{"id":439,"date":"2019-10-10T13:27:00","date_gmt":"2019-10-10T13:27:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=439"},"modified":"2026-03-05T19:49:29","modified_gmt":"2026-03-05T19:49:29","slug":"probability-rules-applications-multiplication-addition-total","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-rules-applications-multiplication-addition-total\/","title":{"rendered":"Application of Probability Rules"},"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 that a bond defaulted given that it was non-investment grade?\",\n    \"text\": \"An analyst studied 150 corporate bonds. Among the 75 non-investment grade bonds, 60 defaulted and 15 did not. Among the 75 investment grade bonds, 10 defaulted and 65 did not. If a bond is selected at random, what is the probability that the bond defaulted given that it was non-investment grade?\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. The conditional probability is calculated as the number of non-investment grade bonds that defaulted divided by the total number of non-investment grade bonds. Therefore, P(Default | Non-investment grade) = 60 \/ 75 = 0.80 or 80%.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"46.67%.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"80.00%.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"81.23%.\"\n      }\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\/cfa-level-1-addition-rule.png\",\n  \"caption\": \"Addition Rule\",\n  \"width\": 1632,\n  \"height\": 972,\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-76.jpg\",\n  \"caption\": \"Addition Rule\",\n  \"width\": 1632,\n  \"height\": 972,\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<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/PsZrsg3ZUDE\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Probability rules are the concepts and facts that must be taken into account while evaluating the probabilities of various events. The CFA curriculum requires candidates to master 3 main rules of probability. These are the multiplication rule, the addition rule, and the law of total probability. Let&#8217;s now look at each rule in detail.<\/p>\n<p><!--more--><\/p>\n<h2><strong>Multiplication Rule<\/strong><\/h2>\n<p>We use the multiplication rule to determine the <strong>joint probability<\/strong> of two events, \\(P(AB)\\). A joint probability is the probability of two events happening together. For example, we may be interested in the probability that both gas prices and bus fares will increase simultaneously.<\/p>\n<h3 data-tadv-p=\"keep\"><strong>Mutually Exclusive Events<\/strong><\/h3>\n<p>If two events are mutually exclusive, then they cannot occur together. Therefore, we say that such events have zero joint probability.<\/p>\n<h3 data-tadv-p=\"keep\"><strong>Non-mutually Exclusive Events<\/strong><\/h3>\n<p>For non-mutually exclusive events, the multiplication rule states that:<\/p>\n<p>$$ P(AB) = P(A | B)P(B) $$<\/p>\n<h4><strong>Example: Multiplication Rule<\/strong><\/h4>\n<p>Let\u2019s assume we have a bag containing 16 blue and 14 yellow balls. Further, assume that we randomly draw two balls, one after the other, without replacement. Let\u2019s define:<\/p>\n<p>Event B = The first ball is blue.<\/p>\n<p>Event A = The second ball is blue.<\/p>\n<p>What will be the joint probability of A and B?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>From the wording of the question, we can calculate the conditional probability \\(P(A|B)\\) as:<\/p>\n<p>The probability that the first ball to be drawn is blue, P(B) = \\(\\cfrac {16}{30}\\).<\/p>\n<p>The probability that the second ball to be drawn is blue, given that the first one is also blue, P(A | B) = \\(\\cfrac {15}{29}\\).<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\begin{align*} P(AB) &amp; = P(A | B)P(B) \\\\ &amp; = \\cfrac {15}{29} \u00d7 \\cfrac {16}{30} \\\\ &amp; =\\cfrac {240}{870} \\\\ &amp; =\\cfrac {8}{29} \\\\ \\end{align*} $$<\/p>\n<h2><strong>Addition Rule<\/strong><\/h2>\n<p>We use the addition rule to assess the probability that events A <strong>or<\/strong> B occur.<\/p>\n<h3 data-tadv-p=\"keep\"><strong>Mutually Exclusive Events<\/strong><\/h3>\n<p>If A and B are mutually exclusive events:<\/p>\n<p>$$ \\text P(\\text{A or B}) =\\text P(\\text A) + \\text P(\\text B) $$<\/p>\n<h3 data-tadv-p=\"keep\"><strong>Non-Mutually Exclusive Events<\/strong><\/h3>\n<p>If A and B are non-mutually exclusive events:<\/p>\n<p>$$ \\text P(\\text{A or B}) =\\text P(\\text A) + \\text P(\\text B) \u2013 \\text P(\\text {AB}) $$<\/p>\n<p>This is because we have to remove one of the two instances of the intersection of A and B, as shown in the following figure.