{"id":197123,"date":"2025-03-06T16:55:51","date_gmt":"2025-03-06T16:55:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=197123"},"modified":"2025-03-06T16:55:53","modified_gmt":"2025-03-06T16:55:53","slug":"computer-keyboard-failures-can-be-attributed-to-electrical-defects-or-mechanical-defects","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/06\/computer-keyboard-failures-can-be-attributed-to-electrical-defects-or-mechanical-defects\/","title":{"rendered":"Computer keyboard failures can be attributed to electrical defects or mechanical defects"},"content":{"rendered":"\n<p>Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 13 of which have electrical defects and 12 of which have mechanical defects.<br>(a) How many ways are there to randomly select 6 of these keyboards for a thorough inspection (without regard to order)?<br>ways<br>(b) In how many ways can a sample of 6 keyboards be selected so that exactly two have an electrical defect?<br>ways<br>(c) If a sample of 6 keyboards is randomly selected, what is the probability that at least 5 of these will have a mechanical defect? (Round your answer to four decimal places.)<\/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<h3 class=\"wp-block-heading\">Answers:<\/h3>\n\n\n\n<p>(a) The number of ways to randomly select 6 keyboards from 25 is <strong>177,100<\/strong>.<\/p>\n\n\n\n<p>(b) The number of ways to select a sample of 6 keyboards such that exactly 2 have electrical defects is <strong>38,610<\/strong>.<\/p>\n\n\n\n<p>(c) The probability that at least 5 of the selected keyboards have a mechanical defect is <strong>0.0634<\/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:<\/h3>\n\n\n\n<p>In this problem, we are working with combinatorial selections and probability.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Part (a):<\/h4>\n\n\n\n<p>The total number of ways to choose 6 keyboards from 25 is determined using the combination formula:<\/p>\n\n\n\n<p>[<br>C(n, k) = \\frac{n!}{k!(n-k)!}<br>]<\/p>\n\n\n\n<p>where ( n = 25 ) and ( k = 6 ):<\/p>\n\n\n\n<p>[<br>C(25,6) = \\frac{25!}{6!(25-6)!} = 177,100<br>]<\/p>\n\n\n\n<p>This gives the total number of ways to select any 6 keyboards.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Part (b):<\/h4>\n\n\n\n<p>To find the number of ways to select 6 keyboards such that exactly 2 have electrical defects, we:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Choose 2 keyboards from the 13 electrical defect ones: ( C(13,2) ).<\/li>\n\n\n\n<li>Choose 4 keyboards from the 12 mechanical defect ones: ( C(12,4) ).<\/li>\n<\/ul>\n\n\n\n<p>[<br>C(13,2) = \\frac{13!}{2!(13-2)!} = 78<br>]<\/p>\n\n\n\n<p>[<br>C(12,4) = \\frac{12!}{4!(12-4)!} = 4,095<br>]<\/p>\n\n\n\n<p>Multiplying these gives:<\/p>\n\n\n\n<p>[<br>78 \\times 4,095 = 38,610<br>]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Part (c):<\/h4>\n\n\n\n<p>To find the probability that at least 5 of the selected keyboards have mechanical defects, we consider two cases:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Case 1:<\/strong> Exactly 5 mechanical defects and 1 electrical defect.<\/li>\n\n\n\n<li><strong>Case 2:<\/strong> All 6 are mechanical defects.<\/li>\n<\/ul>\n\n\n\n<p>For <strong>Case 1<\/strong>, we choose:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>5 keyboards from the 12 mechanical ones: ( C(12,5) ).<\/li>\n\n\n\n<li>1 keyboard from the 13 electrical ones: ( C(13,1) ).<\/li>\n<\/ul>\n\n\n\n<p>[<br>C(12,5) = \\frac{12!}{5!(12-5)!} = 792<br>]<\/p>\n\n\n\n<p>[<br>C(13,1) = \\frac{13!}{1!(13-1)!} = 13<br>]<\/p>\n\n\n\n<p>[<br>792 \\times 13 = 10,296<br>]<\/p>\n\n\n\n<p>For <strong>Case 2<\/strong>, we choose all 6 from the 12 mechanical ones:<\/p>\n\n\n\n<p>[<br>C(12,6) = \\frac{12!}{6!(12-6)!} = 924<br>]<\/p>\n\n\n\n<p>Total ways for at least 5 mechanical:<\/p>\n\n\n\n<p>[<br>10,296 + 924 = 11,220<br>]<\/p>\n\n\n\n<p>Probability:<\/p>\n\n\n\n<p>[<br>P = \\frac{11,220}{177,100} = 0.0634<br>]<\/p>\n\n\n\n<p>This means that if we randomly select 6 keyboards, the probability that at least 5 have mechanical defects is <strong>6.34%<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/03\/image-389-1024x706.png\" alt=\"\" class=\"wp-image-197124\"\/><\/figure>\n\n\n\n<p>Here is a pie chart illustrating the probability of selecting at least 5 mechanical defect keyboards. The portion labeled <strong>&#8220;At Least 5 Mechanical&#8221; (6.34%)<\/strong> represents the probability of this scenario occurring, while the remaining portion represents all other possible selections. This visualization helps to understand the likelihood of drawing mostly mechanical defect keyboards in a random selection.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 13 of which have electrical defects and 12 of which have mechanical defects.(a) How many ways are there to randomly select 6 of these keyboards for a thorough inspection (without regard to order)?ways(b) In how [&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-197123","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197123","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=197123"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197123\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=197123"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=197123"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=197123"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}