There are 3 urns containing coins. Urn I contains 7 gold coins, urn II contains 3 gold coins and 3 silver coins, and urn III contains 3 silver coins. An urn is selected and a coin is drawn from the urn. If the coin is silver, what is the probability that urn III was selected?<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n We can solve this problem using Bayes’ Theorem<\/strong>, which helps us find the probability of an event given prior knowledge of related conditions.<\/p>\n\n\n\n Let:<\/p>\n\n\n\n The problem provides:<\/p>\n\n\n\n Since an urn is selected at random, the probability of choosing any urn is: The probability of drawing a silver coin (( P(S) )) is given by:<\/p>\n\n\n\n [ [ [ We want to find ( P(U_3 | S) ):<\/p>\n\n\n\n [ [ [ There are 3 urns containing coins. Urn I contains 7 gold coins, urn II contains 3 gold coins and 3 silver coins, and urn III contains 3 silver coins. An urn is selected and a coin is drawn from the urn. If the coin is silver, what is the probability that urn III was selected? […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-203577","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203577","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=203577"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203577\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=203577"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=203577"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=203577"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Define the Events<\/h3>\n\n\n\n
\n
\n
[
P(U_1) = P(U_2) = P(U_3) = \\frac{1}{3}
]<\/p>\n\n\n\nStep 2: Apply the Law of Total Probability<\/h3>\n\n\n\n
P(S) = P(S | U_1) P(U_1) + P(S | U_2) P(U_2) + P(S | U_3) P(U_3)
]<\/p>\n\n\n\n
P(S) = (0 \\times \\frac{1}{3}) + \\left(\\frac{1}{2} \\times \\frac{1}{3}\\right) + \\left(1 \\times \\frac{1}{3}\\right)
]<\/p>\n\n\n\n
P(S) = 0 + \\frac{1}{6} + \\frac{1}{3} = \\frac{1}{6} + \\frac{2}{6} = \\frac{3}{6} = \\frac{1}{2}
]<\/p>\n\n\n\nStep 3: Apply Bayes’ Theorem<\/h3>\n\n\n\n
P(U_3 | S) = \\frac{P(S | U_3) P(U_3)}{P(S)}
]<\/p>\n\n\n\n
P(U_3 | S) = \\frac{1 \\times \\frac{1}{3}}{\\frac{1}{2}} = \\frac{\\frac{1}{3}}{\\frac{1}{2}} = \\frac{2}{3}
]<\/p>\n\n\n\nFinal Answer:<\/h3>\n\n\n\n
\\boxed{\\frac{2}{3}}
]<\/p>\n\n\n\nExplanation (Summary)<\/h3>\n\n\n\n
\n