{"id":219744,"date":"2025-05-26T18:23:35","date_gmt":"2025-05-26T18:23:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=219744"},"modified":"2025-05-26T18:23:37","modified_gmt":"2025-05-26T18:23:37","slug":"in-a-clinical-study-volunteers-are-tested-for-a-gene-that-has-been-found-to-increase-the-risk-for-a-disease","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/26\/in-a-clinical-study-volunteers-are-tested-for-a-gene-that-has-been-found-to-increase-the-risk-for-a-disease\/","title":{"rendered":"In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a disease."},"content":{"rendered":"\n

. In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a disease. The probability that a person carries the gene (independently of other people) is 0.1. What is the probability that we will have to test at least 7 people in order to find 4 people who carry the gene? (b) .00255 (c) .9987 (d) 1.458 x 10-6 (a) .9982 (e) .00346<\/p>\n\n\n\n

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

To solve this problem, we need to understand the Negative Binomial distribution<\/strong>, which models the number of trials needed to achieve a certain number of successes<\/strong> in independent Bernoulli trials.<\/p>\n\n\n\n

\n\n\n\n

Given:<\/strong><\/h3>\n\n\n\n
    \n
  • The probability that a person carries the gene (a \u201csuccess\u201d) is p=0.1p = 0.1.<\/li>\n\n\n\n
  • We are interested in finding 4<\/strong> people who carry the gene.<\/li>\n\n\n\n
  • We want to know the probability that we must test at least 7<\/strong> people to find these 4 carriers.<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    Restating the question:<\/strong><\/h3>\n\n\n\n

    Let XX be the number of people tested to find the 4th carrier<\/strong>.
    We want to compute: P(X\u22657)P(X \\geq 7)<\/p>\n\n\n\n

    This is the complement<\/strong> of the cumulative probability up to 6 people: P(X\u22657)=1\u2212P(X\u22646)P(X \\geq 7) = 1 – P(X \\leq 6)<\/p>\n\n\n\n

    Here, X\u223cNegative Binomial(r=4,p=0.1)X \\sim \\text{Negative Binomial}(r=4, p=0.1), which counts the number of trials until the 4th success<\/strong>.<\/p>\n\n\n\n

    We now calculate P(X\u22646)P(X \\leq 6), i.e., the probability that the 4th person who carries the gene is found on or before<\/strong> the 6th person.<\/p>\n\n\n\n

    Use the Negative Binomial probability mass function<\/strong>: P(X=x)=(x\u22121r\u22121)pr(1\u2212p)x\u2212rP(X = x) = \\binom{x-1}{r-1} p^r (1-p)^{x – r}<\/p>\n\n\n\n

    For r=4r = 4, we calculate for x=4,5,6x = 4, 5, 6:<\/p>\n\n\n\n

      \n
    • P(X=4)=(33)(0.1)4(0.9)0=1\u22c50.0001=0.0001P(X=4) = \\binom{3}{3} (0.1)^4 (0.9)^0 = 1 \\cdot 0.0001 = 0.0001<\/li>\n\n\n\n
    • P(X=5)=(43)(0.1)4(0.9)1=4\u22c50.0001\u22c50.9=0.00036P(X=5) = \\binom{4}{3} (0.1)^4 (0.9)^1 = 4 \\cdot 0.0001 \\cdot 0.9 = 0.00036<\/li>\n\n\n\n
    • P(X=6)=(53)(0.1)4(0.9)2=10\u22c50.0001\u22c50.81=0.00081P(X=6) = \\binom{5}{3} (0.1)^4 (0.9)^2 = 10 \\cdot 0.0001 \\cdot 0.81 = 0.00081<\/li>\n<\/ul>\n\n\n\n

      So, P(X\u22646)=0.0001+0.00036+0.00081=0.00127P(X \\leq 6) = 0.0001 + 0.00036 + 0.00081 = 0.00127 P(X\u22657)=1\u22120.00127=0.99873P(X \\geq 7) = 1 – 0.00127 = 0.99873<\/p>\n\n\n\n

      \n\n\n\n

      Correct answer: (c) 0.9987<\/strong><\/h3>\n\n\n\n
      \n\n\n\n

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

      This problem involves using the Negative Binomial distribution<\/strong>, which is appropriate when you want to determine the number of independent trials required to achieve a fixed number of successes. In this scenario, a \u201csuccess\u201d is defined as finding a person who carries a particular gene, and each person is tested independently with a 10% chance (0.1 probability) of being a carrier.<\/p>\n\n\n\n

      We are told to find the probability that at least 7 people must be tested to find the 4th<\/strong> carrier of the gene. That is, the first 3 carriers are found somewhere in the first 6 people, but the 4th has not yet appeared by the 6th trial. This means the 4th carrier appears in the 7th trial or later.<\/p>\n\n\n\n

      The Negative Binomial distribution gives us the probability that the r-th success<\/strong> occurs on the x-th<\/strong> trial. Here, r = 4 (we\u2019re looking for the 4th carrier), and we want the cumulative probability up to x = 6. We compute the probabilities of getting the 4th success on trials 4, 5, and 6 using the negative binomial probability mass function, and subtract their sum from 1 to get the probability of needing at least 7<\/strong> trials.<\/p>\n\n\n\n

      After computing these values, we find that the total probability of getting the 4th success within the first 6 trials is approximately 0.00127. Therefore, the chance that we will need to test 7 or more people<\/strong> is: 1\u22120.00127=0.998731 – 0.00127 = 0.99873<\/p>\n\n\n\n

      Thus, the correct answer is (c) 0.9987<\/strong>.<\/p>\n\n\n\n

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

      . In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a disease. The probability that a person carries the gene (independently of other people) is 0.1. What is the probability that we will have to test at least 7 people in order to find 4 […]<\/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-219744","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219744","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=219744"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219744\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219744"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219744"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219744"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}