{"id":776,"date":"2024-06-11T10:58:35","date_gmt":"2024-06-11T17:58:35","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=776"},"modified":"2019-04-20T01:37:51","modified_gmt":"2019-04-20T08:37:51","slug":"gmat-probability-difficult-dice-questions","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-probability-difficult-dice-questions\/","title":{"rendered":"GMAT Probability: Difficult Dice Questions"},"content":{"rendered":"<p><strong>This post was updated in 2024 for the new GMAT.<\/strong><\/p>\n<p>Dice problems aren\u2019t too common on GMAT quant. As for the exact probability of getting a dice problem is something only privy to those over at GMAC.<\/p>\n<p>While it is a good idea to know your dice basics, doing so will only help you in the case of an easy dice problem. And one thing is for sure \u2013 if you are looking to break a 705 on the GMAT the probability of an easy dice problem (if you end up getting a dice problem) is very, very low.<\/p>\n<p>Below are three excruciatingly difficult dice problems. They will pop up only if you are heading towards a high score. For those still brushing up on their math fundamental don\u2019t waste time on such challenge questions.<\/p>\n<h2>GMAT Dice Probability Practice Questions<\/h2>\n<p>1. What is the probability of rolling the same number exactly three times with five six-sided dice?<\/p>\n<ol>\n<li>1\/5<\/li>\n<li>5\/18<\/li>\n<li>35\/216<\/li>\n<li>125\/648<\/li>\n<li>225\/1296<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>2.\u00a0What is the probability of rolling three six-sided dice, and getting a different number on each die?<\/p>\n<ol>\n<li>1\/12<\/li>\n<li>1\/3<\/li>\n<li>4\/9<\/li>\n<li>5\/9<\/li>\n<li>7\/18<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>3. A magician holds one six-sided die in his left hand and two in his right. What is the probability the number on the dice in his left hand is greater than the sum of the dice in his right?<\/p>\n<ol>\n<li>\u00a07\/108<\/li>\n<li>5\/54<\/li>\n<li>1\/9<\/li>\n<li>2\/17<\/li>\n<li>1\/4<\/li>\n<\/ol>\n<div><\/div>\n<h2>Answer Explanations:<\/h2>\n<h3>Question 1 <\/h3>\n<p>Rolling any number on a dice three times in a row is equal to the number of throws <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_0785fc2c15e6562d708dd231a26194d0.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"{{1\/6}^3}*6 = 1\/36\" title=\"{{1\/6}^3}*6 = 1\/36\"\/>, where 3 represents the number of throws and 6 is the number of different ways to get three of the same number (e.g. 1, 1, 1\u00a0 2, 2, 2\u00a0 3, 3, 3\u2026).<\/p>\n<p>Next we have to use the combinations formula to determine how many ways three out of five can be the same. <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_42a741a12074d75023fa74e59e0c68c0.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"{5!}\/{3!*2!}=10\" title=\"{5!}\/{3!*2!}=10\"\/>. We want to multiply this number to <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_5406f209497bb542272c43545fee7a79.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"{1\/36}*10=10\/36\" title=\"{1\/36}*10=10\/36\"\/><\/p>\n<p>We are not finished yet \u2013 there is one little twist to the problem. The question says \u201cexactly three times.\u201d Meaning we have to discount all the instances in which we roll the same number four times and those instance in which we roll the same number five times.<\/p>\n<p>One way to do so is by multiplying by the probability that the fourth and fifth dice will NOT land on the same number as the three dice. Because there are five other possibilities on dice that will not compromise \u201cexactly three numbers\u201d we can multiply by <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_1002_4a391a39a39f270d61d5bba2d328bb40.png\" style=\"vertical-align:2px; display: inline-block ;\" alt=\"{5\/6{*{5\/6}=25\/36\" title=\"{5\/6{*{5\/6}=25\/36\"\/><\/p>\n<p>Finally this gives us <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_b5301122e2e702d4d7c69dcb87d4b7b7.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"{10\/36}*{25\/36}={125\/648}\" title=\"{10\/36}*{25\/36}={125\/648}\"\/><\/p>\n<h3>Question 2<\/h3>\n<p>For the first die, we can roll any one of six numbers. For the second die, we can roll any number save for the number we rolled on the first die, giving us 5 possibilities. For the third die, we can roll four different numbers (we can\u2019t roll the number we rolled on the first or second die.<\/p>\n<p>6 x 5 x 4 = 120 possibilities out of 216 total possibilities. (For total possibilities we get 6 x 6 x 6 = 216).<\/p>\n<p>120\/216 = 5\/9<\/p>\n<h3>Question 3<\/h3>\n<p>Perhaps the hardest of the bunch, this question requires that we find a solution that doesn\u2019t take much longer than 2 minutes. A quick way \u2013 or at least relatively quick way \u2013 is to determine the number of instances in which our roll of one die will yield more than two die. Intuitively, this is a good place to start because the number of instances in which rolling a great number with one die than with two is much smaller than vice versa.<\/p>\n<p>The only way to roll higher on one die is if the magicians rolls between 2 and 5, inclusive, with two dice. Were he to roll a six with two dice than there is no way he could eclipse that number by rolling one die.<\/p>\n<p>Below is the probability of rolling a certain number with two dice.