{"id":555,"date":"2019-10-10T13:27:00","date_gmt":"2019-10-10T13:27:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=555"},"modified":"2026-03-20T19:47:12","modified_gmt":"2026-03-20T19:47:12","slug":"tree-diagram-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/tree-diagram-example\/","title":{"rendered":"Tree Diagram"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Probability Concepts (2021 Level I CFA\u00ae Exam \u2013 Reading 8)\",\n  \"description\": \"This video covers key probability concepts, including definitions, types of probabilities (conditional, unconditional), odds, the multiplication and total probability rules, Bayes\u2019 formula, and counting principles. It explains covariance, correlation, portfolio variance, and tools like tree diagrams and factorials, with practical examples to aid understanding.\",\n  \"uploadDate\": \"2020-10-01T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/hu47ZbsskEw\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/hu47ZbsskEw\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw\",\n  \"duration\": \"PT48M33S\"\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-88.jpg\",\n  \"caption\": \"Tree Diagram Image\",\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<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Probability of drawing two red cards without replacement\",\n    \"text\": \"A card is picked from an ordinary deck of 52 playing cards without replacement, and then another card is picked. What is the probability of drawing two red cards?\\n\\nA. 25\/102\\n\\nB. 13\/51\\n\\nC. 26\/51\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A.\\n\\nProbability of first red card = 26\/52.\\nProbability of second red card = 25\/51.\\n\\nTotal probability = (26\/52) \u00d7 (25\/51) = 25\/102.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/hu47ZbsskEw?si=I9nLpVhnGrSYB0Q_\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\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 can help you record all the possibilities in a clear, uncomplicated manner. Each branch in a tree diagram represents an outcome. Consider the following example:<\/p>\n<p><!--more--><\/p>\n<h2><strong>Example<\/strong><\/h2>\n<p>If we tossed a fair coin twice, the first outcome would not affect the second outcome (the two events are independent). The outcome of the first toss can either be a head or a tail. Similarly, the outcome of the second toss can be a head or a tail, regardless of the outcome of the first toss, since the coin has <strong>no memory<\/strong>! We can represent the probabilities in a tree diagram as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-16964 size-full\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-88.jpg\" alt=\"tree diagram Image\" 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 data-tadv-p=\"keep\">the tree diagram must include <strong>all<\/strong> the possible outcomes;<\/li>\n<li data-tadv-p=\"keep\">the sum of the probabilities <strong>must<\/strong> add up to 1;<\/li>\n<li data-tadv-p=\"keep\">the number of <strong>branches<\/strong> is the number of different possibilities; and<\/li>\n<li data-tadv-p=\"keep\">probabilities are represented by the <strong>numbers <\/strong>on the branches.<\/li>\n<\/ul>\n<p>To calculate probabilities, we go along the tree branches <strong>from left to right<\/strong> to get to the end and then multiply any probabilities that we have passed together. Therefore, the probability of getting two heads, i.e., HH, is:<\/p>\n<p>$$ \\begin{align*}<br \/>\\text P(\\text{HH})&amp; =\\cfrac {1}{2} * \\cfrac {1}{2} \\\\<br \/>&amp;= \\cfrac {1}{4} \\\\<br \/>\\end{align*} $$<\/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>A card is picked from an ordinary pack of 52 playing cards, <strong>without replacement<\/strong>, and then another one is picked. Draw a probability tree and use it to calculate the probability of picking two red cards.<\/p>\n<p>A. 25\/102<\/p>\n<p>B. 13\/51<\/p>\n<p>C. 26\/51<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is A.<\/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>\n<\/blockquote>\n<p><em>Reading 8 LOS 8j<\/em><\/p>\n<p><em>Explain the use of a tree diagram to represent an investment problem.<\/em><\/p>\n<div class=\"notes_inv\"><hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum-quantitative-methods\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>","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 can help you record all the possibilities in&#8230;<\/p>\n","protected":false},"author":2,"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-555","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>Tree Diagram Examples for Probability Problems<\/title>\n<meta name=\"description\" content=\"Learn how to use tree diagrams to solve probability problems, with step-by-step examples and how to interpret branches and outcomes.\" \/>\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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