{"id":196335,"date":"2025-03-04T17:12:29","date_gmt":"2025-03-04T17:12:29","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=196335"},"modified":"2025-03-04T17:12:34","modified_gmt":"2025-03-04T17:12:34","slug":"many-lottery-games-for-very-large-prizes-include-guessing-which-numbers-will-appear-from-a-random-sample-of-the-numbered-balls","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/04\/many-lottery-games-for-very-large-prizes-include-guessing-which-numbers-will-appear-from-a-random-sample-of-the-numbered-balls\/","title":{"rendered":"Many lottery games for very large prizes include guessing which numbers will appear from a random sample of the numbered balls"},"content":{"rendered":"\n<p>Many lottery games for very large prizes include guessing which numbers will appear from a random sample of the numbered balls. In Minnesota&#8217;s Gopher 5 lottery, there are five balls numbered 1-47. If a player matches at least two balls, she wins a prize (see table below). If the cost of playing is $ t least how large must the jackpot be before the expected return of playing Gopher 5 becomes favorable to the player? Assume there is only one winner, and also ignore the fact that winnings are also taxed. Please provide just a numerical value for your answer, do not include a dollar sign. MatchWin Odds 4 of 5 $500 $15 $1 5 of 5 Jackpot 1 in 1,533,939 1 in 7,304 1 in 178 1 in 13 3 of 53 2 of 5<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To determine the minimum jackpot size at which the expected return from playing Minnesota&#8217;s Gopher 5 lottery becomes favorable to the player, we need to calculate the expected value (EV) of a single ticket purchase and find the jackpot amount that makes this EV equal to or greater than the ticket price of $1.\ue206<\/p>\n\n\n\n<p><strong>Understanding the Prize Structure and Odds:<\/strong><\/p>\n\n\n\n<p>In Gopher 5, players select five numbers from 1 to 47. The prize tiers, corresponding odds, and prize amounts are as follows:\ue206<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Match 5 of 5 (Jackpot):<\/strong> Odds of 1 in 1,533,939.\ue206<\/li>\n\n\n\n<li><strong>Match 4 of 5:<\/strong> Odds of 1 in 7,304, with a prize of $500.\ue206<\/li>\n\n\n\n<li><strong>Match 3 of 5:<\/strong> Odds of 1 in 178, with a prize of $15.\ue206<\/li>\n\n\n\n<li><strong>Match 2 of 5:<\/strong> Odds of 1 in 13, with a prize of $1.\ue206<\/li>\n<\/ul>\n\n\n\n<p>\ue200cite\ue202turn0search0\ue201<\/p>\n\n\n\n<p><strong>Calculating the Expected Value (EV):<\/strong><\/p>\n\n\n\n<p>The EV is calculated by multiplying the probability of each outcome by its respective prize and summing these products:\ue206<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Jackpot Prize:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Probability: 1 \/ 1,533,939\ue206<\/li>\n\n\n\n<li>Prize: $J (the jackpot amount)\ue206<\/li>\n\n\n\n<li>Contribution to EV: (1 \/ 1,533,939) * J\ue206<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Match 4 of 5:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Probability: 1 \/ 7,304\ue206<\/li>\n\n\n\n<li>Prize: $500\ue206<\/li>\n\n\n\n<li>Contribution to EV: (1 \/ 7,304) * 500\ue206<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Match 3 of 5:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Probability: 1 \/ 178\ue206<\/li>\n\n\n\n<li>Prize: $15\ue206<\/li>\n\n\n\n<li>Contribution to EV: (1 \/ 178) * 15\ue206<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Match 2 of 5:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Probability: 1 \/ 13\ue206<\/li>\n\n\n\n<li>Prize: $1\ue206<\/li>\n\n\n\n<li>Contribution to EV: (1 \/ 13) * 1\ue206<\/li>\n<\/ul>\n\n\n\n<p>Summing these contributions:<\/p>\n\n\n\n<p>EV = (1 \/ 1,533,939) * J + (1 \/ 7,304) * 500 + (1 \/ 178) * 15 + (1 \/ 13) * 1\ue206<\/p>\n\n\n\n<p>For the expected return to be favorable, EV must be at least equal to the ticket price of $1:\ue206<\/p>\n\n\n\n<p>1 \u2264 (1 \/ 1,533,939) * J + (1 \/ 7,304) * 500 + (1 \/ 178) * 15 + (1 \/ 13) * 1\ue206<\/p>\n\n\n\n<p><strong>Solving for the Jackpot Amount (J):<\/strong><\/p>\n\n\n\n<p>First, calculate the fixed components:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(1 \/ 7,304) * 500 \u2248 0.0685\ue206<\/li>\n\n\n\n<li>(1 \/ 178) * 15 \u2248 0.0842\ue206<\/li>\n\n\n\n<li>(1 \/ 13) * 1 \u2248 0.0769\ue206<\/li>\n<\/ul>\n\n\n\n<p>Summing these:<\/p>\n\n\n\n<p>0.0685 + 0.0842 + 0.0769 \u2248 0.2296\ue206<\/p>\n\n\n\n<p>Now, the inequality becomes:<\/p>\n\n\n\n<p>1 \u2264 (1 \/ 1,533,939) * J + 0.2296\ue206<\/p>\n\n\n\n<p>Subtract 0.2296 from both sides:<\/p>\n\n\n\n<p>0.7704 \u2264 (1 \/ 1,533,939) * J\ue206<\/p>\n\n\n\n<p>Multiply both sides by 1,533,939 to solve for J:<\/p>\n\n\n\n<p>J \u2265 0.7704 * 1,533,939 \u2248 1,181,000\ue206<\/p>\n\n\n\n<p><strong>Conclusion:<\/strong><\/p>\n\n\n\n<p>Therefore, the jackpot must be at least <strong>1,181,000<\/strong> for the expected return from playing Gopher 5 to be favorable to the player. This means that if the jackpot is below this amount, the expected value of purchasing a ticket is less than the cost of the ticket, making it an unfavorable bet.\ue206<\/p>\n\n\n\n<p><em>Note: This calculation assumes there is only one winner and does not account for taxes or other potential deductions.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Many lottery games for very large prizes include guessing which numbers will appear from a random sample of the numbered balls. In Minnesota&#8217;s Gopher 5 lottery, there are five balls numbered 1-47. If a player matches at least two balls, she wins a prize (see table below). If the cost of playing is $ t [&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-196335","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196335","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=196335"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196335\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196335"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196335"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196335"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}