{"id":2978,"date":"2019-06-28T13:30:34","date_gmt":"2019-06-28T13:30:34","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=2978"},"modified":"2026-01-16T08:45:05","modified_gmt":"2026-01-16T08:45:05","slug":"state-bayes-theorem-and-use-it-to-calculate-conditional-probabilities","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/general-probability\/state-bayes-theorem-and-use-it-to-calculate-conditional-probabilities\/","title":{"rendered":"State Bayes Theorem and use it to calculate conditional probabilities"},"content":{"rendered":"<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/study-notes\/images\/law-of-total-probability-example\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Img_13.jpg\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Img_13.jpg\",\n  \"caption\": \"Example: law of total probability\",\n  \"width\": 1590,\n  \"height\": 1091,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\",\n    \"url\": \"https:\/\/analystprep.com\/\"\n  }\n}\n<\/script><\/p>\n<h2><strong>Bayes Theorem<\/strong><\/h2>\n<p>Before we move on to Bayes Theorem, we need to learn about the\u00a0<strong>law of total probability.<\/strong><\/p>\n<h3><strong>The Law of Total Probability<\/strong><\/h3>\n<p>The law of total probability states that if E is an event, and \\(A_1, A_2, \\cdots A_n\\) are the partition of the sample space, then<\/p>\n<p>$$P(E)=P(A_1 \\cap E)+P(A_2 \\cap E)+\\cdots P(A_n \\cap E)$$<\/p>\n<p>We can use the law intuitively by recalling that:<\/p>\n<p>$$A=(A\\cap B)\\cup (A\\cap B^c)$$<\/p>\n<p>Consider the following Venn Diagram:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-14804\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Img_13.jpg\" alt=\"Example: Law of Total Probability\" width=\"1590\" height=\"1091\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Img_13.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Img_13-300x206.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Img_13-1024x703.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Img_13-768x527.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Img_13-1536x1054.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Img_13-400x274.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Note that\u00a0 \\(B^c\\) (also be written as \\(B&#8217;\\)) is the complement of an event B.<\/p>\n<p>Therefore,<\/p>\n<p>$$P(A)=P(A\\cap B)+P(A\\cap B^c)$$<\/p>\n<p>Recall the conditional probability that:<\/p>\n<p>$$P(A|B)=\\frac{P(A\\cap B)}{P(B)}$$<\/p>\n<p>So that:<\/p>\n<p>$$P(A\\cap B)=P(B)\\bullet\u00a0P(A|B)$$<\/p>\n<p>\u00a0Similarly,<\/p>\n<p>$$P(A\\cap B^c)=P(B^c)\\bullet\u00a0P(A|B^c)$$<\/p>\n<p>Thus,<\/p>\n<p>$$ P\\left( A \\right) =P\\left( A|B \\right) \\bullet\u00a0 P\\left( B \\right) +P\\left(A|B^c \\right)\\bullet \\left( P\\left( B^c \\right) \\right) $$<\/p>\n<p>At this point, we can write the expression for \\(P(B|A)\\) as:<\/p>\n<p>$$P(B|A)=\\frac{P(A\\cap B)}{P(A)}=\\frac{P(A)\\bullet P(A|B)}{P\\left( A|B \\right) \\bullet P\\left( B \\right) +P\\left(A|B^c \\right)\\bullet \\left( P\\left( B^c \\right) \\right)}$$<\/p>\n<p>The last expression is referred to as\u00a0<strong>Bayes&#8217; Theorem.\u00a0<\/strong><\/p>\n<p>Similarly,<\/p>\n<p>$$P(B^c |A)=\\frac{P(A\\cap B^c)}{P(A)}=\\frac{P(A)\\bullet P(A|B^c)}{P\\left( A|B \\right) \\bullet\u00a0 P\\left( B \\right) +P\\left(A|B^c \\right)\\bullet \\left( 1-P\\left( B \\right) \\right)}$$<\/p>\n<h3><strong>Definition: Bayes&#8217; Theorem<\/strong><\/h3>\n<p>If \\(E\\) is an event and \\(A_1 , A_2 ,&#8230;, A_3\\) are the partition of a sample space, then:<\/p>\n<p>$$\\begin{align} P(A_i |E)&amp;=\\frac{A_i \\cap E}{P(E)}\\\\ &amp;=\\frac{P(A_i)\\bullet P(E|A_i)}{P(A_1)\\bullet P(E|A_1)+P(A_2)\\bullet P(E|A_2)+\\cdots +P(A_n)\\bullet P(E|A_n)} \\end{align}$$<\/p>\n<p>Consider the following examples:<\/p>\n<h4><strong>Example: Law of Total Probability<\/strong><\/h4>\n<p>In a given population of students, 25% play baseball, and 30% play basketball. Also, the probability of a student playing baseball given that they also play basketball is 10%.