{"id":27466,"date":"2021-08-25T04:39:21","date_gmt":"2021-08-25T04:39:21","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27466"},"modified":"2026-03-10T19:09:13","modified_gmt":"2026-03-10T19:09:13","slug":"calculating-unconditional-probability-using-total-probability","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-unconditional-probability-using-total-probability\/","title":{"rendered":"Calculating Unconditional Probability Using Total Probability"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"What is the total probability that the stock price will rise given the probabilities of economic states?\",\n    \"text\": \"Suppose you assess the performance of a stock under different economic circumstances with the following probabilities:\\n\\nNo recession P(RC) = 0.7\\nP(Stock Rise | RC) = 0.8\\nP(Stock Fall | RC) = 0.2\\n\\nRecession P(R) = 0.3\\nP(Stock Rise | R) = 0.3\\nP(Stock Fall | R) = 0.7\\n\\nThe total probability that the stock price will rise is closest to:\\n\\nA. 0.50\\nB. 0.65\\nC. 0.70\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B (0.65). The unconditional probability of a stock rise is calculated using the law of total probability: P(SR) = P(SR|RC)P(RC) + P(SR|R)P(R). Substituting the values gives: (0.8 \u00d7 0.7) + (0.3 \u00d7 0.3) = 0.56 + 0.09 = 0.65.\"\n    }\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/PsZrsg3ZUDE\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We can use the total probability rule to determine the unconditional probability of an event in terms of conditional probabilities in certain scenarios.<\/p>\n<p><!--more--><\/p>\n<p>In general, if we have a set of mutually exclusive and exhaustive events S<sub>1<\/sub>, S<sub>2<\/sub> \u2026S<sub>n<\/sub>, then:<\/p>\n<p>$$ \\text P(\\text T) = \\text P(\\text T | \\text S_1) \\text P( \\text S_1) + \\text P(\\text T | \\text S_2) \\text P(\\text S_2) + \u2026 + \\text P(\\text T | \\text S_{n}) \\text P(\\text S_n) $$<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Suppose you assess the performance of a stock under different circumstances and come up with the following probabilities.<\/p>\n<p>$$ \\begin{array}{c|c|c|c} {\\textbf{State of}} &amp; {\\textbf{Probability of}} &amp; {\\textbf{Stock}} &amp; {\\textbf{Probability}} \\\\ {\\textbf{Economy}} &amp; {\\textbf{Economic State}} &amp; {\\textbf{Performance}} &amp; {} \\\\ \\hline {} &amp; {} &amp; {\\text{Rise } \\text P(\\text {SR}|\\text R^{\\text C}) } &amp; { 0.8} \\\\ {\\text {No recession } \\text P(\\text R^{\\text C}) } &amp; {0.7} &amp; { \\text{Fall } \\text P(\\text {SR}^{\\text C}|\\text R^{\\text C}) } &amp; {0.2} \\\\ \\hline {} &amp; {} &amp; {\\text{Rise }\\text P(\\text {SR}|\\text R)} &amp; { 0.3} \\\\ {\\text {Recession } \\text P(\\text R) } &amp; {0.3} &amp; { \\text{Fall } \\text P(\\text {SR}^{\\text C}|\\text R) } &amp; {0.7} \\\\ \\end{array} $$<\/p>\n<p>The total probability of a stock rise is <em>closest<\/em> to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>0.50.<\/li>\n<li>0.65.<\/li>\n<li>0.70.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is\u00a0<strong>B<\/strong>.<\/p>\n<p>You need to find the unconditional probability of a stock rise under all circumstances. Therefore,<\/p>\n<p>$$ \\begin{align*} \\text P(\\text{SR}) &amp; = \\text P(\\text {SR} | \\text R^\\text{C}) \\text P(\\text R^\\text{C}) + \\text P(\\text{SR} | \\text R) \\text P(\\text R) \\\\ &amp; = 0.8 \u00d7 0.7 + 0.3 \u00d7 0.3 \\\\ &amp; = 0.65 \\\\ \\end{align*} $$<\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>We can use the total probability rule to determine the unconditional probability of an event in terms of conditional probabilities in certain scenarios.<\/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-27466","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 - 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