{"id":1891,"date":"2019-03-26T12:00:00","date_gmt":"2019-03-26T11:00:00","guid":{"rendered":"https:\/\/kindsonthegenius.com\/blog\/conditional-probability-multiplication-law\/"},"modified":"2026-07-05T03:22:19","modified_gmt":"2026-07-05T01:22:19","slug":"conditional-probability-multiplication-law","status":"publish","type":"post","link":"https:\/\/kindsonthegenius.com\/blog\/conditional-probability-multiplication-law\/","title":{"rendered":"Conditional Probability \u2013 Multiplication Law"},"content":{"rendered":"<p>In this short lesson we would understand how to use the Multiplication Law in Conditional Probability.<\/p>\n<p>You may want to review <a href=\"https:\/\/kindsonthegenius.com\/tempsite\/basics-of-conditional-probability\/\">Basics of Conditional Probability<\/a>.<\/p>\n<p>The Multiplication Law helps us to find conditional probability and is given by the formula:<\/p>\n<p style=\"text-align: center;\"><strong>P(A\u00a0\u2229 B) = P(B | A) P(A)<\/strong><\/p>\n<p>Of course you can also interchange the order of A and B giving:<\/p>\n<p style=\"text-align: center;\">P(B | A) = P(A | B) P(B)<\/p>\n<p>Let&#8217;s take and example to illustrate this.<\/p>\n<hr \/>\n<p><strong>Example 1<\/strong><\/p>\n<p>A bag contains 3 red balls and 1 blue ball. Two balls are selected without replacement. What is the probability that both balls are red?<\/p>\n<hr \/>\n<p>Let R1 be the probability that the first ball is red<\/p>\n<p>R2 be the probability that the second ball is red<\/p>\n<p>So we can find P(R1) and P(R2)<\/p>\n<p>Then we use the results to find\u00a0P(R1\u00a0\u2229 R2) as well.<\/p>\n<p>P(R1) = 3\/4\u00a0 \u00a0 \u00a0 \u00a0 \u00a0that is 3 red balls over total ball.<\/p>\n<p>P(R2 | R1) = 2\/3\u00a0 \u00a0 \u00a0 \u00a0 \u00a0since only 2 red balls is left and total balls is now 3<\/p>\n<p>Using the multiplication Law, we can now find P(R1\u00a0\u2229 R2)<\/p>\n<p style=\"text-align: center;\"><strong>P(R1\u00a0\u2229 R2) = 3\/4 x 2\/3 = 1\/2 <\/strong><\/p>\n<p>This is the probability that both balls are red<\/p>\n<p>&nbsp;<\/p>\n<p>Let&#8217;s take another example<\/p>\n<hr \/>\n<p><strong>Example 2<\/strong><\/p>\n<p>Assuming that if the weather is cloudy, the probability that it is raining (A) is 0.3. Also the probability that it is cloudy is P(B) is 0.2.<\/p>\n<p>What is the probability that it is cloudy and raining:<\/p>\n<hr \/>\n<p>P(B) = 0.2<\/p>\n<p>P(A | B) = 0.3<\/p>\n<p>Therefore:<\/p>\n<p>P(A\u00a0\u2229 B) = P(A | B) P(B) = 0.2 * 0.3 = 0.6<\/p>\n<p>&nbsp;<\/p>\n<p>In the next lesson, we would examine Law of Total Probability<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this short lesson we would understand how to use the Multiplication Law in Conditional Probability. You may want to review Basics of Conditional Probability. &hellip; <\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pagelayer_contact_templates":[],"_pagelayer_content":"","footnotes":""},"categories":[414],"tags":[],"class_list":["post-1891","post","type-post","status-publish","format-standard","hentry","category-programming"],"_links":{"self":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1891","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/comments?post=1891"}],"version-history":[{"count":1,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1891\/revisions"}],"predecessor-version":[{"id":2059,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/posts\/1891\/revisions\/2059"}],"wp:attachment":[{"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/media?parent=1891"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/categories?post=1891"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kindsonthegenius.com\/blog\/wp-json\/wp\/v2\/tags?post=1891"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}