{"id":194517,"date":"2025-02-24T08:31:35","date_gmt":"2025-02-24T08:31:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=194517"},"modified":"2025-02-24T08:31:38","modified_gmt":"2025-02-24T08:31:38","slug":"the-massachusetts-state-lottery-game-cash-winfall-used-to-have-a-way-that-anyone-with-enough-money-and-time-could-stand-a-good-chance-of-getting-rich-and-it-is-reported-that-an-mit-computer-scienti","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/24\/the-massachusetts-state-lottery-game-cash-winfall-used-to-have-a-way-that-anyone-with-enough-money-and-time-could-stand-a-good-chance-of-getting-rich-and-it-is-reported-that-an-mit-computer-scienti\/","title":{"rendered":"The Massachusetts state lottery game, Cash WinFall, used to have a way that anyone with enough money and time could stand a good chance of getting rich, and it is reported that an MIT computer scientist did just that"},"content":{"rendered":"\n

The Massachusetts state lottery game, Cash WinFall, used to have a way that anyone with enough money and time could stand a good chance of getting rich, and it is reported that an MIT computer scientist did just that. In this game, a player picks 6 numbers from the range from 1 to 46. If he matches all 6, then he could win as much as $2 million, but the odds of that payout don’t justify a bet, so let us ignore the possibility of winning this jackpot. Nevertheless, there were times when matching just 5 of the 6 numbers in a $2 lottery ticket would pay $100,000. Suppose in this scenario that you were able to bet $600,000.<\/p>\n\n\n\n

(a) What is the expected amount that you would win?<\/p>\n\n\n\n

(b) Derive a bound on the probability that you would lose $300,000 or more in this scenario, that is, that you would have 3 or fewer of the 5 of the 6 winning tickets.<\/p>\n\n\n\n

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

Let’s break this down carefully.<\/p>\n\n\n\n

(a) Expected Amount Won<\/strong><\/h3>\n\n\n\n

Each Cash WinFall ticket costs $2<\/strong>, and you are betting $600,000<\/strong>, meaning you are purchasing:<\/p>\n\n\n\n

[
\\frac{600,000}{2} = 300,000
]<\/p>\n\n\n\n

tickets. You win $100,000<\/strong> if you match 5 out of 6 numbers<\/strong>. The probability of this happening can be computed using combinatorial counting:<\/p>\n\n\n\n

[
P(\\text{5 out of 6}) = \\frac{\\binom{5}{5} \\binom{40}{1}}{\\binom{46}{6}}
]<\/p>\n\n\n\n

[
= \\frac{1 \\times 40}{9,366,819} = \\frac{40}{9,366,819} \\approx 4.27 \\times 10^{-6}
]<\/p>\n\n\n\n

The expected number of winning tickets is:<\/p>\n\n\n\n

[
300,000 \\times 4.27 \\times 10^{-6} = 1.281
]<\/p>\n\n\n\n

Each such winning ticket earns $100,000<\/strong>, so the expected winnings are:<\/p>\n\n\n\n

[
1.281 \\times 100,000 = 128,100
]<\/p>\n\n\n\n

Thus, the expected amount won is $128,100<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

(b) Probability of Losing $300,000 or More<\/strong><\/h3>\n\n\n\n

To lose at least $300,000<\/strong>, you must win 3 or fewer<\/strong> of these 5 out of 6 matches<\/strong>.<\/p>\n\n\n\n

Let ( X ) be the number of such wins. Since we are playing 300,000<\/strong> independent trials with probability ( p = 4.27 \\times 10^{-6} )<\/strong> per ticket, ( X ) follows a Binomial distribution<\/strong>:<\/p>\n\n\n\n

[
X \\sim \\text{Binomial}(300,000, 4.27 \\times 10^{-6})
]<\/p>\n\n\n\n

By normal approximation:<\/p>\n\n\n\n

[
\\mu = 1.281, \\quad \\sigma^2 = 300,000 \\times p \\times (1 – p) \\approx 1.280
]<\/p>\n\n\n\n

Thus, ( \\sigma \\approx 1.13 ). Using the normal approximation, we compute:<\/p>\n\n\n\n

[
P(X \\leq 3) \\approx P\\left( \\frac{3 – 1.281}{1.13} \\right) = P(Z \\leq 1.52)
]<\/p>\n\n\n\n

Using the standard normal table:<\/p>\n\n\n\n

[
P(Z \\leq 1.52) \\approx 0.935
]<\/p>\n\n\n\n

Thus, there is about a 93.5% chance<\/strong> that you will lose $300,000 or more<\/strong> in this scenario.<\/p>\n\n\n\n

Now, let’s generate an image illustrating the probability distribution of winnings.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The image illustrates the probability distribution of the number of $100,000 winning tickets in a Cash WinFall scenario where 300,000 tickets are purchased. The binomial probability distribution (blue bars) is approximated by a normal curve (red dashed line). The green dotted line represents the threshold of 3 winning tickets, beyond which a loss of $300,000 or more occurs. As seen, the probability mass is concentrated around 1-2 tickets, aligning with our calculated 93.5% chance of significant loss.<\/p>\n","protected":false},"excerpt":{"rendered":"

The Massachusetts state lottery game, Cash WinFall, used to have a way that anyone with enough money and time could stand a good chance of getting rich, and it is reported that an MIT computer scientist did just that. In this game, a player picks 6 numbers from the range from 1 to 46. If […]<\/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-194517","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194517","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=194517"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194517\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=194517"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=194517"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=194517"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}