{"id":206350,"date":"2025-03-27T19:34:47","date_gmt":"2025-03-27T19:34:47","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=206350"},"modified":"2025-03-27T19:34:50","modified_gmt":"2025-03-27T19:34:50","slug":"in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-4","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/27\/in-a-clinic-70-of-patients-are-vaccinated-against-the-flu-4\/","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<\/strong>?<\/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 was unvaccinated, we use Bayes’ Theorem<\/strong>.<\/p>\n\n\n\n

Step 1: Define Events<\/h3>\n\n\n\n
    \n
  • Let VV be the event that a patient is vaccinated.<\/li>\n\n\n\n
  • Let UU be the event that a patient is unvaccinated.<\/li>\n\n\n\n
  • Let FF be the event that a patient contracts the flu.<\/li>\n<\/ul>\n\n\n\n

    From the problem:<\/p>\n\n\n\n

      \n
    • P(V)=0.7P(V) = 0.7, so P(U)=1\u22120.7=0.3P(U) = 1 – 0.7 = 0.3.<\/li>\n\n\n\n
    • The probability of contracting the flu given vaccination:
      P(F\u2223V)=1\u22120.9=0.1P(F | V) = 1 – 0.9 = 0.1.<\/li>\n\n\n\n
    • The probability of contracting the flu given no vaccination:
      P(F\u2223U)=1\u22120.4=0.6P(F | U) = 1 – 0.4 = 0.6.<\/li>\n<\/ul>\n\n\n\n

      Step 2: Find the Total Probability of Contracting the Flu<\/h3>\n\n\n\n

      Using the Law of Total Probability<\/strong>: P(F)=P(F\u2223V)P(V)+P(F\u2223U)P(U)P(F) = P(F | V) P(V) + P(F | U) P(U) P(F)=(0.1\u00d70.7)+(0.6\u00d70.3)P(F) = (0.1 \\times 0.7) + (0.6 \\times 0.3) P(F)=0.07+0.18=0.25P(F) = 0.07 + 0.18 = 0.25<\/p>\n\n\n\n

      Step 3: Apply Bayes’ Theorem<\/h3>\n\n\n\n

      We need to find P(U\u2223F)P(U | F): P(U\u2223F)=P(F\u2223U)P(U)P(F)P(U | F) = \\frac{P(F | U) P(U)}{P(F)} P(U\u2223F)=(0.6\u00d70.3)0.25P(U | F) = \\frac{(0.6 \\times 0.3)}{0.25} P(U\u2223F)=0.180.25=0.72P(U | F) = \\frac{0.18}{0.25} = 0.72<\/p>\n\n\n\n

      Conclusion<\/h3>\n\n\n\n

      If a patient has the flu, the probability that they were unvaccinated is 0.72 (or 72%)<\/strong>.<\/p>\n\n\n\n

      \n\n\n\n

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

      This result shows that even though only 30% of the clinic\u2019s patients are unvaccinated, they make up a disproportionately high percentage of flu cases due to their higher infection rate. This highlights the effectiveness of vaccines in preventing the flu.<\/p>\n\n\n\n

      \n\n\n\n

      Now, I will generate an image representing this probability scenario.<\/p>\n\n\n\n

      Here is the infographic illustrating the probability scenario. It visually represents the vaccination rates, flu contraction probabilities, and the final result that 72% of flu cases come from unvaccinated patients. Let me know if you need any modifications!<\/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: To […]<\/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-206350","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/206350","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=206350"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/206350\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=206350"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=206350"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=206350"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}