{"id":27401,"date":"2021-08-24T17:11:38","date_gmt":"2021-08-24T17:11:38","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27401"},"modified":"2026-01-21T18:01:48","modified_gmt":"2026-01-21T18:01:48","slug":"conditional-probability","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/conditional-probability\/","title":{"rendered":"Conditional 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\": \"Conditional probability of buying bonds given stocks\",\n    \"text\": \"In a group of 100 investors:\\n\\n40 buy stocks.\\n30 purchase bonds.\\n20 purchase stocks and bonds.\\n\\nIf an investor chosen randomly bought stocks, the probability they also bought bonds is closest to:\\n\\nA. 0.50.\\nB. 0.67.\\nC. 0.70.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A. 0.50.\\n\\nThe conditional probability that an investor bought bonds given that they bought stocks is:\\n\\nP(B | A) = P(A \u2229 B) \/ P(A) = 0.20 \/ 0.40 = 0.50.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"B. 0.67.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C. 0.70.\"\n      }\n    ]\n  }\n}\n<\/script>\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/PsZrsg3ZUDE\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><strong>Unconditional Probability<\/strong><\/h2>\n<p>Unconditional probability (also known as marginal probability) is simply the probability that the occurrence of an event does not, in any way, depend on any other preceding events. In other words, unconditional probabilities are <strong>not <\/strong>conditioned on the occurrence of any other events; they are &#8216;stand-alone&#8217; events. Therefore, if we are interested in the probability of, say, event A, the standard annotation is P(A). Let\u2019s look at a few examples:<\/p>\n<p><!--more--><\/p>\n<ol>\n<li>The probability that it will rain on a given day without considering the rainfall pattern of a given area or any other climatic factor.<\/li>\n<li>The probability that a given stock will earn a 10% annual return without considering the preceding annual returns.<\/li>\n<\/ol>\n<p>In a future learning outcome statement, we will learn to calculate and interpret an unconditional probability using the total probability rule.<\/p>\n<h2><strong>Conditional Probability<\/strong><\/h2>\n<p>Conditional probability is the probability of one event occurring with some relationship to one or more other events. Our interest lies in the probability of event &#8216;A&#8217; <strong>given<\/strong>\u00a0that another event &#8216;B &#8216;<strong>has already occurred<\/strong>. Here\u2019s what you should ask yourself:<\/p>\n<p>&#8220;What is the probability of one event occurring\u00a0<strong>if\u00a0<\/strong>another event has already taken place?&#8221;<\/p>\n<p>We pronounce P(A | B) as &#8220;the probability of A given B.,&#8221; and it is given by:<\/p>\n<p>$$P(A\u2502B)=\\frac{P(A\\cap B)}{P(B)}$$<\/p>\n<p>The bar sandwiched between A and B simply indicates &#8220;given.&#8221;<\/p>\n<h4><b>Example: Groups of Investors<\/b><\/h4>\n<p>In a group of 100 investors,<\/p>\n<ul>\n<li>40 buy stocks.<\/li>\n<li>30 purchase bonds.<\/li>\n<li>20 purchase stocks and bonds.<\/li>\n<\/ul>\n<p>If an investor chosen randomly bought bonds, what is the probability they also bought stocks?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$<br \/>\n\\begin{array}{c|c|c}<br \/>\n\\textbf{Event} &amp; \\textbf{Notation} &amp; \\textbf{Probability} \\\\<br \/>\n\\hline<br \/>\n\\text{Buys stocks} &amp; \\text{A} &amp; \\text{0.4(=40\/100)} \\\\<br \/>\n\\text{Buys bonds} &amp; \\text{B} &amp; \\text{0.3 (=30\/100)} \\\\<br \/>\n\\text{Buys stocks and bonds} &amp; \\text{A and B} &amp; \\text{0.2 (=20\/100)} \\\\<br \/>\n\\end{array}<br \/>\n$$<\/p>\n<p>We want the probability of an investor buying stocks, given that they have already bought bonds. This is P(A | B):<\/p>\n<p>$$ \\begin{align} P\\left( A|B \\right) &amp; =\\frac { P\\left( A\\cap B \\right) }{ P\\left( B \\right) } \\\\ &amp; =\\frac { 0.2 }{ 0.3 }\u00a0 = 0.67 \\end{align}$$<\/p>\n<h3>Rearranging the Formula<\/h3>\n<p>Note that we can also make the numerator the subject so that:<\/p>\n<p>$$ P\\left( A\\cap B \\right) =P\\left( A|B \\right) P\\left( B \\right) $$<\/p>\n<p>For independent events, which we will see later, however,<\/p>\n<p>$$ P\\left( A|B \\right) =\\frac{P(A)P(B)}{P(B)}=P\\left( A \\right) $$<\/p>\n<p>This is also true for \\(P(B\u2502A)=P(B)\\).<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>In a group of 100 investors,<\/p>\n<ul>\n<li>40 buy stocks.<\/li>\n<li>30 purchase bonds.<\/li>\n<li>20 purchase stocks and bonds.<\/li>\n<\/ul>\n<p>If an investor chosen randomly bought stocks, the probability they also bought bonds is <em>closest to:<\/em><\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>0.50.<\/li>\n<li>0.67.<\/li>\n<li>0.70.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>Consider the following table:<\/p>\n<p>$$<br \/>\n\\begin{array}{c|c|c}<br \/>\n\\textbf{Event} &amp; \\textbf{Notation} &amp; \\textbf{Probability} \\\\<br \/>\n\\hline<br \/>\n\\text{Buys stocks} &amp; \\text{A} &amp; \\text{0.4(=40\/100)} \\\\<br \/>\n\\text{Buys bonds} &amp; \\text{B} &amp; \\text{0.3 (=30\/100)} \\\\<br \/>\n\\text{Buys stocks and bonds} &amp; \\text{A and B} &amp; \\text{0.2 (=20\/100)} \\\\<br \/>\n\\end{array}<br \/>\n$$<\/p>\n<p>The probability of an investor buying stocks given that they have already bought bonds. This is P(B|A):<\/p>\n<p>$$ \\begin{align} P\\left( B|A \\right) &amp; =\\frac { P\\left( A\\cap B \\right) }{ P\\left( A \\right) } \\\\ &amp; =\\frac { 0.2 }{ 0.4 }\u00a0 = 0.50 \\end{align}$$<\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Unconditional Probability Unconditional probability (also known as marginal probability) is simply the probability that the occurrence of an event does not, in any way, depend on any other preceding events. In other words, unconditional probabilities are not conditioned on the&#8230;<\/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-27401","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>Conditional vs Unconditional Probability | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Understand conditional probability and how it differs from unconditional probability. 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