{"id":208315,"date":"2025-04-26T13:07:40","date_gmt":"2025-04-26T13:07:40","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=208315"},"modified":"2025-04-26T13:07:43","modified_gmt":"2025-04-26T13:07:43","slug":"determine-whether-events-a-and-b-are-independent-or-dependent","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/04\/26\/determine-whether-events-a-and-b-are-independent-or-dependent\/","title":{"rendered":"Determine whether events A and B are independent or dependent"},"content":{"rendered":"\n<p>Determine whether events A and B are independent or dependent, then calculate P(A and B). <\/p>\n\n\n\n<p>a. A: When a baby is born, it is a girl. 8: When a six-sided die is rolled, the result is a six. <\/p>\n\n\n\n<p>b. A: A queen is drawn from a standard deck of cards. B: A second queen is drawn without replacement<\/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\">a. Events A and B<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>A<\/strong>: When a baby is born, it is a girl.<\/li>\n\n\n\n<li><strong>B<\/strong>: When a six-sided die is rolled, the result is a six.<\/li>\n<\/ul>\n\n\n\n<p><strong>Independence of Events A and B<\/strong>:<br>Two events are independent if the occurrence of one event does not affect the occurrence of the other. In this case, the events A and B are <strong>independent<\/strong> because the outcome of one does not influence the other. The birth of a baby being a girl is a biological event and does not affect the result of a die roll, which is a random event.<\/p>\n\n\n\n<p>To calculate <strong>P(A and B)<\/strong>, which is the probability that both events occur simultaneously, we multiply the probabilities of each event:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>P(A)<\/strong>: The probability that a baby is a girl is approximately 0.5 (since there is roughly an equal chance for a baby to be born a girl or a boy).<\/li>\n\n\n\n<li><strong>P(B)<\/strong>: The probability of rolling a six on a six-sided die is 1\/6.<\/li>\n<\/ul>\n\n\n\n<p>Thus, P(A&nbsp;and&nbsp;B)=P(A)\u00d7P(B)=0.5\u00d716=112.P(A \\text{ and } B) = P(A) \\times P(B) = 0.5 \\times \\frac{1}{6} = \\frac{1}{12}.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">b. Events A and B<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>A<\/strong>: A queen is drawn from a standard deck of cards.<\/li>\n\n\n\n<li><strong>B<\/strong>: A second queen is drawn <strong>without replacement<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p><strong>Independence of Events A and B<\/strong>:<br>In this case, events A and B are <strong>dependent<\/strong> because the outcome of the second draw (event B) is affected by the outcome of the first draw (event A). When a queen is drawn from the deck (event A), there are fewer queens remaining in the deck, so the probability of drawing a second queen changes.<\/p>\n\n\n\n<p>To calculate <strong>P(A and B)<\/strong>, we need to find the probabilities of both events occurring:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>P(A)<\/strong>: The probability of drawing a queen from a standard deck (which has 52 cards and 4 queens) is: P(A)=452=113.P(A) = \\frac{4}{52} = \\frac{1}{13}.<\/li>\n\n\n\n<li><strong>P(B | A)<\/strong>: The probability of drawing a second queen, given that the first queen has already been drawn (without replacement), is: P(B\u2223A)=351\u00a0(since\u00a0one\u00a0queen\u00a0is\u00a0already\u00a0removed,\u00a0leaving\u00a03\u00a0queens\u00a0out\u00a0of\u00a051\u00a0remaining\u00a0cards).P(B | A) = \\frac{3}{51} \\text{ (since one queen is already removed, leaving 3 queens out of 51 remaining cards)}.<\/li>\n<\/ul>\n\n\n\n<p>Thus, P(A&nbsp;and&nbsp;B)=P(A)\u00d7P(B\u2223A)=113\u00d7351=3663=1221.P(A \\text{ and } B) = P(A) \\times P(B | A) = \\frac{1}{13} \\times \\frac{3}{51} = \\frac{3}{663} = \\frac{1}{221}.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For part <strong>a<\/strong>, the events are independent, and the probability P(A\u00a0and\u00a0B)=112P(A \\text{ and } B) = \\frac{1}{12}.<\/li>\n\n\n\n<li>For part <strong>b<\/strong>, the events are dependent, and the probability P(A\u00a0and\u00a0B)=1221P(A \\text{ and } B) = \\frac{1}{221}.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Determine whether events A and B are independent or dependent, then calculate P(A and B). a. A: When a baby is born, it is a girl. 8: When a six-sided die is rolled, the result is a six. b. A: A queen is drawn from a standard deck of cards. B: A second queen is [&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-208315","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/208315","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=208315"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/208315\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=208315"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=208315"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=208315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}