannot both occur at the same time. For example, when a coin is flipped, it cannot land on both the head and tail simultaneously. Therefore, we consider the events \u201clanding heads\u201d and \u201clanding tails\u201d as mutually exclusive events.<\/p>\n<p>If events are mutually exclusive, then their intersection \\(\\left( A\\cap B \\right) \\) is equal to \\(\\emptyset\\) (null set).<\/p>\n<p>In the diagram below, \\(A\\) and \\(B\\) are mutually exclusive events.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1590\" height=\"1091\" class=\"alignnone size-full wp-image-32399\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Mutually-Exclusive-Events-2.jpg\" alt=\"Mutually Exclusive Events\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Mutually-Exclusive-Events-2.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Mutually-Exclusive-Events-2-300x206.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Mutually-Exclusive-Events-2-1024x703.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Mutually-Exclusive-Events-2-768x527.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Mutually-Exclusive-Events-2-1536x1054.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Mutually-Exclusive-Events-2-400x274.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>If \\(A\\) and \\(B\\) are mutually exclusive, then the probability of \\(A\\) or \\(B\\) occurring is the sum of the probability of event \\(A\\) occurring and the probability of event \\(B\\) occurring, as shown in the formula below:<\/p>\n<p>$$ P(A \\cup B) = P(A) + P(B) $$<\/p>\n<h3><strong>Example: Mutually Exclusive Events<\/strong><\/h3>\n<p>Consider drawing 2 cards from a standard deck. Let event \\(A\\) be drawing a Jack and event \\(B\\) be drawing a Queen. The probability of event \\(A\\) occurring is \\({4}\/{52}\\) or \\({1}\/{13}\\) and the probability of event \\(B\\) is also \\({4}\/{52}\\) or \\({1}\/{13}\\). Therefore, we can calculate:<\/p>\n<p>$$ P(A\\cup B)=P(A)+P(B)={ 1 }\/{ 13 }+{ 1 }\/{ 13 }=2\/{ 13 } $$<\/p>\n<p>Events \\(A\\) and \\(B\\) are <strong>exhaustive<\/strong> events if \\(A\\cup B\\) is equal to the entire sample space, \\(S\\). It follows then that if \\(A\\) and \\(B\\) are mutually exclusive and exhaustive events then \\(A\\cap B=\\emptyset \\) and \\(A\\cup B=S\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1590\" height=\"1091\" class=\"alignnone size-full wp-image-32400\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Exhaustive-Events.jpg\" alt=\"Exhaustive Events\" style=\"max-width:100%\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Exhaustive-Events.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Exhaustive-Events-300x206.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Exhaustive-Events-1024x703.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Exhaustive-Events-768x527.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Exhaustive-Events-1536x1054.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Exhaustive-Events-400x274.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Consider event \\(A\\) and its\u00a0 complement\u00a0 \\(A^{c}\\) (or \\(A^{\\prime}\\)). These events are mutually exclusive and because they make up the entire sample space, \\(S\\), they are also exhaustive. Therefore,<\/p>\n<p>$$ \\begin{align*} &amp; P\\left( A \\right) +P\\left( { A }^{ \u2019 } \\right) =1,\\quad \\text{and} \\\\ &amp; P\\left( A \\right) =1-P\\left( { A }^{ \u2019 } \\right) \\\\ \\end{align*} $$<\/p>\n<h3 style=\"text-align: left;\"><strong>Example: Mutually Exclusive Events<\/strong><\/h3>\n<p>Consider the experiment rolling a 6-sided die until a 1 is rolled. If we define event \\(A\\) as \u201cNOT rolling a 1 before the 3<sup>rd<\/sup> roll,\u201d then we can define event A\u2019 as \u201crolling a 1 before the 3<sup>rd<\/sup> roll\u201d which is the same as rolling a 1 on either the 1<sup>st<\/sup> or 2<sup>nd<\/sup> roll.<\/p>\n<p>The probability of rolling a 1 on the 1<sup>st<\/sup> roll is \\({1}\/{6}\\).<\/p>\n<p>The probability of rolling a 1 on the 2<sup>nd<\/sup> roll is \\({5}\/{6}\\) (because if we rolled a 1 on the 1<sup>st<\/sup> roll we would have stopped) * \\({1}\/{6}\\) = \\({5}\/{36}\\)<\/p>\n<p>$$ \\begin{align*} &amp; P\\left( { A }^{ \u2019 } \\right) ={ 1 }\/{ 6 }+{ 5 }\/{ 36 }={ 11 }\/{ 36 } \\\\ &amp; P\\left( A \\right) =1-P\\left( { A }^{ \u2019 } \\right) =1-{ 11 }\/{ 36 }={ 25 }\/{ 36 } \\\\ \\end{align*} $$<\/p>\n<p><em><strong>Learning Outcome<\/strong><\/em><\/p>\n<p><em><strong>1.d Topic: General Probability &#8211;\u00a0Calculate probabilities of mutually exclusive events.\u00a0<\/strong><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Mutually Exclusive Events \\(A\\) and \\(B\\) are mutually exclusive events if \\(A\\) and \\(B\\) cannot both occur at the same time. For example, when a coin is flipped, it cannot land on both the head and tail simultaneously. Therefore, we&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[97],"tags":[],"class_list":["post-2939","post","type-post","status-publish","format-standard","hentry","category-general-probability","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>Calculate probabilities of mutually exclusive events - CFA, FRM, and Actuarial Exams Study Notes<\/title>\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\/study-notes\/actuarial-exams\/soa\/p-probability\/general-probability\/calculate-probabilities-of-mutually-exclusive-events\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Calculate probabilities of mutually exclusive events - CFA, FRM, and Actuarial Exams Study Notes\" \/>\n<meta property=\"og:description\" content=\"Mutually Exclusive Events (A) and (B) are mutually exclusive events if (A) and (B) cannot both occur at the same time. 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