{"id":217483,"date":"2025-05-22T03:57:33","date_gmt":"2025-05-22T03:57:33","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217483"},"modified":"2025-05-22T03:57:35","modified_gmt":"2025-05-22T03:57:35","slug":"in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-7","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/22\/in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-7\/","title":{"rendered":"In a clinic, 70% of patients are vaccinated against the flu"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\">If a patient is known to have contracted the flu, what is the probability that they were unvaccinated?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We are given the following probabilities:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>P(V)=0.70P(V) = 0.70: Probability that a patient is vaccinated.<\/li>\n\n\n\n<li>P(U)=0.30P(U) = 0.30: Probability that a patient is unvaccinated (1 \u2212 0.70).<\/li>\n\n\n\n<li>P(No\u00a0Flu\u2223V)=0.90P(\\text{No Flu} \\mid V) = 0.90 \u27f9 P(Flu\u2223V)=0.10P(\\text{Flu} \\mid V) = 0.10<\/li>\n\n\n\n<li>P(No\u00a0Flu\u2223U)=0.40P(\\text{No Flu} \\mid U) = 0.40 \u27f9 P(Flu\u2223U)=0.60P(\\text{Flu} \\mid U) = 0.60<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">We are asked to find:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\">P(U\u2223Flu)P(U \\mid \\text{Flu}): The probability that a patient was unvaccinated <strong>given<\/strong> they contracted the flu.<\/p>\n<\/blockquote>\n\n\n\n<p class=\"wp-block-paragraph\">This is a <strong>conditional probability<\/strong> problem. We will use <strong>Bayes&#8217; Theorem<\/strong>: P(U\u2223Flu)=P(Flu\u2223U)\u22c5P(U)P(Flu)P(U \\mid \\text{Flu}) = \\frac{P(\\text{Flu} \\mid U) \\cdot P(U)}{P(\\text{Flu})}<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We already have:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>P(Flu\u2223U)=0.60P(\\text{Flu} \\mid U) = 0.60<\/li>\n\n\n\n<li>P(U)=0.30P(U) = 0.30<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">We need to compute P(Flu)P(\\text{Flu}), the total probability that a patient contracts the flu. We use the <strong>Law of Total Probability<\/strong>: P(Flu)=P(Flu\u2223V)\u22c5P(V)+P(Flu\u2223U)\u22c5P(U)P(\\text{Flu}) = P(\\text{Flu} \\mid V) \\cdot P(V) + P(\\text{Flu} \\mid U) \\cdot P(U) P(Flu)=(0.10)(0.70)+(0.60)(0.30)=0.07+0.18=0.25P(\\text{Flu}) = (0.10)(0.70) + (0.60)(0.30) = 0.07 + 0.18 = 0.25<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now plug into Bayes&#8217; Theorem: P(U\u2223Flu)=(0.60)(0.30)0.25=0.180.25=0.72P(U \\mid \\text{Flu}) = \\frac{(0.60)(0.30)}{0.25} = \\frac{0.18}{0.25} = 0.72<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 Final Answer: <strong>0.72<\/strong> (or <strong>72%<\/strong>)<\/h3>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd0d Explanation (300+ words):<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">This problem is a classic application of <strong>Bayes\u2019 Theorem<\/strong>, which helps us reverse conditional probabilities \u2014 specifically, finding the probability that a person belongs to a particular group given an outcome. In this case, we want to know the chance that a flu-infected person was <strong>unvaccinated<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We start by organizing the given data into probabilities. We know that 70% of the clinic\u2019s patients are vaccinated, and only 10% of these vaccinated individuals contract the flu. Among the 30% who are unvaccinated, a much higher 60% get the flu. We are trying to determine, <em>if someone ends up getting the flu<\/em>, how likely it is that they were unvaccinated.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Bayes&#8217; Theorem uses the <strong>likelihood of the outcome for each group<\/strong>, along with the <strong>prior probabilities of being in each group<\/strong>, to calculate the <strong>posterior probability<\/strong> \u2014 the probability of the group given the outcome.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The total probability of getting the flu is found by summing the contributions from both groups:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>70% are vaccinated, with 10% flu rate \u21d2 0.07<\/li>\n\n\n\n<li>30% are unvaccinated, with 60% flu rate \u21d2 0.18<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">So 25% of all patients get the flu. Of these flu cases, 0.18 come from the unvaccinated group. Dividing 0.18 by the total 0.25 gives 0.72 \u2014 meaning <strong>72% of all flu cases come from the unvaccinated group<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This result emphasizes how effective the flu vaccine is. Even though only 30% of people are unvaccinated, they make up <strong>the vast majority of flu cases<\/strong> \u2014 demonstrating a key concept in public health: vaccines greatly reduce disease burden, not just in individuals but in populations.<\/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-217483","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217483","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=217483"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217483\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217483"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217483"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217483"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}