{"id":216492,"date":"2025-05-19T10:58:20","date_gmt":"2025-05-19T10:58:20","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=216492"},"modified":"2025-05-19T10:58:23","modified_gmt":"2025-05-19T10:58:23","slug":"in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-6","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/19\/in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-6\/","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>, which helps us find the probability of a condition (unvaccinated) given an outcome (contracted the flu).<\/p>\n\n\n\n

\n\n\n\n

Given Data:<\/strong><\/h3>\n\n\n\n
    \n
  • P(V)=0.70P(V) = 0.70: Probability a patient is vaccinated<\/li>\n\n\n\n
  • P(U)=0.30P(U) = 0.30: Probability a patient is unvaccinated<\/li>\n\n\n\n
  • P(No\u00a0Flu\u2223V)=0.90P(\\text{No Flu} | V) = 0.90 \u21d2 P(Flu\u2223V)=0.10P(\\text{Flu} | V) = 0.10<\/li>\n\n\n\n
  • P(No\u00a0Flu\u2223U)=0.40P(\\text{No Flu} | U) = 0.40 \u21d2 P(Flu\u2223U)=0.60P(\\text{Flu} | U) = 0.60<\/li>\n<\/ul>\n\n\n\n

    We are asked:
    What is P(U\u2223Flu)P(U | \\text{Flu})?<\/strong>
    This is the probability a patient is unvaccinated<\/strong> given that they contracted the flu<\/strong>.<\/p>\n\n\n\n

    \n\n\n\n

    Step 1: Apply Bayes\u2019 Theorem<\/strong><\/h3>\n\n\n\n

    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

    We already know:<\/p>\n\n\n\n

      \n
    • P(Flu\u2223U)=0.60P(\\text{Flu} | U) = 0.60<\/li>\n\n\n\n
    • P(U)=0.30P(U) = 0.30<\/li>\n<\/ul>\n\n\n\n

      We need to compute the total probability of contracting the flu<\/strong>, P(Flu)P(\\text{Flu}), using the Law of Total Probability<\/strong>: P(Flu)=P(Flu\u2223V)\u22c5P(V)+P(Flu\u2223U)\u22c5P(U)P(\\text{Flu}) = P(\\text{Flu} | V) \\cdot P(V) + P(\\text{Flu} | U) \\cdot P(U) P(Flu)=(0.10\u22c50.70)+(0.60\u22c50.30)=0.07+0.18=0.25P(\\text{Flu}) = (0.10 \\cdot 0.70) + (0.60 \\cdot 0.30) = 0.07 + 0.18 = 0.25<\/p>\n\n\n\n

      \n\n\n\n

      Step 2: Plug into Bayes\u2019 Theorem<\/strong><\/h3>\n\n\n\n

      P(U\u2223Flu)=0.60\u22c50.300.25=0.180.25=0.72P(U | \\text{Flu}) = \\frac{0.60 \\cdot 0.30}{0.25} = \\frac{0.18}{0.25} = 0.72<\/p>\n\n\n\n

      \n\n\n\n

      \u2705 Final 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 means that if a patient is known to have contracted the flu, there is a 72% chance<\/strong> that they were unvaccinated<\/strong>. Even though only 30% of the clinic population is unvaccinated, they represent a disproportionately large part of flu cases because their flu risk (60%) is much higher than that of vaccinated individuals (10%). This reflects how effective vaccination is in preventing flu and how Bayes\u2019 Theorem allows us to reverse conditions (from outcome to cause).<\/p>\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-216492","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216492","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=216492"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216492\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=216492"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=216492"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=216492"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}