{"id":2971,"date":"2019-06-28T13:22:35","date_gmt":"2019-06-28T13:22:35","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=2971"},"modified":"2026-01-02T13:26:43","modified_gmt":"2026-01-02T13:26:43","slug":"define-and-calculate-conditional-probabilities","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/general-probability\/define-and-calculate-conditional-probabilities\/","title":{"rendered":"Define and calculate conditional probabilities"},"content":{"rendered":"<h2><strong>Conditional Probability<\/strong><\/h2>\n<p><script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Conditional-Probability.jpg\",\n  \"caption\": \"Conditional 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  }\n}\n<\/script><\/p>\n<p>Given two events \\(A\\) and \\(B\\), the<strong> conditional probability<\/strong> of event \\(A\\) occurring, given that event \\(B\\) has occurred, is the probability of event \\(A\\) and \\(B\\) occurring over the probability of event \\(B\\) occurring as shown by the formula below:<\/p>\n<p>$$ P\\left( A|B \\right) =\\frac { P\\left( A\\cap B \\right) }{ P\\left( B \\right) } $$<br \/>\nas long as \\(P (B) &gt; 0\\).<\/p>\n<p>As shown in the diagram below, the probability of \\(A\\) given \\(B\\) occurred is the shaded region \\(A \\cap B\\).<\/p>\n<h4><strong><img loading=\"lazy\" decoding=\"async\" width=\"1590\" height=\"1091\" class=\"alignnone size-full wp-image-32396\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Conditional-Probability.jpg\" alt=\"Conditional Probability\" srcset=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Conditional-Probability.jpg 1590w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Conditional-Probability-300x206.jpg 300w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Conditional-Probability-1024x703.jpg 1024w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Conditional-Probability-768x527.jpg 768w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Conditional-Probability-1536x1054.jpg 1536w, https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2019\/06\/Conditional-Probability-400x274.jpg 400w\" sizes=\"auto, (max-width: 1590px) 100vw, 1590px\" \/>Example: Conditional Probability<\/strong><\/h4>\n<p>Suppose 50% of the students in a school take geometry, 30% take history, and 15% take both geometry and history. Given that a student selected at random takes geometry, what is the probability he or she also takes history?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Let event \\(A\\) be \u201ctakes geometry\u201d and event \\(B\\) be \u201ctakes history.\u201d<\/p>\n<p>\\(P \\left(B \\right) = 0.30\\)<\/p>\n<p>\\(P \\left(A \\cap B \\right) = 0.15\\)<\/p>\n<p>$$ P\\left( B|A \\right) =\\frac { P\\left( A\\cap B \\right) }{ P\\left( A \\right) } ={ 0.15 }\/{ 0.50 }=0.30\\quad or\\quad 30\\% $$<\/p>\n<p>This concept can be extended to more than 2 events as shown in the formula for 3 events below:<\/p>\n<p>$$ P\\left( C|A\\cap B \\right) =\\frac { P\\left( A\\cap B\\cap C \\right) }{ P\\left( A \\right) \\times P\\left( B|A \\right) } $$<\/p>\n<h4><strong>Example: Conditional Probability<\/strong><\/h4>\n<p>Suppose that the students at a university are grouped such that:<\/p>\n<p>\u200345% take chemistry<\/p>\n<p>\u200335% take biology<\/p>\n<p>\u200325% take math<\/p>\n<p>\u200310% take chemistry and math<\/p>\n<p>\u20035% take chemistry and biology<\/p>\n<p>\u20033% take biology and math<\/p>\n<p>\u20031% take all three subjects<\/p>\n<p>Calculate the probability that a student takes math, given that they take chemistry and biology.<\/p>\n<p>$$P\\left( \\text{Math|Chemistry and Biology} \\right) =\\frac{{ P\\left( \\text{Math and Chemistry and Biology} \\right) }}{{ P\\left( \\text{Chemistry} \\right) \\times P\\left( \\text{Biology|Chemistry} \\right) }}$$<\/p>\n<p>$$\\begin{align*} P\\left( \\text{Biology|Chemistry} \\right) &amp; =\\frac{{ P\\left( \\text{Biology and Chemistry} \\right) }}{{ P\\left( \\text{Chemistry} \\right) }} \\\\ &amp; ={ .05 }\/{ .45 }={ 1 }\/{ 9 } \\\\ \\end{align*} $$<\/p>\n<p>$$P(\\text{Math|Chemistry and Biology})=\\frac{{ .01 }}{{ \\left( { .45\\times .05 }\/{ .45 } \\right) }}={ .01 }\/{ .05 }={ 1 }\/{ 5 }=.20\\quad or\\quad 20\\%$$<\/p>\n<p>The definition of conditional probability can be rewritten to come up with multiplication rule for probabilities:<\/p>\n<p>$$P(A\\cap B)=P(A|B)\\bullet P(B)$$<\/p>\n<p>Also, if the events A and B are independent, then:<\/p>\n<p>$$P(A|B)=P(A)$$<\/p>\n<p><em><strong>Learning Outcome<\/strong><\/em><\/p>\n<p><em><strong>Topic 1.e: General Probability &#8211; Define and calculate conditional probabilities.<\/strong><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Conditional Probability Given two events \\(A\\) and \\(B\\), the conditional probability of event \\(A\\) occurring, given that event \\(B\\) has occurred, is the probability of event \\(A\\) and \\(B\\) occurring over the probability of event \\(B\\) occurring as shown by&#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-2971","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 - 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