Two cards are randomly selected from a standard 52-card deck. What is the probability of getting 2 hearts or 2 numbers less than 4 (count aces as 1)<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n To find the probability of getting either two hearts or two cards with numbers less than 4 (where aces count as 1), we will use the principle of inclusion-exclusion.<\/p>\n\n\n\n The total number of ways to choose 2 cards from a standard 52-card deck is given by the combination formula:<\/p>\n\n\n\n [ For our case, ( n = 52 ) and ( k = 2 ):<\/p>\n\n\n\n [ There are 13 hearts in a deck. The number of ways to choose 2 hearts is:<\/p>\n\n\n\n [ The cards with numbers less than 4 are the aces, 2s, and 3s, totaling 12 cards (4 of each suit). The number of ways to choose 2 cards from these is:<\/p>\n\n\n\n [ Since the events “getting 2 hearts” and “getting 2 cards with numbers less than 4” are mutually exclusive (you cannot have 2 hearts that are also < 4, as they are not less than 4), we simply add the probabilities of both events.<\/p>\n\n\n\n The total number of favorable outcomes for either event is:<\/p>\n\n\n\n [ Finally, the probability of getting either 2 hearts or 2 cards with numbers less than 4 is given by the formula:<\/p>\n\n\n\n [ Thus, the probability of drawing either 2 hearts or 2 cards with values less than 4 is approximately 0.1087<\/strong> or 10.87%<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":" Two cards are randomly selected from a standard 52-card deck. What is the probability of getting 2 hearts or 2 numbers less than 4 (count aces as 1) The Correct Answer and Explanation is : To find the probability of getting either two hearts or two cards with numbers less than 4 (where aces count […]<\/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-146199","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/146199","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=146199"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/146199\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=146199"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=146199"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=146199"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Total Possible Outcomes<\/h3>\n\n\n\n
\\binom{n}{k} = \\frac{n!}{k!(n-k)!}
]<\/p>\n\n\n\n
\\text{Total outcomes} = \\binom{52}{2} = \\frac{52 \\times 51}{2} = 1326
]<\/p>\n\n\n\nProbability of Getting 2 Hearts<\/h3>\n\n\n\n
\\text{Ways to choose 2 hearts} = \\binom{13}{2} = \\frac{13 \\times 12}{2} = 78
]<\/p>\n\n\n\nProbability of Getting 2 Cards with Numbers Less than 4<\/h3>\n\n\n\n
\\text{Ways to choose 2 cards < 4} = \\binom{12}{2} = \\frac{12 \\times 11}{2} = 66
]<\/p>\n\n\n\nProbability of Getting Both Events (2 Hearts and 2 Cards < 4)<\/h3>\n\n\n\n
Combined Probability<\/h3>\n\n\n\n
\\text{Favorable outcomes} = \\text{Ways to choose 2 hearts} + \\text{Ways to choose 2 cards < 4} = 78 + 66 = 144
]<\/p>\n\n\n\nProbability Calculation<\/h3>\n\n\n\n
P = \\frac{\\text{Favorable outcomes}}{\\text{Total outcomes}} = \\frac{144}{1326} \\approx 0.1087
]<\/p>\n\n\n\nConclusion<\/h3>\n\n\n\n