The probability a gambler will win money on a certain slot machine is 2.5%. Determine the probability that a gambler will win at least 5 times if they play 100 times.<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n To solve this problem, we are dealing with a binomial distribution<\/strong>. In a binomial distribution, there are a fixed number of independent trials (in this case, 100 plays), each with two possible outcomes (win or not win), and a constant probability of success (win) in each trial.<\/p>\n\n\n\n We have the following parameters:<\/p>\n\n\n\n The number of wins $X$ follows a binomial distribution<\/strong>:<\/p>\n\n\n\n $$ The probability mass function (PMF) for a binomial distribution is given by:<\/p>\n\n\n\n $$ where $\\binom{n}{k}$ is the binomial coefficient, and $k$ is the number of successes (wins).<\/p>\n\n\n\n The problem asks for $P(X \\geq 5)$, which is the probability of having 5 or more wins. We can calculate this by finding the complement, which is the probability of having fewer than 5 wins, and subtracting it from 1:<\/p>\n\n\n\n $$ This requires calculating the individual probabilities for $P(X = 0), P(X = 1), P(X = 2), P(X = 3),$ and $P(X = 4)$, then summing them up and subtracting from 1.<\/p>\n\n\n\n For large $n$, a binomial distribution can be approximated by a normal distribution. Since $n$ is large, we can use the normal approximation to the binomial distribution:<\/p>\n\n\n\n $$ where:<\/p>\n\n\n\n Using the normal approximation, we convert the discrete probability to a continuous probability using a continuity correction. We want $P(X \\geq 5)$, which is equivalent to $P(X \\geq 4.5)$ in the normal distribution.<\/p>\n\n\n\n We can standardize this to a z-score:<\/p>\n\n\n\n $$ Using standard normal tables or a calculator, we find the probability corresponding to a z-score of 1.28, which is approximately 0.8997.<\/p>\n\n\n\n Thus:<\/p>\n\n\n\n $$ The probability that a gambler will win at least 5 times if they play 100 times is approximately 0.1003<\/strong>, or 10.03%<\/strong>. This means there is about a 10% chance the gambler will win at least 5 times after 100 plays, based on the given probability of winning in each play.<\/p>\n","protected":false},"excerpt":{"rendered":" The probability a gambler will win money on a certain slot machine is 2.5%. Determine the probability that a gambler will win at least 5 times if they play 100 times. The correct answer and explanation is : To solve this problem, we are dealing with a binomial distribution. In a binomial distribution, there are […]<\/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-211327","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/211327","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=211327"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/211327\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=211327"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=211327"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=211327"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Defining the parameters of the binomial distribution<\/h3>\n\n\n\n
\n
Step 2: Applying the binomial distribution<\/h3>\n\n\n\n
X \\sim B(n=100, p=0.025)
$$<\/p>\n\n\n\n
P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k}
$$<\/p>\n\n\n\nStep 3: Calculating the probability of winning at least 5 times<\/h3>\n\n\n\n
P(X \\geq 5) = 1 – P(X < 5) = 1 – (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4))
$$<\/p>\n\n\n\nStep 4: Using a normal approximation<\/h3>\n\n\n\n
X \\sim N(\\mu = np, \\sigma^2 = np(1-p))
$$<\/p>\n\n\n\n\n
z = \\frac{X – \\mu}{\\sigma} = \\frac{4.5 – 2.5}{1.56} \\approx 1.28
$$<\/p>\n\n\n\n
P(X \\geq 5) \\approx 1 – 0.8997 = 0.1003
$$<\/p>\n\n\n\nStep 5: Conclusion<\/h3>\n\n\n\n