{"id":219342,"date":"2025-05-26T04:09:02","date_gmt":"2025-05-26T04:09:02","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=219342"},"modified":"2025-05-26T04:09:05","modified_gmt":"2025-05-26T04:09:05","slug":"in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-8","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/26\/in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-8\/","title":{"rendered":"In a clinic, 70% of patients are vaccinated against the flu."},"content":{"rendered":"\n

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

If a patient is known to have contracted the flu, what is the probability that they were unvaccinated?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To solve this problem, we use Bayes\u2019 Theorem<\/strong>.<\/p>\n\n\n\n

Given:<\/h3>\n\n\n\n
    \n
  • P(V)=0.7P(V) = 0.7: Probability a patient is vaccinated.<\/li>\n\n\n\n
  • P(U)=0.3P(U) = 0.3: Probability a patient is unvaccinated (1 – 0.7).<\/li>\n\n\n\n
  • P(No\u00a0Flu\u2223V)=0.9P(\\text{No Flu} | V) = 0.9: Probability a vaccinated patient does not<\/strong> get the flu.
    \u2192 So, P(Flu\u2223V)=1\u22120.9=0.1P(\\text{Flu} | V) = 1 – 0.9 = 0.1.<\/li>\n\n\n\n
  • P(No\u00a0Flu\u2223U)=0.4P(\\text{No Flu} | U) = 0.4: Probability an unvaccinated patient does not<\/strong> get the flu.
    \u2192 So, P(Flu\u2223U)=1\u22120.4=0.6P(\\text{Flu} | U) = 1 – 0.4 = 0.6.<\/li>\n<\/ul>\n\n\n\n

    We are asked to find:<\/p>\n\n\n\n

    \n

    What is the probability that a patient was unvaccinated given that they have contracted the flu?<\/strong><\/p>\n<\/blockquote>\n\n\n\n

    This is P(U\u2223Flu)P(U | \\text{Flu}), and we use Bayes’ Theorem: P(U\u2223Flu)=P(Flu\u2223U)\u22c5P(U)P(Flu)P(U | \\text{Flu}) = \\frac{P(\\text{Flu} | U) \\cdot P(U)}{P(\\text{Flu})}<\/p>\n\n\n\n

    First, compute the total probability that a patient gets the flu: P(Flu)=P(Flu\u2223V)\u22c5P(V)+P(Flu\u2223U)\u22c5P(U)=(0.1)(0.7)+(0.6)(0.3)=0.07+0.18=0.25P(\\text{Flu}) = P(\\text{Flu} | V) \\cdot P(V) + P(\\text{Flu} | U) \\cdot P(U) = (0.1)(0.7) + (0.6)(0.3) = 0.07 + 0.18 = 0.25<\/p>\n\n\n\n

    Now apply Bayes’ Theorem: P(U\u2223Flu)=(0.6)(0.3)0.25=0.180.25=0.72P(U | \\text{Flu}) = \\frac{(0.6)(0.3)}{0.25} = \\frac{0.18}{0.25} = 0.72<\/p>\n\n\n\n

    \n\n\n\n

    Answer:<\/strong><\/h3>\n\n\n\n

    0.72\\boxed{0.72}<\/p>\n\n\n\n

    \n\n\n\n

    Explanation <\/strong><\/h3>\n\n\n\n

    This problem involves conditional probability, where we are trying to determine the likelihood that a patient was unvaccinated given<\/strong> that they contracted the flu<\/strong>. Bayes\u2019 Theorem is the standard approach in such scenarios.<\/p>\n\n\n\n

    Bayes\u2019 Theorem allows us to “reverse” conditional probabilities. Instead of looking at the probability of getting the flu given<\/strong> vaccination status (which we\u2019re given), we want to find the probability of being unvaccinated<\/strong> given a flu diagnosis.<\/p>\n\n\n\n

    The clinic data tells us that vaccinated patients are far less likely to get the flu\u2014only 10% do. In contrast, 60% of unvaccinated patients catch the flu. While only 30% of patients are unvaccinated, they represent a disproportionate number of flu cases.<\/p>\n\n\n\n

    To find the probability that a flu patient is unvaccinated, we calculate:<\/p>\n\n\n\n

      \n
    1. The overall flu rate (from both vaccinated and unvaccinated patients).<\/li>\n\n\n\n
    2. How much of that flu rate is specifically from unvaccinated individuals.<\/li>\n\n\n\n
    3. The proportion of flu cases that come from unvaccinated patients.<\/li>\n<\/ol>\n\n\n\n

      By Bayes\u2019 Theorem, this gives us 72%. This means that if a patient shows up with the flu, there\u2019s a 72% chance<\/strong> they weren\u2019t vaccinated. This insight highlights the vaccine’s effectiveness\u2014though a minority are unvaccinated, they make up most flu cases.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      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: […]<\/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-219342","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219342","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=219342"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219342\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219342"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219342"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219342"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}