{"id":216818,"date":"2025-05-20T12:39:50","date_gmt":"2025-05-20T12:39:50","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=216818"},"modified":"2025-05-20T12:39:53","modified_gmt":"2025-05-20T12:39:53","slug":"what-is-the-probability-that-a-witness-will-refuses-to-serve-alcoholic-beverages-to-only-two-minor-if-she-randomly-check-id-of-five-students-from-among-nine-students-of-which-four-are-not-of-legal-a-8","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/20\/what-is-the-probability-that-a-witness-will-refuses-to-serve-alcoholic-beverages-to-only-two-minor-if-she-randomly-check-id-of-five-students-from-among-nine-students-of-which-four-are-not-of-legal-a-8\/","title":{"rendered":"What is the probability that a witness will refuses to serve alcoholic beverages to only two minor if she randomly check Id of five students from among nine students of which four are not of legal age"},"content":{"rendered":"\n

What is the probability that a witness will refuses to serve alcoholic beverages to only two minor if she randomly check Id of five students from among nine students of which four are not of legal age ?<\/p>\n\n\n\n

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

,<\/p>\n\n\n\n

To solve this probability problem, we use combinatorics<\/strong> and probability concepts<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

\ud83d\udd39 Problem Summary<\/strong><\/h3>\n\n\n\n
    \n
  • There are 9 students<\/strong> in total.<\/li>\n\n\n\n
  • 4 are minors<\/strong> (not of legal age).<\/li>\n\n\n\n
  • The witness checks IDs of 5 students<\/strong> at random<\/strong>.<\/li>\n\n\n\n
  • We want the probability that exactly 2 minors<\/strong> are among those 5 students checked.<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    \u2705 Correct Answer<\/strong>:<\/h3>\n\n\n\n

    P=(42)\u22c5(53)(95)=6\u22c510126=60126=1021P = \\frac{{\\binom{4}{2} \\cdot \\binom{5}{3}}}{{\\binom{9}{5}}} = \\frac{6 \\cdot 10}{126} = \\frac{60}{126} = \\frac{10}{21}<\/p>\n\n\n\n

    \n\n\n\n

    \ud83d\udd0d Step-by-Step Explanation (300+ words)<\/strong><\/h3>\n\n\n\n

    This is a classic hypergeometric probability<\/strong> problem. We are choosing a sample (of 5 students) from a population (of 9 students) without replacement, and we are interested in the probability of getting exactly 2 minors<\/strong> in this sample.<\/p>\n\n\n\n

    Step 1: Understand the groups<\/h4>\n\n\n\n
      \n
    • Total students: 9<\/li>\n\n\n\n
    • Minors (unqualified to drink): 4<\/li>\n\n\n\n
    • Adults (legal age): 9 \u2013 4 = 5<\/li>\n\n\n\n
    • Sample size: 5<\/li>\n\n\n\n
    • Desired outcome: exactly 2 minors in the sample<\/li>\n<\/ul>\n\n\n\n

      Step 2: Use the hypergeometric probability formula<\/strong>:<\/h4>\n\n\n\n

      P(X=k)=(Kk)\u22c5(N\u2212Kn\u2212k)(Nn)P(X = k) = \\frac{\\binom{K}{k} \\cdot \\binom{N – K}{n – k}}{\\binom{N}{n}}<\/p>\n\n\n\n

      Where:<\/p>\n\n\n\n

        \n
      • N=9N = 9: total population<\/li>\n\n\n\n
      • K=4K = 4: total minors (successes)<\/li>\n\n\n\n
      • n=5n = 5: number selected<\/li>\n\n\n\n
      • k=2k = 2: number of minors we want<\/li>\n<\/ul>\n\n\n\n

        P=(42)\u22c5(53)(95)P = \\frac{\\binom{4}{2} \\cdot \\binom{5}{3}}{\\binom{9}{5}}<\/p>\n\n\n\n

        Step 3: Calculate each combination<\/h4>\n\n\n\n
          \n
        • (42)=6\\binom{4}{2} = 6: Ways to choose 2 minors out of 4<\/li>\n\n\n\n
        • (53)=10\\binom{5}{3} = 10: Ways to choose 3 adults out of 5<\/li>\n\n\n\n
        • (95)=126\\binom{9}{5} = 126: Total ways to choose 5 students out of 9<\/li>\n<\/ul>\n\n\n\n

          P=6\u22c510126=60126=1021P = \\frac{6 \\cdot 10}{126} = \\frac{60}{126} = \\frac{10}{21}<\/p>\n\n\n\n

          Step 4: Interpret the result<\/h4>\n\n\n\n

          The probability that the witness checks the ID of exactly 2 minors<\/strong> (out of the 5 randomly selected students) is 10\/21<\/strong>, or approximately 47.62%<\/strong>.<\/p>\n\n\n\n

          This means that under these random conditions, there is nearly a 48% chance that exactly two minors are identified, helping the witness prevent them from being served alcohol.<\/p>\n\n\n\n

          \n\n\n\n

          \u2705 Final Answer:<\/h3>\n\n\n\n

          1021\\boxed{\\frac{10}{21}}<\/p>\n","protected":false},"excerpt":{"rendered":"

          What is the probability that a witness will refuses to serve alcoholic beverages to only two minor if she randomly check Id of five students from among nine students of which four are not of legal age ? The Correct Answer and Explanation is: , To solve this probability problem, we use combinatorics and probability […]<\/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-216818","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216818","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=216818"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/216818\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=216818"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=216818"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=216818"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}