{"id":45465,"date":"2023-08-06T16:01:20","date_gmt":"2023-08-06T16:01:20","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45465"},"modified":"2026-05-12T09:51:40","modified_gmt":"2026-05-12T09:51:40","slug":"probability-trees","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/","title":{"rendered":"Probability Trees"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Probability Trees and Conditional Expectations (2025 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 Module 4)\",\n  \"description\": \"CFA\u00ae Level I Quantitative Methods video lesson from AnalystPrep on Probability Trees and Conditional Expectations (Module 4). This lesson covers probability-weighted expected value, variance and standard deviation, how to build and interpret probability and decision trees, conditional and unconditional expectations, and updating beliefs using Bayes\u2019 theorem, with practical exam-style examples.\",\n  \"uploadDate\": \"2024-01-12\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/Ciu1kP6q9d0\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=Ciu1kP6q9d0\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/Ciu1kP6q9d0\",\n  \"duration\": \"PT33M31S\"\n}\n<\/script>\n\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@graph\": [\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG3-768x665.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG3-768x665.jpg\",\n      \"caption\": \"Tree diagram example\",\n      \"width\": 768,\n      \"height\": 665,\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        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG2-1024x660.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG2-1024x660.jpg\",\n      \"caption\": \"Probability tree\",\n      \"width\": 1024,\n      \"height\": 660,\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        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    },\n    {\n      \"@type\": \"ImageObject\",\n      \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg\",\n      \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg\",\n      \"caption\": \"Tree diagram\",\n      \"width\": 1312,\n      \"height\": 977,\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        \"url\": \"https:\/\/analystprep.com\/\"\n      }\n    }\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\": \"Someone picks a card from an ordinary pack of 52 playing cards, without replacement. He then picks another one. What is the probability of picking two red cards?\",\n    \"text\": \"Someone picks a card from an ordinary pack of 52 playing cards, without replacement. He then picks another one. What is the probability of picking two red cards?\\n\\nA. 25\/102\\nB. 13\/51\\nC. 26\/51\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"A: 25\/102\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"B: 13\/51\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C: 26\/51\"\n      }\n    ]\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\": \"There is a 20% chance that the government will impose a tariff on imported cars. A company that assembles cars locally expects returns of 14% if the tariff is imposed and returns of 11% otherwise. The (unconditional) expected return is closest to:\",\n    \"text\": \"There is a 20% chance that the government will impose a tariff on imported cars. A company that assembles cars locally expects returns of 14% if the tariff is imposed and returns of 11% otherwise. The (unconditional) expected return is closest to:\\n\\n1. 11.6%\\n2. 12.8%\\n3. 12.5%\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A: 11.6%. The unconditional expected return is calculated as: E(X) = 0.11(0.8) + 0.14(0.2) = 0.116 = 11.6%.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"B: 12.8% \u2014 incorrect.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"C: 12.5% \u2014 incorrect.\"\n      }\n    ]\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/Ciu1kP6q9d0\" 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 handy when we have several possible outcomes.<\/p>\n<p>They facilitate the recording of all the possibilities in a clear, uncomplicated manner. Each branch in a tree diagram represents an outcome.<\/p>\n<p><strong>Example: Probability Tree<\/strong><\/p>\n<p>Let&#8217;s consider a scenario where we toss a fair coin twice. The outcomes of these tosses are independent, meaning the first toss doesn&#8217;t influence the second one.<\/p>\n<p>For the first toss, we have two possibilities: It can result in either a head or a tail. Similarly, we still have the same two possibilities for the second toss: head or tail. Importantly, the outcome of the second toss is not affected by what happened in the first toss because coins don&#8217;t have memory.