{"id":30384,"date":"2025-06-21T18:04:55","date_gmt":"2025-06-21T18:04:55","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=30384"},"modified":"2025-06-21T18:04:56","modified_gmt":"2025-06-21T18:04:56","slug":"scrabble-in-the-game-of-scrabble-each-player-begins-by-drawing-7-tiles-from-a-bag-containing-100-tiles-2","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/scrabble-in-the-game-of-scrabble-each-player-begins-by-drawing-7-tiles-from-a-bag-containing-100-tiles-2\/","title":{"rendered":"Scrabble In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles."},"content":{"rendered":"\n<p>Scrabble In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses her 7 tiles and is surprised to discover that all of them are vowels. Can we use a binomial distribution to approximate this probability? Justify your answer.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p><strong>Correct Answer:<\/strong><\/p>\n\n\n\n<p>No, we <strong>cannot<\/strong> use a <strong>binomial distribution<\/strong> to accurately approximate this probability because the <strong>selection of Scrabble tiles is done without replacement<\/strong>, which <strong>violates a key condition<\/strong> of the binomial model.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Explanation<\/strong><\/p>\n\n\n\n<p>In probability, the <strong>binomial distribution<\/strong> is used to model the number of successes in a fixed number of <strong>independent<\/strong> trials, where each trial has only <strong>two possible outcomes<\/strong> (commonly labeled success and failure), and the probability of success remains <strong>constant<\/strong> throughout.<\/p>\n\n\n\n<p>To determine whether the binomial distribution can be used in Cait&#8217;s situation, we must examine the key assumptions of the binomial model:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Fixed number of trials<\/strong>: Cait draws 7 tiles. This condition is met.<\/li>\n\n\n\n<li><strong>Only two outcomes per trial<\/strong>: Each tile can be a vowel (success) or not a vowel (failure). This condition is also satisfied.<\/li>\n\n\n\n<li><strong>Probability of success stays the same<\/strong>: This is where the problem arises. Since Cait draws tiles <strong>without replacement<\/strong>, the total number of tiles in the bag changes with each draw. For example, if the first tile is a vowel, there are now fewer vowels and fewer total tiles in the bag. This changes the probability of drawing another vowel on the next trial.<\/li>\n\n\n\n<li><strong>Trials are independent<\/strong>: In this case, each draw affects the next one. Since tiles are not replaced, the outcome of one draw influences the probabilities of subsequent draws. This means the trials are <strong>not independent<\/strong>.<\/li>\n<\/ol>\n\n\n\n<p>Because the <strong>third<\/strong> and <strong>fourth conditions<\/strong> are violated, we <strong>cannot<\/strong> use a binomial distribution. Instead, a more appropriate method would be the <strong>hypergeometric distribution<\/strong>, which is specifically designed to model the probability of successes in draws <strong>without replacement<\/strong> from a finite population.<\/p>\n\n\n\n<p>Therefore, the probability that Cait draws 7 vowels from the bag should be calculated using a <strong>hypergeometric model<\/strong>, not a binomial one.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-143.jpeg\" alt=\"\" class=\"wp-image-30389\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-143.jpeg 1024w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-143-300x300.jpeg 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-143-150x150.jpeg 150w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-143-768x768.jpeg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Scrabble In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses her 7 tiles and is surprised to discover that all of them are vowels. Can we use a binomial distribution to [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-30384","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/30384","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=30384"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/30384\/revisions"}],"predecessor-version":[{"id":30390,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/30384\/revisions\/30390"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=30384"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=30384"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=30384"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}