{"id":215371,"date":"2025-05-16T19:31:36","date_gmt":"2025-05-16T19:31:36","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=215371"},"modified":"2025-05-16T19:31:38","modified_gmt":"2025-05-16T19:31:38","slug":"in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/16\/in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-5\/","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 find the probability that a patient who contracted the flu<\/strong> was unvaccinated<\/strong>, we can use Bayes\u2019 Theorem<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Step 1: Define the Events<\/strong><\/h3>\n\n\n\n

Let\u2019s define:<\/p>\n\n\n\n

    \n
  • VV: Patient is vaccinated<\/strong><\/li>\n\n\n\n
  • UU: Patient is unvaccinated<\/strong><\/li>\n\n\n\n
  • FF: Patient contracted the flu<\/strong><\/li>\n\n\n\n
  • FcF^c: Patient did not contract the flu<\/strong><\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    Step 2: Given Information<\/strong><\/h3>\n\n\n\n
      \n
    • P(V)=0.70\u21d2P(U)=1\u22120.70=0.30P(V) = 0.70 \\Rightarrow P(U) = 1 – 0.70 = 0.30<\/li>\n\n\n\n
    • P(Fc\u2223V)=0.90\u21d2P(F\u2223V)=1\u22120.90=0.10P(F^c|V) = 0.90 \\Rightarrow P(F|V) = 1 – 0.90 = 0.10<\/li>\n\n\n\n
    • P(Fc\u2223U)=0.40\u21d2P(F\u2223U)=1\u22120.40=0.60P(F^c|U) = 0.40 \\Rightarrow P(F|U) = 1 – 0.40 = 0.60<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

      Step 3: Use the Law of Total Probability to Find P(F)P(F)<\/strong><\/h3>\n\n\n\n

      P(F)=P(F\u2223V)\u22c5P(V)+P(F\u2223U)\u22c5P(U)P(F) = P(F|V) \\cdot P(V) + P(F|U) \\cdot P(U) P(F)=(0.10\u22c50.70)+(0.60\u22c50.30)=0.07+0.18=0.25P(F) = (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 4: Use Bayes\u2019 Theorem to Find P(U\u2223F)P(U|F)<\/strong><\/h3>\n\n\n\n

      P(U\u2223F)=P(F\u2223U)\u22c5P(U)P(F)=0.60\u22c50.300.25=0.180.25=0.72P(U|F) = \\frac{P(F|U) \\cdot P(U)}{P(F)} = \\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: 0.72\\boxed{0.72} or 72%<\/strong><\/h3>\n\n\n\n
      \n\n\n\n

      Explanation (300+ Words)<\/strong><\/h3>\n\n\n\n

      This problem is a classic application of Bayes\u2019 Theorem<\/strong>, which allows us to reverse conditional probabilities \u2014 in this case, determining the likelihood that a patient was unvaccinated<\/strong>, given that they contracted the flu<\/strong>.<\/p>\n\n\n\n

      We begin by breaking the population into two groups: vaccinated (70%)<\/strong> and unvaccinated (30%)<\/strong>. Among those vaccinated, only 10%<\/strong> contract the flu, thanks to the vaccine’s effectiveness. Conversely, 60%<\/strong> of unvaccinated individuals get the flu, highlighting the higher risk in this group.<\/p>\n\n\n\n

      To find the overall chance<\/strong> that any patient gets the flu, we use the law of total probability<\/strong>. This combines the probabilities of flu in both groups, weighted by how common each group is. We calculate: P(F)=0.07(vaccinatedflucases)+0.18(unvaccinatedflucases)=0.25P(F) = 0.07 (vaccinated flu cases) + 0.18 (unvaccinated flu cases) = 0.25<\/p>\n\n\n\n

      This means 25% of all patients<\/strong> contract the flu.<\/p>\n\n\n\n

      Now, to find the likelihood that a flu patient was unvaccinated<\/strong>, we apply Bayes\u2019 Theorem<\/strong>: P(U\u2223F)=P(F\u2223U)\u22c5P(U)P(F)=0.180.25=0.72P(U|F) = \\frac{P(F|U) \\cdot P(U)}{P(F)} = \\frac{0.18}{0.25} = 0.72<\/p>\n\n\n\n

      So, there is a 72% chance<\/strong> that a patient who contracted the flu was unvaccinated.<\/p>\n\n\n\n

      This result makes intuitive sense. Even though unvaccinated patients are the minority (30%)<\/strong>, they account for a disproportionate number<\/strong> of flu cases (60% chance of infection), so they form the majority (72%)<\/strong> of those who end up with the flu. This highlights the protective effect of vaccination<\/strong> and supports the importance of increasing vaccine coverage to reduce overall flu incidence.<\/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-215371","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215371","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=215371"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215371\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=215371"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=215371"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=215371"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}