{"id":27503,"date":"2021-08-25T06:14:23","date_gmt":"2021-08-25T06:14:23","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27503"},"modified":"2025-12-08T21:07:45","modified_gmt":"2025-12-08T21:07:45","slug":"probability-tree-and-its-application-to-investment-problems","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-tree-and-its-application-to-investment-problems\/","title":{"rendered":"Probability Tree and its Application to Investment Problems"},"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 probability of picking two red cards from a standard deck without replacement?\",\n    \"text\": \"Someone picks a card from an ordinary pack of 52 playing cards, without replacement. He then picks another one. Draw a probability tree and use it to calculate the probability of picking two red cards.\\n\\nA. 25\/102\\nB. 13\/51\\nC. 26\/51\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A.\\n\\nThere are 26 red cards out of 52 on the first draw. The probability of drawing a red card first is 26\/52 = 1\/2.\\nAfter one red card is removed, 25 red cards remain out of 51 total cards. The probability of the second red card is 25\/51.\\n\\nTherefore, the probability of two red cards is:\\n(26\/52) \u00d7 (25\/51) = 25\/102.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-88.jpg\",\n  \"caption\": \"Tree Diagram\",\n  \"width\": 1632,\n  \"height\": 1414,\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>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-89.jpg\",\n  \"caption\": \"Tree Diagram Example\",\n  \"width\": 1632,\n  \"height\": 1414,\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>\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/PsZrsg3ZUDE\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>A tree diagram is a visual representation of all possible future outcomes and the associated probabilities of a random variable. Tree diagrams are particularly useful when we have several possible outcomes. They facilitate the recording of all the possibilities in a clear, uncomplicated manner. Each branch in a tree diagram represents an outcome.<!--more--><\/p>\n<h4><strong>Example: Probability Tree<\/strong><\/h4>\n<p>Suppose we toss a fair coin twice. The first outcome will <em><strong>not<\/strong> <\/em>affect the second outcome (the two events are independent). The first toss&#8217; outcome can either be a head or a tail. Similarly, the outcome of the second toss can either be a head or a tail, regardless of the outcome of the first toss, since the coin has no memory! We can represent the probabilities in a tree diagram as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-16964\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-88.jpg\" alt=\"tree-diagram\" width=\"1632\" height=\"1414\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-88.jpg 1632w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-88-300x260.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-88-768x665.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-88-1024x887.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-88-400x347.jpg 400w\" sizes=\"auto, (max-width: 1632px) 100vw, 1632px\" \/><\/p>\n<p>Please, note the following:<\/p>\n<ul>\n<li>The tree diagram must include all the possible outcomes.<\/li>\n<li>The sum of the probabilities must<span style=\"font-size: revert; color: initial;\"> add up to 1.<\/span><\/li>\n<li>The number of branches<span style=\"font-size: revert; color: initial;\"> is the number of different possibilities. <\/span><\/li>\n<li>The numbers on the branches present probabilities.<\/li>\n<\/ul>\n<p>To calculate probabilities, we go along the tree branches from left to right to get to the end and then multiply together any probabilities that we have passed. Therefore, the probability of getting two heads, i.e., \\(HH\\), is:<\/p>\n<p>$$ \\begin {align*}<br \/>\n\\text P ( \\text{HH}) &amp;\u00a0 =\\cfrac\u00a0 {1} {2} * \\cfrac{1} {2} \\\\<br \/>\n&amp; =\u00a0 \\cfrac {1} {4} \\\\<br \/>\n\\end {align*} $$<\/p>\n<p>If we sum up the probabilities of all possibilities, we get 1. (The candidate can prove this.)<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Someone picks a card from an ordinary pack of 52 playing cards, <strong>without replacement<\/strong>. He then picks another one. Draw a probability tree and use it to calculate the probability of picking two red cards.<\/p>\n<p>A. \\( \\frac{25} {102} \\)<\/p>\n<p>B. \\( \\frac{13} {51} \\)<\/p>\n<p>C. \\( \\frac{26} {51} \\)<\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-16965\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-89.jpg\" alt=\"tree-diagram-example\" width=\"1632\" height=\"1414\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-89.jpg 1632w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-89-300x260.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-89-768x665.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-89-1024x887.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-89-400x347.jpg 400w\" sizes=\"auto, (max-width: 1632px) 100vw, 1632px\" \/><\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>A tree diagram is a visual representation of all possible future outcomes and the associated probabilities of a random variable. Tree diagrams are particularly useful when we have several possible outcomes. They facilitate the recording of all the possibilities in&#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-27503","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>Probability Tree in Investment Analysis | CFA Level 1<\/title>\n<meta name=\"description\" content=\"A probability tree visually maps possible outcomes and their probabilities, aiding investment decision-making and risk assessment.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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