{"id":217207,"date":"2025-05-21T07:30:16","date_gmt":"2025-05-21T07:30:16","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217207"},"modified":"2025-05-21T07:30:18","modified_gmt":"2025-05-21T07:30:18","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-11","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/21\/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-11\/","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<p>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<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let&#8217;s carefully analyze the problem step-by-step.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Problem Restatement:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>There are <strong>9 students total<\/strong>.<\/li>\n\n\n\n<li>Among these, <strong>4 are minors (not of legal age)<\/strong>.<\/li>\n\n\n\n<li>A witness <strong>randomly checks IDs of 5 students<\/strong> from the 9.<\/li>\n\n\n\n<li>We want the <strong>probability that the witness refuses to serve alcoholic beverages to exactly 2 minors<\/strong> among those 5 checked.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Key points:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Refusing service means identifying minors.<\/li>\n\n\n\n<li>We want exactly 2 minors in the sample of 5.<\/li>\n\n\n\n<li>Total students = 9.<\/li>\n\n\n\n<li>Minors = 4.<\/li>\n\n\n\n<li>Legal age students = 9 &#8211; 4 = 5.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Approach:<\/h3>\n\n\n\n<p>This is a classic <strong>hypergeometric probability problem<\/strong>.<\/p>\n\n\n\n<p><strong>Hypergeometric distribution<\/strong> describes the probability of k successes (minors identified) in n draws (students checked), without replacement, from a finite population of size N that contains K successes.<\/p>\n\n\n\n<p>Parameters:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Population size, N=9N = 9<\/li>\n\n\n\n<li>Number of successes in population, K=4K = 4 (minors)<\/li>\n\n\n\n<li>Number of draws, n=5n = 5<\/li>\n\n\n\n<li>Number of observed successes, k=2k = 2<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Hypergeometric Probability Formula:<\/h3>\n\n\n\n<p>P(X=k)=(Kk)\u00d7(N\u2212Kn\u2212k)(Nn)P(X = k) = \\frac{\\binom{K}{k} \\times \\binom{N-K}{n-k}}{\\binom{N}{n}}<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(Kk)\\binom{K}{k} = ways to choose k minors from 4 minors.<\/li>\n\n\n\n<li>(N\u2212Kn\u2212k)\\binom{N-K}{n-k} = ways to choose remaining students who are not minors (legal age) from the 5 legal age students.<\/li>\n\n\n\n<li>(Nn)\\binom{N}{n} = total ways to choose any 5 students from 9 students.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Calculate each term:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(42)=4!2!\u00d72!=244=6\\binom{4}{2} = \\frac{4!}{2! \\times 2!} = \\frac{24}{4} = 6<\/li>\n\n\n\n<li>(53)=5!3!\u00d72!=1206\u00d72=10\\binom{5}{3} = \\frac{5!}{3! \\times 2!} = \\frac{120}{6 \\times 2} = 10<\/li>\n\n\n\n<li>(95)=9!5!\u00d74!=362,880120\u00d724=126\\binom{9}{5} = \\frac{9!}{5! \\times 4!} = \\frac{362,880}{120 \\times 24} = 126<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Substitute into formula:<\/h3>\n\n\n\n<p>P(X=2)=6\u00d710126=60126=1021\u22480.4762P(X=2) = \\frac{6 \\times 10}{126} = \\frac{60}{126} = \\frac{10}{21} \\approx 0.4762<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p><strong>The probability that exactly 2 minors are checked (and thus refused) among the 5 students randomly selected is 1021\\boxed{\\frac{10}{21}} or approximately 0.4762 (47.62%).<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation (300 words):<\/h3>\n\n\n\n<p>This problem involves the hypergeometric distribution, which applies when sampling is done <strong>without replacement<\/strong> from a finite population that has a mix of two types of elements\u2014in this case, minors (underage students) and legal-age students.<\/p>\n\n\n\n<p>Here, the total number of students is 9, with 4 minors and 5 legal-age students. The witness randomly checks 5 students from these 9, and we want to know the probability that <strong>exactly 2 of the 5 checked students are minors<\/strong>.<\/p>\n\n\n\n<p>To solve this, we calculate how many ways the witness can pick exactly 2 minors from the 4 minors available, and at the same time pick the remaining 3 students from the 5 legal-age students. The combination formula (nk)\\binom{n}{k} tells us the number of ways to select k elements from n elements without regard to order.<\/p>\n\n\n\n<p>The number of ways to pick exactly 2 minors from 4 is (42)=6\\binom{4}{2} = 6. The number of ways to pick the remaining 3 students from the 5 legal-age students is (53)=10\\binom{5}{3} = 10. Thus, the number of favorable outcomes (exactly 2 minors among the 5 checked) is 6\u00d710=606 \\times 10 = 60.<\/p>\n\n\n\n<p>The total number of ways to choose any 5 students from 9 is (95)=126\\binom{9}{5} = 126. Hence, the probability is the ratio of favorable outcomes to total outcomes: P=60126=1021\u22480.4762P = \\frac{60}{126} = \\frac{10}{21} \\approx 0.4762<\/p>\n\n\n\n<p>This means that if the witness randomly selects 5 students, there is about a 47.6% chance that exactly 2 of them will be minors who should be refused alcohol. This problem is a straightforward application of hypergeometric probability, which is useful whenever sampling is done without replacement and categories are involved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>If you want, I can also help you understand this concept visually or with other examples!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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: Let&#8217;s carefully analyze the problem step-by-step. Problem Restatement: Key points: Approach: [&hellip;]<\/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-217207","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217207","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=217207"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217207\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217207"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217207"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}