<\/p>\n<p>Calculate the probability that a student plays baseball given that they do not play basketball.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Let event \\(A\\) be \u201cplays baseball\u201d and event \\(B\\) be \u201cplays basketball.\u201d Using the law of total probability, we have:<\/p>\n<p>$$ \\begin{align} P\\left( A \\right) &amp; =P\\left( A|B \\right) \\bullet P\\left( B \\right) +P\\left(A|B^c\\right)\\bullet \\left( 1-P\\left( B \\right) \\right) \\\\ 0.25 &amp;= 0.10 \\bullet 0.30 + x \\left(1-0.30 \\right) \\\\ 0.22 &amp;= x\u00a0 \\left(0.70 \\right) \\\\ x &amp;= \\frac{0.22}{0.70} = 0.314 \\ or \\ 31.4\\%\u00a0 \\end{align} $$<\/p>\n<h4><strong>Example: Bayes&#8217; Theorem<\/strong><\/h4>\n<p>An insurance company deals with three insurance policies: 40% of life insurance, 25% of car insurance, and 35% of health insurance. The probability that a life insurance policyholder will file a claim in a given year is 0.50. The probability that a car insurance policyholder will file a claim in a given year is 0.20. Lastly, the probability that a health insurance policyholder will file a claim in a given year is 0.10.<\/p>\n<p>At the course of this year, a policyholder files a claim.<\/p>\n<p>Calculate the probability that the claim comes from the car insurance policyholder.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Let:<\/p>\n<p>\\(L\\) = proportion of life insurance policyholders.<\/p>\n<p>\\(R\\) = proportion of car insurance policyholders.<\/p>\n<p>\\(H\\) = proportion of health insurance policyholders.<\/p>\n<p>\\(C\\) = event that a claim is made.<\/p>\n<p>From the information given in the question, we have:<\/p>\n<p>$$ P(L) = 0.40,\\\u00a0 P(R) = 0.25,\\ P(H) = 0.35 $$<\/p>\n<p>Also, we have:<\/p>\n<p>$$ P(C|L) = 0.50,\\\u00a0 P(C|R) = 0.20,\\ P(C|H) = 0.10 $$<\/p>\n<p>We need, \\(P(R|C)\\). Using the Bayes&#8217; theorem, we know that:<\/p>\n<p>$$\\begin{align}P(R|C)&amp;=\\frac{P(R)\\bullet\u00a0P(C|R)} {P(R)\\bullet\u00a0P(C|R)+P(L)\\bullet\u00a0P(C|L)+P(H)\\bullet\u00a0P(C|H)}\\\\ &amp;=\\frac{0.25\\times 0.20}{0.25\\times 0.20+0.40\\times 0.50 +0.35\\times 0.10}\\\\ &amp;=0.17543 \\approx 17.54\\%\\end{align}$$<\/p>\n<p><em><strong>Learning Outcome<\/strong><\/em><\/p>\n<p><em><strong>Topic 1.g: General Probability &#8211; State Bayes Theorem and use it to calculate conditional probabilities.<\/strong><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bayes Theorem Before we move on to Bayes Theorem, we need to learn about the\u00a0law of total probability. The Law of Total Probability The law of total probability states that if E is an event, and \\(A_1, A_2, \\cdots A_n\\)&#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-2978","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 v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Bayes\u2019 Theorem and Conditional Probability | Study Notes<\/title>\n<meta name=\"description\" content=\"Learn how Bayes\u2019 theorem and the law of total probability are used to calculate conditional probabilities, with clear formulas and step-by-step examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, 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