<\/p>\n<p>We can visualize these probabilities using a tree diagram like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1312\" height=\"977\" class=\"aligncenter wp-image-45466 size-medium_large\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg 1312w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1-300x223.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1-1024x763.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1-768x572.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1-400x298.jpg 400w\" sizes=\"auto, (max-width: 1312px) 100vw, 1312px\" \/><\/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 add up to 1.<\/li>\n<li>The number of branches is the number of different possibilities.<\/li>\n<li>The numbers on the branches present probabilities.<\/li>\n<\/ul>\n<div class=\"flex-1 overflow-hidden\">\n<div class=\"react-scroll-to-bottom--css-atgos-79elbk h-full dark:bg-gray-800\">\n<div class=\"react-scroll-to-bottom--css-atgos-1n7m0yu\">\n<div class=\"flex flex-col text-sm dark:bg-gray-800\">\n<div class=\"group w-full text-token-text-primary border-b border-black\/10 gizmo:border-0 dark:border-gray-900\/50 gizmo:dark:border-0 bg-gray-50 gizmo:bg-transparent dark:bg-[#444654] gizmo:dark:bg-transparent\" data-testid=\"conversation-turn-583\">\n<div class=\"p-4 justify-center text-base md:gap-6 md:py-6 m-auto\">\n<div class=\"flex flex-1 gap-4 text-base mx-auto md:gap-6 gizmo:gap-3 gizmo:md:px-5 gizmo:lg:px-1 gizmo:xl:px-5 md:max-w-2xl lg:max-w-[38rem] gizmo:md:max-w-3xl gizmo:lg:max-w-[40rem] gizmo:xl:max-w-[48rem] xl:max-w-3xl }\">\n<div class=\"relative flex w-[calc(100%-50px)] flex-col gap-1 gizmo:w-full md:gap-3 lg:w-[calc(100%-115px)] agent-turn\">\n<div class=\"flex flex-grow flex-col gap-3 max-w-full\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-3 whitespace-pre-wrap break-words overflow-x-auto\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<p>To calculate probabilities, we follow the tree branches from left to right and multiply any probabilities we encounter.<\/p>\n<p>So, to find the probability of getting two heads (HH), we multiply the probabilities along the path.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>$$ P(HH)=\\frac {1}{2} \\times \\frac {1}{2}=\\frac {1}{4} $$<\/p>\n<p>If we sum up the probabilities of all possibilities, we get 1.<\/p>\n<div style=\"text-align: center; margin: 24px 0;\"><a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 10px 24px; border: 1.5px solid #1a73e8; border-radius: 999px; color: #1a73e8; text-decoration: none; font-size: 16px; font-weight: 600;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Practice probability tree concepts with our CFA Free Trial. <\/a><\/div>\n<h2>Conditional Expectations<\/h2>\n<p>In investments, conditional expectation means predicting the expected value of an investment based on specific real-world events. Analysts consider the probability and impact of future events when calculating this value.<\/p>\n<p>Competitors, governments, and other financial institutions keep releasing new information. Such pieces of information may have a positive or a negative impact on investment. This means that a project&#8217;s expected value must be based on real-world dynamics.<\/p>\n<p>In statistics, the conditional expected value is the expected value of a random variable \\(X\\) given an event or scenario \\(S\\), denoted by \\(E(X \\mid S)\\).<\/p>\n<p>Now, assume that \\(X\\) can take on any of \\(n\\) different outcomes \\(X_1, X_2,\\ldots, X_n\\), which are outcomes from a set of mutually exclusive and exhaustive events. Then,<\/p>\n<p>$$ \\begin{align*}<br \/>E(X \\mid S)&amp; =X_1\\cdot P(X_1 \\mid S)+X_2\\cdot P(X_2\\mid S)+\\cdots +X_n \\cdot P(X_n \\mid S) \\\\<br \/>&amp; =\\sum_{i=1}^n X_i \\cdot P(X_i \\mid S)<br \/>\\end{align*} $$<\/p>\n<h2>Stating Unconditional Expected Value in Terms of Conditional Expected Value<\/h2>\n<p>To state the unconditional expected values in terms of conditional expected value, we use the total probability rule for expected value, which is built from the following formula:<\/p>\n<p>$$ E(X)=P(S)\\cdot E(X \\mid S)+P(S^C )\\cdot E(X \\mid S^C) $$<\/p>\n<p>Where \\(S^C\\) is the complement (event or scenario \u201cS\u201d does not occur) of \\(S\\).<\/p>\n<p>Now assume that \\(S\\) can take on any of \\(S_1, S_2,\\ldots, S_n\\) mutually exclusive and exhaustive scenarios or events, then.