{"id":201056,"date":"2025-03-14T18:29:28","date_gmt":"2025-03-14T18:29:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=201056"},"modified":"2025-03-14T18:29:30","modified_gmt":"2025-03-14T18:29:30","slug":"in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/14\/in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-2\/","title":{"rendered":"In a clinic, 70% of patients are vaccinated against the flu"},"content":{"rendered":"\n<p>In a clinic, 70% of patients are vaccinated against the flu. Among vaccinated patients, 90% do not contract the flu, while among unvaccinated patients, only 40% do not contract the flu.<\/p>\n\n\n\n<p>If a patient is known to have contracted the flu, what is the probability that they were unvaccinated?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>We can use <strong>Bayes&#8217; Theorem<\/strong> to solve this problem. Let&#8217;s define the events:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( V ) = Patient is <strong>vaccinated<\/strong>.<\/li>\n\n\n\n<li>( U ) = Patient is <strong>unvaccinated<\/strong>.<\/li>\n\n\n\n<li>( F ) = Patient <strong>contracts the flu<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>From the problem statement:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( P(V) = 0.7 ), meaning 70% of patients are vaccinated.<\/li>\n\n\n\n<li>( P(U) = 1 &#8211; P(V) = 0.3 ), meaning 30% are unvaccinated.<\/li>\n\n\n\n<li>( P(F | V) = 1 &#8211; 0.9 = 0.1 ), meaning 10% of vaccinated patients contract the flu.<\/li>\n\n\n\n<li>( P(F | U) = 1 &#8211; 0.4 = 0.6 ), meaning 60% of unvaccinated patients contract the flu.<\/li>\n<\/ul>\n\n\n\n<p>We need to find <strong>( P(U | F) )<\/strong>, the probability that a patient who contracted the flu was unvaccinated. Using <strong>Bayes&#8217; Theorem<\/strong>:<\/p>\n\n\n\n<p>[<br>P(U | F) = \\frac{P(F | U) P(U)}{P(F)}<br>]<\/p>\n\n\n\n<p>First, calculate ( P(F) ), the total probability that a patient contracts the flu, using the <strong>Law of Total Probability<\/strong>:<\/p>\n\n\n\n<p>[<br>P(F) = P(F | V) P(V) + P(F | U) P(U)<br>]<\/p>\n\n\n\n<p>[<br>P(F) = (0.1 \\times 0.7) + (0.6 \\times 0.3)<br>]<\/p>\n\n\n\n<p>[<br>P(F) = 0.07 + 0.18 = 0.25<br>]<\/p>\n\n\n\n<p>Now, compute ( P(U | F) ):<\/p>\n\n\n\n<p>[<br>P(U | F) = \\frac{(0.6 \\times 0.3)}{0.25}<br>]<\/p>\n\n\n\n<p>[<br>P(U | F) = \\frac{0.18}{0.25} = 0.72<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>The probability that a patient who contracted the flu was unvaccinated is <strong>0.72 (or 72%)<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The calculation is based on <strong>Bayes&#8217; Theorem<\/strong>, which allows us to update probabilities based on given evidence. While only 30% of patients are unvaccinated, they are much more likely to contract the flu (60% vs. 10%). Since more flu cases come from unvaccinated patients, the likelihood that a flu patient was unvaccinated is <strong>72%<\/strong>, even though they are a minority in the total population.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In a clinic, 70% of patients are vaccinated against the flu. Among vaccinated patients, 90% do not contract the flu, while among unvaccinated patients, only 40% do not contract the flu. If a patient is known to have contracted the flu, what is the probability that they were unvaccinated? The correct answer and explanation is [&hellip;]<\/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-201056","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201056","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=201056"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201056\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=201056"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=201056"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=201056"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}