Apps My Blue Hen Home Carvas Math QuizletHandshake, Pat Math 114 College Math and Statistics – Thompson – Fall 2020 Homework: Math 114 Week 12 Homework Score: 0 of 1 pt 8.5.1 Suppose a single fair die is rolled. Find the probability that it is a 1, given that it is an even number. The probability is (Type an integer or a simplified fraction.) Enter your answer in the answer box and then click Check Answer, All parts showing<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To solve this probability problem, we need to apply conditional probability<\/strong>.<\/p>\n\n\n\n A fair six-sided die has the numbers:<\/p>\n\n\n\n 1, 2, 3, 4, 5, 6<\/strong><\/p>\n<\/blockquote>\n\n\n\n The even numbers<\/strong> among these are:<\/p>\n\n\n\n 2, 4, 6<\/strong><\/p>\n<\/blockquote>\n\n\n\n So, the new sample space<\/strong> (since the condition is that the result is even) is:<\/p>\n\n\n\n {2, 4, 6}<\/p>\n<\/blockquote>\n\n\n\n There are 3 outcomes<\/strong> in this conditional sample space.<\/p>\n\n\n\n We are asked to find the probability that the number is 1<\/strong>, given<\/strong> that the number is even.<\/p>\n\n\n\n But looking at the even numbers (2, 4, 6), we see:<\/p>\n\n\n\n 1 is not an even number<\/strong>, and therefore it is not<\/strong> in the new sample space.<\/p>\n<\/blockquote>\n\n\n\n That means:<\/p>\n\n\n\n It is impossible<\/strong> to roll a 1 if we already know the number is even.<\/p>\n<\/blockquote>\n\n\n\n So, the event “the number is 1” cannot occur<\/strong> under the condition that the number is even.<\/p>\n\n\n\n The formula for conditional probability<\/strong> is: P(A\u2223B)=P(A\u2229B)P(B)P(A \\mid B) = \\frac{P(A \\cap B)}{P(B)}<\/p>\n\n\n\n Let:<\/p>\n\n\n\n Since 1 is not even, A\u2229B=\u2205A \\cap B = \\emptyset, so: P(A\u2229B)=0P(A \\cap B) = 0 P(B)=36=12P(B) = \\frac{3}{6} = \\frac{1}{2}<\/p>\n\n\n\n So: P(1\u2223even)=01\/2=0P(\\text{1} \\mid \\text{even}) = \\frac{0}{1\/2} = 0<\/p>\n\n\n\n 0\\boxed{0}<\/p>\n\n\n\n This problem is an example of conditional probability<\/strong>, where we are given that one event has occurred and asked to find the probability of another event happening under that condition. In this case, we are told that a fair die is rolled and the outcome is an even number. The even numbers on a standard six-sided die are 2, 4, and 6. This means we now only consider these numbers when calculating probabilities.<\/p>\n\n\n\n We are then asked to find the probability that the number rolled is a 1, given<\/strong> that we already know it is an even number. Since the number 1 is not even, and is not part of the subset {2, 4, 6}, it is not possible<\/strong> for the outcome to be a 1 under the given condition. When an event cannot happen within the restricted sample space, the probability of that event is 0.<\/p>\n\n\n\n This situation shows why understanding the sample space is critical when dealing with conditional probabilities. The conditional probability formula is: P(A\u2223B)=P(A\u2229B)P(B)P(A \\mid B) = \\frac{P(A \\cap B)}{P(B)}<\/p>\n\n\n\n Here, P(A\u2229B)P(A \\cap B) represents the probability that both events A and B occur simultaneously. In our case, “rolling a 1” and “rolling an even number” cannot both happen at the same time, so their intersection is empty and the probability is 0. Dividing 0 by any non-zero probability (such as P(B)=0.5P(B) = 0.5) still gives 0.<\/p>\n\n\n\n Therefore, the final answer is 0<\/strong>, meaning it is impossible to roll a 1 if we already know the outcome is even.<\/p>\n","protected":false},"excerpt":{"rendered":" Apps My Blue Hen Home Carvas Math QuizletHandshake, Pat Math 114 College Math and Statistics – Thompson – Fall 2020 Homework: Math 114 Week 12 Homework Score: 0 of 1 pt 8.5.1 Suppose a single fair die is rolled. Find the probability that it is a 1, given that it is an even number. The […]<\/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-215588","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215588","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=215588"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/215588\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=215588"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=215588"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=215588"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Given:<\/strong><\/h3>\n\n\n\n
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\n\n\n\nStep 1: Understand the Sample Space<\/strong><\/h3>\n\n\n\n
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\n\n\n\nStep 2: Define the Event of Interest<\/strong><\/h3>\n\n\n\n
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\n\n\n\nStep 3: Write the Conditional Probability<\/strong><\/h3>\n\n\n\n
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\n\n\n\nFinal Answer:<\/strong><\/h3>\n\n\n\n
\n\n\n\n300-Word Explanation:<\/strong><\/h3>\n\n\n\n