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-37602\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/cfa-level-1-addition-rule.png\" alt=\"\" width=\"1632\" height=\"972\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/cfa-level-1-addition-rule.png 1632w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/cfa-level-1-addition-rule-300x179.png 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/cfa-level-1-addition-rule-1024x610.png 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/cfa-level-1-addition-rule-768x457.png 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/cfa-level-1-addition-rule-1536x915.png 1536w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/cfa-level-1-addition-rule-400x238.png 400w\" sizes=\"auto, (max-width: 1632px) 100vw, 1632px\" \/><\/p>\n<h4><strong>Example: Addition Rule<\/strong><\/h4>\n<p>Assume that the probability of relaxed import restrictions is 0.5 and the probability of a price war is 0.2. If the joint probability of relaxed import restrictions and a price war is 0.4, what is the probability of relaxed trade restrictions <strong>or <\/strong>a price war?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\begin{align} \\text P(\\text{A or B}) &amp;= \\text P(\\text A) + \\text P(\\text B) \u2013 \\text P(\\text {AB}) \\\\ &amp; = 0.5 + 0.2 \u2013 0.4 = 0.3 \\end{align} $$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1632\" height=\"972\" class=\"aligncenter size-full wp-image-16956\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-76.jpg\" alt=\"addition_rule\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-76.jpg 1632w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-76-300x179.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-76-768x457.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-76-1024x610.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-76-400x238.jpg 400w\" sizes=\"auto, (max-width: 1632px) 100vw, 1632px\" \/><\/p>\n<h2><strong>Total Probability Rule<\/strong><\/h2>\n<p>Suppose we have a set of mutually exclusive and exhaustive events B<sub>1<\/sub>, B<sub>2<\/sub>, B<sub>3<\/sub>\u2026B<sub>n<\/sub>.<\/p>\n<p>We can determine the unconditional probability of an event, given the conditional probabilities:<\/p>\n<p>$$ \\text P(\\text A) = \\text P(\\text A | \\text B_1)\\text P(\\text B_1) + \\text P(\\text A | \\text B_3)\\text P(\\text B_3) + \u2026 + \\text P(\\text A | \\text B_n)\\text P(\\text B_n) $$<\/p>\n<p>Perhaps an example will help you understand the concept:<\/p>\n<h4><strong>Example:\u00a0<\/strong><strong>Total Probability Rule<\/strong><\/h4>\n<p>Suppose a local authority subdivides a forest into three regions, B<sub>1<\/sub>, B<sub>2<\/sub>, and B<sub>3<\/sub>. Assume that each region covers 50km<sup>2<\/sup>, 65km<sup>2<\/sup>, and 74km<sup>2<\/sup>, respectively. What is the total forest area?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>We simply add the respective areas:<\/p>\n<p>$$ 50 + 65 + 74 = 189 \\text{ km}^2 $$<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>An analyst analyzed defaults of one hundred and fifty corporate bonds, investment grade, and non-investment grade. The following table summarizes the results. \u00a0\u00a0<\/p>\n<p>$$ \\begin{array}{l|c|c|c}\\\\ {}&amp;\\textbf{Bonds Default}&amp;\\textbf{Bonds did not}&amp;\\textbf{Total}\\\\ {}&amp;{}&amp;\\textbf{default}&amp;{}\\\\ \\hline \\text{Non-investment}&amp;60&amp;15&amp;75\\\\ \\text{grade bonds}&amp;{}&amp;{}&amp;{}\\\\ \\hline \\text{Investment grade}&amp;10&amp;65&amp;75\\\\ \\text{bonds}&amp;{}&amp;{}&amp;{}\\\\ \\hline \\text{Total}&amp;70&amp;80&amp;150\\\\ \\end{array} $$<\/p>\n<p class=\"ListLevel1\" style=\"margin-left: 0in; text-indent: 0in;\">If one bond from the sample of 150 bonds is selected at random, determine the probability that the bond defaulted given that it was non-investment grade:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>46.67%.<\/li>\n<li>80.00%.<\/li>\n<li>81.23%.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <b>B<\/b>.<\/p>\n<p>The number of bonds that were defaulted and were non-investment grade was 60. The number of bonds that were non-investment grades was 75.<\/p>\n<p>So, the probability that the bond defaulted, given that the bond did not default = 60 \/ 75 = 0.8 = 80%.<\/p>\n<p><strong>A is incorrect<\/strong>. It represents the probability of bonds defaulted out of the total sample of 150 bonds (70 \/ 150).<\/p>\n<p><strong>C is incorrect<\/strong>. It represents the probability that the bond was investment grade, given that the bond did not default. The number of bonds that did not default and were investment grade is 65. The number of bonds that did not default is 80.\u00a0 So the probability that the bond was investment grade given that the bond did not default = 65 \/ 80 = 81.23%.<\/p>\n<\/blockquote>\n\n","protected":false},"excerpt":{"rendered":"<p>Probability rules are the concepts and facts that must be taken into account while evaluating the probabilities of various events. The CFA curriculum requires candidates to master 3 main rules of probability. These are the multiplication rule, the addition rule,&#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-439","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 Rules &amp; Applications | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn key probability rules, including addition and multiplication, and their applications in financial analysis 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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