{"id":204128,"date":"2025-03-21T14:56:19","date_gmt":"2025-03-21T14:56:19","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=204128"},"modified":"2025-03-21T14:56:21","modified_gmt":"2025-03-21T14:56:21","slug":"in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/21\/in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-3\/","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>To find the probability that a patient was unvaccinated given that they contracted the flu, we can use <strong>Bayes&#8217; Theorem<\/strong>. Bayes&#8217; Theorem helps us update the probability of a hypothesis (unvaccinated) based on new evidence (the patient contracted the flu). The formula for Bayes&#8217; Theorem is:<\/p>\n\n\n\n<p>[<br>P(\\text{Unvaccinated | Flu}) = \\frac{P(\\text{Flu | Unvaccinated}) \\cdot P(\\text{Unvaccinated})}{P(\\text{Flu})}<br>]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( P(\\text{Unvaccinated | Flu}) ) is the probability the patient is unvaccinated given they contracted the flu (what we want to find).<\/li>\n\n\n\n<li>( P(\\text{Flu | Unvaccinated}) ) is the probability that a patient contracts the flu given they are unvaccinated.<\/li>\n\n\n\n<li>( P(\\text{Unvaccinated}) ) is the probability that a patient is unvaccinated.<\/li>\n\n\n\n<li>( P(\\text{Flu}) ) is the total probability that a patient contracts the flu, regardless of vaccination status.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Identify the known probabilities<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( P(\\text{Vaccinated}) = 0.70 ) (70% of patients are vaccinated).<\/li>\n\n\n\n<li>( P(\\text{Unvaccinated}) = 1 &#8211; 0.70 = 0.30 ).<\/li>\n\n\n\n<li>Among vaccinated patients, 90% do not contract the flu, so 10% do. Hence, ( P(\\text{Flu | Vaccinated}) = 0.10 ).<\/li>\n\n\n\n<li>Among unvaccinated patients, 40% do not contract the flu, so 60% do. Hence, ( P(\\text{Flu | Unvaccinated}) = 0.60 ).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Calculate ( P(\\text{Flu}) )<\/h3>\n\n\n\n<p>The total probability of contracting the flu can be found by considering both vaccinated and unvaccinated groups:<\/p>\n\n\n\n<p>[<br>P(\\text{Flu}) = P(\\text{Flu | Vaccinated}) \\cdot P(\\text{Vaccinated}) + P(\\text{Flu | Unvaccinated}) \\cdot P(\\text{Unvaccinated})<br>]<\/p>\n\n\n\n<p>Substituting the values:<\/p>\n\n\n\n<p>[<br>P(\\text{Flu}) = (0.10 \\cdot 0.70) + (0.60 \\cdot 0.30) = 0.07 + 0.18 = 0.25<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Apply Bayes&#8217; Theorem<\/h3>\n\n\n\n<p>Now, we can plug the values into Bayes&#8217; Theorem:<\/p>\n\n\n\n<p>[<br>P(\\text{Unvaccinated | Flu}) = \\frac{0.60 \\cdot 0.30}{0.25} = \\frac{0.18}{0.25} = 0.72<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>The probability that a patient who contracted the flu was unvaccinated is <strong>0.72<\/strong>, or <strong>72%<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>This result shows that, given the flu, it is more likely that the patient was unvaccinated. This is because unvaccinated patients have a much higher probability of contracting the flu (60%) compared to vaccinated patients (10%). The prior probability of being unvaccinated (30%) combined with the higher likelihood of contracting the flu among unvaccinated individuals increases the conditional probability to 72%.<\/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-204128","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204128","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=204128"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/204128\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=204128"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=204128"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=204128"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}