<\/p>\n<p>$$ \\begin{align*}<br \/>E(X) &amp; =P(S_1)\\cdot E(X \\mid S_1)+P(S_2)\\cdot E(X \\mid S_2)+\\cdots+P(S_n )\\cdot E(X \\mid S_n) \\\\<br \/>&amp; =\\sum_{i=1}^n P(S_i)\\cdot E(X \\mid S_i) \\end{align*} $$<\/p>\n<p><strong>Example: Conditional Expectation<\/strong><\/p>\n<p>The probability of relaxed trade restrictions in a given country is 40%. Therefore, shareholders of XYZ Company Limited expect a 5% share return if trade restrictions are maintained and a loss of 8% if they are relaxed. The expected change in return is <em>closest to<\/em>:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>We must take every possibility into account. We have a 40% chance of relaxed trade restrictions in this case. Intuitively, this means there is a 60% chance that the current restrictions will be maintained. Therefore:<\/p>\n<p>$$ \\begin{align*}<br \/>E(X)&amp; =\\sum_{i=1}^n P(S_i)\\cdot E(X \\mid S_i) \\\\<br \/>&amp; =0.6(0.05) + 0.4(-0.08) \\\\<br \/>&amp; =-0.2\\%<br \/>\\end{align*} $$<\/p>\n<p><strong>Example: Total Probability Rule for Expected Value<\/strong><\/p>\n<p>BlueChip Inc.&#8217;s profits are sensitive to economic growth, benefitting significantly during periods of high economic growth. Suppose there is a 0.70 probability that BlueChip Inc. will operate in a high-growth economic environment in the next fiscal year and a 0.30 probability that it will operate in a moderate-growth environment. Assume that the chance of a recession is negligible.<\/p>\n<p>If a high-growth economic environment occurs, the probability that EPS will be USD 3.00 is estimated at 0.40, and the probability that EPS will be USD 2.80 is estimated at 0.60.<\/p>\n<p>On the other hand, if the company operates in a moderate-growth environment, the probabilities that the EPS will be USD 2.50 and USD 2.25 are 20% and 80%, respectively.<\/p>\n<p>Calculate the expected value of EPS for BlueChip Inc. in the next fiscal year.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>We start by drawing a probability tree:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"768\" height=\"495\" class=\"alignnone size-medium_large wp-image-45471\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG2-768x495.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG2-768x495.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG2-300x193.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG2-1024x660.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG2-400x258.jpg 400w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG2.jpg 1430w\" sizes=\"auto, (max-width: 768px) 100vw, 768px\" \/><\/p>\n<p>We first need to calculate the conditional expectations of EPS for each scenario: High-growth environment and moderate-growth environment. That is,<\/p>\n<p>$$ \\begin{align*}<br \/>E(EPS\\mid \\text{High-growth environment})&amp;=0.40 \\times 3.00+0.60 \\times 2.80 \\\\<br \/>&amp;=\\text{USD } 2.88 \\\\<br \/>E(EPS \\mid \\text{Moderate-growth environment})&amp;=0.20 \\times 2.50+0.80 \\times 2.25 \\\\<br \/>&amp;=\\text{USD } 2.30<br \/>\\end{align*} $$<\/p>\n<p>Using the total probability for the expected value, we have:<\/p>\n<p>$$ \\begin{align*}<br \/>E(EPS) &amp; =P(\\text{High-growth environment}) \\\\ &amp; \\cdot E(EPS \\mid \\text{High-growth environment}) \\\\ &amp; +P(\\text{High-growth environment}) \\\\ &amp; \\cdot E(EPS?\\text{Moderate-growth environment}) \\\\<br \/>&amp;=0.70 \\times 2.88+0.30 \\times 2.30 \\\\<br \/>&amp; =\\text{USD } 2.71<br \/>\\end{align*} $$<\/p>\n<blockquote>\n<h2>Question 1<\/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<ol type=\"A\">\n<li>\\(\\frac {25}{102}\\).<\/li>\n<li>\\(\\frac {13}{51}\\).<\/li>\n<li>\\(\\frac {26}{51}\\).<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>The correct answer is A.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"768\" height=\"665\" class=\"aligncenter size-medium_large wp-image-45474\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG3-768x665.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG3-768x665.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG3-300x260.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG3-1024x887.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG3-400x347.jpg 400w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG3.jpg 1251w\" sizes=\"auto, (max-width: 768px) 100vw, 768px\" \/><\/p>\n<h2>Question 2<\/h2>\n<p>There is a 20% chance that the government will impose a tariff on imported cars. A company that assembles cars locally expects returns of 14% if the tariff is imposed and returns of 11% otherwise. The (unconditional) expected return is <em>closest to<\/em>:<\/p>\n<ol type=\"A\">\n<li>11.6%.<\/li>\n<li>12.8%.<\/li>\n<li>12.5%.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p><strong>The correct answer is A.<\/strong><\/p>\n<p>The unconditional expected return will be the sum of:<\/p>\n<ol type=\"1\">\n<li style=\"list-style-type: none;\">\n<ol type=\"1\">\n<li>The expected return <strong>given<\/strong> no tariff times the probability that a tariff will not be imposed.