{"id":222342,"date":"2025-05-31T08:54:22","date_gmt":"2025-05-31T08:54:22","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=222342"},"modified":"2025-05-31T08:54:25","modified_gmt":"2025-05-31T08:54:25","slug":"find-the-probability-that-atmost-5-defective-fuses-will-be-found-in-a-box-of-200-fuses-if-experience-shows-that-2-percent-of-such-fuses-are-defective-e-4-0-0183","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/31\/find-the-probability-that-atmost-5-defective-fuses-will-be-found-in-a-box-of-200-fuses-if-experience-shows-that-2-percent-of-such-fuses-are-defective-e-4-0-0183\/","title":{"rendered":"Find the probability that atmost 5 defective fuses will be found in a box of 200 fuses if experience shows that 2 percent of such fuses are defective (e^-4= 0.0183)"},"content":{"rendered":"\n<p>Find the probability that atmost 5 defective fuses will be found in a box of 200 fuses if experience shows that 2 percent of such fuses are defective (e^-4= 0.0183)<\/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>We are given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A box contains 200 fuses.<\/li>\n\n\n\n<li>The probability that any individual fuse is defective is 2% or 0.02.<\/li>\n\n\n\n<li>We are to find the probability that <strong>at most 5 defective fuses<\/strong> are found in the box.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Define the distribution<\/h3>\n\n\n\n<p>Let XX be the number of defective fuses in the box of 200.<br>Since we have a fixed number of trials (200), a constant probability of success (0.02), and independent outcomes, X\u223cBinomial(n=200,p=0.02)X \\sim \\text{Binomial}(n = 200, p = 0.02).<\/p>\n\n\n\n<p>However, calculating binomial probabilities for large nn can be complex. Since nn is large and pp is small, we use the <strong>Poisson approximation<\/strong> to the binomial distribution.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Use Poisson approximation<\/h3>\n\n\n\n<p>If X\u223cBinomial(n,p)X \\sim \\text{Binomial}(n, p), then for large nn, small pp, and \u03bb=np\\lambda = np, we can approximate: X\u223cPoisson(\u03bb)X \\sim \\text{Poisson}(\\lambda)<\/p>\n\n\n\n<p>Here, \u03bb=np=200\u00d70.02=4\\lambda = np = 200 \\times 0.02 = 4<\/p>\n\n\n\n<p>So X\u223cPoisson(4)X \\sim \\text{Poisson}(4)<\/p>\n\n\n\n<p>We want P(X\u22645)P(X \\leq 5), the probability of at most 5 defective fuses.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Calculate P(X\u22645)P(X \\leq 5) using Poisson probabilities<\/h3>\n\n\n\n<p>Poisson probability mass function: P(X=k)=e\u2212\u03bb\u03bbkk!P(X = k) = \\frac{e^{-\\lambda} \\lambda^k}{k!}<\/p>\n\n\n\n<p>Given e\u22124=0.0183e^{-4} = 0.0183, compute: P(X\u22645)=\u2211k=054ke\u22124k!P(X \\leq 5) = \\sum_{k=0}^{5} \\frac{4^k e^{-4}}{k!}<\/p>\n\n\n\n<p>Now compute each term:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>P(0)=40\u22c50.01830!=1\u22c50.01831=0.0183P(0) = \\frac{4^0 \\cdot 0.0183}{0!} = \\frac{1 \\cdot 0.0183}{1} = 0.0183<\/li>\n\n\n\n<li>P(1)=41\u22c50.01831!=4\u22c50.01831=0.0732P(1) = \\frac{4^1 \\cdot 0.0183}{1!} = \\frac{4 \\cdot 0.0183}{1} = 0.0732<\/li>\n\n\n\n<li>P(2)=16\u22c50.01832=0.1464P(2) = \\frac{16 \\cdot 0.0183}{2} = 0.1464<\/li>\n\n\n\n<li>P(3)=64\u22c50.01836=0.1952P(3) = \\frac{64 \\cdot 0.0183}{6} = 0.1952<\/li>\n\n\n\n<li>P(4)=256\u22c50.018324=0.1955P(4) = \\frac{256 \\cdot 0.0183}{24} = 0.1955<\/li>\n\n\n\n<li>P(5)=1024\u22c50.0183120=0.1569P(5) = \\frac{1024 \\cdot 0.0183}{120} = 0.1569<\/li>\n<\/ul>\n\n\n\n<p>Add them: P(X\u22645)\u22480.0183+0.0732+0.1464+0.1952+0.1955+0.1569=0.7855P(X \\leq 5) \\approx 0.0183 + 0.0732 + 0.1464 + 0.1952 + 0.1955 + 0.1569 = \\boxed{0.7855}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>This problem requires calculating the probability that at most 5 fuses are defective in a batch of 200, given that 2% are usually defective. The number of defective fuses in the box can be modeled by a binomial distribution: each fuse has a 2% chance of being defective, and there are 200 independent fuses. This results in a binomial distribution with parameters n=200n = 200 and p=0.02p = 0.02.<\/p>\n\n\n\n<p>However, since nn is large and pp is small, the Poisson distribution is a good approximation. The Poisson approximation to the binomial distribution is suitable when npnp (the expected number of defects) is moderate. Here, np=200\u00d70.02=4np = 200 \\times 0.02 = 4, which meets the condition.<\/p>\n\n\n\n<p>We approximate the binomial with a Poisson distribution where the mean \u03bb=4\\lambda = 4. The Poisson distribution describes the probability of a given number of events happening in a fixed interval and is often used for modeling rare events.<\/p>\n\n\n\n<p>To find the probability of at most 5 defective fuses, we calculate the sum of the Poisson probabilities from 0 to 5. Using the provided value e\u22124=0.0183e^{-4} = 0.0183, we apply the Poisson probability formula to each k=0k = 0 through 55. After computing each term, we sum them to obtain the cumulative probability.<\/p>\n\n\n\n<p>The final result, P(X\u22645)\u22480.7855P(X \\leq 5) \\approx 0.7855, means that there&#8217;s approximately a 78.55% chance that 5 or fewer fuses in the box are defective, based on the given defect rate. This helps in quality control decisions and assessing the likelihood of acceptable batches.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner7-156.jpeg\" alt=\"\" class=\"wp-image-222343\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the probability that atmost 5 defective fuses will be found in a box of 200 fuses if experience shows that 2 percent of such fuses are defective (e^-4= 0.0183) The Correct Answer and Explanation is: We are given: Step 1: Define the distribution Let XX be the number of defective fuses in the box [&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-222342","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222342","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=222342"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222342\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=222342"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=222342"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=222342"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}