<\/li>\n<li>The expected return <strong>given<\/strong> tariff times the probability that the tariff will be imposed. Therefore,<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>$$ \\begin{align*}<br \/>E(X) &amp; =\\sum_{i=1}^n P(S_i ) E(X \\mid S_i) \\\\<br \/>&amp;=0.11(0.8) + 0.14(0.2) \\\\<br \/>&amp; =0.116=11.6\\% \\end{align*} $$<\/p>\n<\/blockquote>\n<div style=\"background-color: #f5f5f5; padding: 40px 25px; border-radius: 18px; text-align: center; margin: 40px 0;\"><a style=\"display: inline-block; background-color: #1a73e8; color: #ffffff; padding: 12px 30px; border-radius: 999px; text-decoration: none; font-size: 16px; font-weight: 600;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener\"> Start Free Trial \u2192 <\/a>\n<p style=\"margin-top: 28px; font-size: 16px; line-height: 1.8; color: #333333; max-width: 920px; margin-left: auto; margin-right: auto;\">Master CFA Level I probability tree concepts including conditional expectations, joint probabilities, Bayes\u2019 formula, discrete random variables, and probability distributions with AnalystPrep\u2019s study resources.<\/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 handy when we have several possible outcomes. They facilitate the recording of all the possibilities in a&#8230;<\/p>\n","protected":false},"author":7,"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-45465","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Probability Trees Explained | CFA Level 1 - AnalystPrep<\/title>\n<meta name=\"description\" content=\"Learn how probability trees help visualize conditional probabilities, calculate expected values, and simplify decision-making in probability analysis.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Probability Trees Explained | CFA Level 1 - AnalystPrep\" \/>\n<meta property=\"og:description\" content=\"Learn how probability trees help visualize conditional probabilities, calculate expected values, and simplify decision-making in probability analysis.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2023-08-06T16:01:20+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-05-12T09:51:40+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1312\" \/>\n\t<meta property=\"og:image:height\" content=\"977\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Kajal\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Kajal\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/\"},\"author\":{\"name\":\"Kajal\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#\\\/schema\\\/person\\\/3cde53d128c8d0cfdd46d8732e8c3048\"},\"headline\":\"Probability Trees\",\"datePublished\":\"2023-08-06T16:01:20+00:00\",\"dateModified\":\"2026-05-12T09:51:40+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/\"},\"wordCount\":1072,\"image\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/wp-content\\\/uploads\\\/2023\\\/08\\\/LM04_IMG1.jpg\",\"articleSection\":[\"Quantitative Methods\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/\",\"url\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/\",\"name\":\"Probability Trees Explained | CFA Level 1 - AnalystPrep\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/wp-content\\\/uploads\\\/2023\\\/08\\\/LM04_IMG1.jpg\",\"datePublished\":\"2023-08-06T16:01:20+00:00\",\"dateModified\":\"2026-05-12T09:51:40+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#\\\/schema\\\/person\\\/3cde53d128c8d0cfdd46d8732e8c3048\"},\"description\":\"Learn how probability trees help visualize conditional probabilities, calculate expected values, and simplify decision-making in probability analysis.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/#primaryimage\",\"url\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/wp-content\\\/uploads\\\/2023\\\/08\\\/LM04_IMG1.jpg\",\"contentUrl\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/wp-content\\\/uploads\\\/2023\\\/08\\\/LM04_IMG1.jpg\",\"width\":1312,\"height\":977},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/quantitative-methods\\\/probability-trees\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Probability Trees\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#website\",\"url\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/\",\"name\":\"AnalystPrep | CFA\u00ae Exam Study Notes\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/#\\\/schema\\\/person\\\/3cde53d128c8d0cfdd46d8732e8c3048\",\"name\":\"Kajal\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/wp-content\\\/uploads\\\/2016\\\/12\\\/tarryn-150x150.png\",\"url\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/wp-content\\\/uploads\\\/2016\\\/12\\\/tarryn-150x150.png\",\"contentUrl\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/wp-content\\\/uploads\\\/2016\\\/12\\\/tarryn-150x150.png\",\"caption\":\"Kajal\"},\"url\":\"https:\\\/\\\/analystprep.com\\\/cfa-level-1-exam\\\/author\\\/kajal\\\/\"}]}<\/script>\n<meta property=\"og:video\" content=\"https:\/\/www.youtube.com\/embed\/Ciu1kP6q9d0\" \/>\n<meta property=\"og:video:type\" content=\"text\/html\" \/>\n<meta property=\"og:video:duration\" content=\"2012\" \/>\n<meta property=\"og:video:width\" content=\"480\" \/>\n<meta property=\"og:video:height\" content=\"270\" \/>\n<meta property=\"ya:ovs:adult\" content=\"false\" \/>\n<meta property=\"ya:ovs:upload_date\" content=\"2023-08-06T16:01:20+00:00\" \/>\n<meta property=\"ya:ovs:allow_embed\" content=\"true\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Probability Trees Explained | CFA Level 1 - AnalystPrep","description":"Learn how probability trees help visualize conditional probabilities, calculate expected values, and simplify decision-making in probability analysis.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/","og_locale":"en_US","og_type":"article","og_title":"Probability Trees Explained | CFA Level 1 - AnalystPrep","og_description":"Learn how probability trees help visualize conditional probabilities, calculate expected values, and simplify decision-making in probability analysis.","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2023-08-06T16:01:20+00:00","article_modified_time":"2026-05-12T09:51:40+00:00","og_image":[{"width":1312,"height":977,"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg","type":"image\/jpeg"}],"author":"Kajal","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Kajal","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/"},"author":{"name":"Kajal","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/3cde53d128c8d0cfdd46d8732e8c3048"},"headline":"Probability Trees","datePublished":"2023-08-06T16:01:20+00:00","dateModified":"2026-05-12T09:51:40+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/"},"wordCount":1072,"image":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/#primaryimage"},"thumbnailUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg","articleSection":["Quantitative Methods"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/","name":"Probability Trees Explained | CFA Level 1 - AnalystPrep","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"primaryImageOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/#primaryimage"},"image":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/#primaryimage"},"thumbnailUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg","datePublished":"2023-08-06T16:01:20+00:00","dateModified":"2026-05-12T09:51:40+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/3cde53d128c8d0cfdd46d8732e8c3048"},"description":"Learn how probability trees help visualize conditional probabilities, calculate expected values, and simplify decision-making in probability analysis.","breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/#primaryimage","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg","contentUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM04_IMG1.jpg","width":1312,"height":977},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-trees\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"Probability Trees"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/","name":"AnalystPrep | CFA\u00ae Exam Study Notes","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/3cde53d128c8d0cfdd46d8732e8c3048","name":"Kajal","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/12\/tarryn-150x150.png","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/12\/tarryn-150x150.png","contentUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/12\/tarryn-150x150.png","caption":"Kajal"},"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/kajal\/"}]},"og_video":"https:\/\/www.youtube.com\/embed\/Ciu1kP6q9d0","og_video_type":"text\/html","og_video_duration":"2012","og_video_width":"480","og_video_height":"270","ya_ovs_adult":"false","ya_ovs_upload_date":"2023-08-06T16:01:20+00:00","ya_ovs_allow_embed":"true"},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/45465","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/comments?post=45465"}],"version-history":[{"count":34,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/45465\/revisions"}],"predecessor-version":[{"id":60872,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/45465\/revisions\/60872"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=45465"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=45465"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=45465"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}