<\/p>\n<p>\u20182\u2019 &#8211; 1\/36<\/p>\n<p>\u20183\u2019 &#8211; 2\/36<\/p>\n<p>\u20184\u2019 &#8211; 3\/36<\/p>\n<p>\u20185\u2019-\u00a0 4\/36<\/p>\n<p>Now the only numbers a magician can roll with the one die and win is between 3 and 6, inclusive. The chances of rolling any are always 1\/6.<\/p>\n<p>Next, we have to combine the probability distribution relating to two dice with that relating to the one.<\/p>\n<p>The chances of a magician rolling any given number with one die are 1\/6. So let\u2019s start with the lowest number he can roll: a \u20183.\u2019 To win with this roll, he will have to roll a \u20182\u2019 with two dice, the odds of which are 1\/36. So <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_fb244e807b824b6165f443031b90e04b.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"{1\/6} * {1\/36} = 1\/216\" title=\"{1\/6} * {1\/36} = 1\/216\"\/>.<\/p>\n<p>So the chances of him rolling a \u20183\u2019 with one die and winning are 1\/216.<\/p>\n<p>Let\u2019s repeat this logic for the next roll, \u20184.\u2019 Chances of rolling are 1\/6. Only way a \u20184\u2019 wins is if he rolls a \u20182\u2019 or a \u20183\u2019 with two dice.<\/p>\n<p>Odds of rolling a \u20182\u2019 with two dice\u00a0 + odds of rolling a \u20183\u2019 with two dice = <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_6b597c2e0481650cf51b124f1bf2998b.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"{1\/36} + {2\/36} = {3\/36}\" title=\"{1\/36} + {2\/36} = {3\/36}\"\/>. Combine this with the odds of rolling a \u20184\u2019 ( which is 1\/6): <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_aec0005dd67caa88bacc920b1fb92e8a.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"{3\/216}\" title=\"{3\/216}\"\/>.<\/p>\n<p>If he rolls a \u20185\u2019 with one die, he can win if he rolls a \u20184\u2019 with two dice, the probability is 3\/36. He can also win if he rolls a \u20182\u2019 or a \u20183\u2019 with two dice, the number outcomes we just found: 3\/36. So we add <img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_6eb1386d4ec41f945faaacd50697afa8.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"{3\/36} + {3\/36} = {6\/36} * {1\/6} (odds of rolling a 5 with one die) = {6\/216}\" title=\"{3\/36} + {3\/36} = {6\/36} * {1\/6} (odds of rolling a 5 with one die) = {6\/216}\"\/><\/p>\n<p>Next, if he rolls a \u20186\u2019 on one die, he can beat \u20182\u2019 through \u20185\u2019 with the two dice. Number of ways to roll a \u20185\u2019 = 4\/36. Combining this with the odds of rolling a \u20186\u2019 on one die with the odds or rolling \u20182\u2019, \u20183\u2019, \u20184\u2019, or \u20185\u2019 with the two dice we get: 10\/216.<\/p>\n<p>You may be wondering why I left the denominator as 216. Well, this allows us to add up all the instances he can possibly win:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/wpmathpub\/phpmathpublisher\/img\/math_984_3e12be7fbacae2c8487c3777f6d4995c.png\" style=\"vertical-align:-16px; display: inline-block ;\" alt=\"1\/216 + 3\/216 + 6\/216 + 10\/216 = 20\/216 = 5\/54\" title=\"1\/216 + 3\/216 + 6\/216 + 10\/216 = 20\/216 = 5\/54\"\/>\u00a0(Answer B)<\/p>\n<p>&nbsp;<\/p>\n<p><em>Improve your GMAT scores with <a href=\"https:\/\/gmat.magoosh.com\/plans?utm_source=gmatblog&#038;utm_medium=blog&#038;utm_campaign=gmatplans&#038;utm_term=inline&#038;utm_content=gmat-probability-difficult-dice-questions\">Magoosh GMAT<\/a>, you can get access to over 200 expert-created video lessons and 800 practice questions, plus full-length practice tests!<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post was updated in 2024 for the new GMAT. Dice problems aren\u2019t too common on GMAT quant. As for the exact probability of getting a dice problem is something only privy to those over at GMAC. While it is a good idea to know your dice basics, doing so will only help you in [&hellip;]<\/p>\n","protected":false},"author":10,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[112],"tags":[],"ppma_author":[13210],"class_list":["post-776","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>GMAT Probability: Difficult Dice Questions - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"description\" content=\"Dice probability problems on the GMAT tend to be very challenging, but uncommon. 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He's been quoted as a subject expert in many publications, including US News, GMAC, and Business Because.","sameAs":["https:\/\/www.facebook.com\/ChrisLeleMagoosh1\/","ttps:\/\/www.linkedin.com\/in\/chris-lele-095005a0","https:\/\/twitter.com\/chrismagoosh","https:\/\/www.youtube.com\/channel\/UCZ4sGYJ6l0cgnVNIHY8GY6w"],"knowsAbout":["GRE","GMAT","SAT","ACT","MCAT (CARS)","LSAT"],"jobTitle":"Principal Curriculum Manager","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/chris\/"}]}},"authors":[{"term_id":13210,"user_id":10,"is_guest":0,"slug":"chris","display_name":"Chris Lele","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/39381a6b4ab74cd4c87c7e678683c33c3b18f1e14f6f0ac762a946185968952b?s=96&d=mm&r=g","user_url":"","last_name":"Lele","first_name":"Chris","description":"Chris Lele is the Principal Curriculum Manager (and <a href=\"https:\/\/www.amazon.com\/Vocabulary-Builder-Workbook-Activities-Must-Know\/dp\/193975481X\" rel=\"noopener noreferrer\">vocabulary wizard<\/a>) at Magoosh. Chris graduated from UCLA with a BA in Psychology and has 20 years of experience in the test prep industry. He's been quoted as a subject expert in many publications, including US News,<a href=\"https:\/\/www.mba.com\/exams-and-exam-prep\/gmat-exam\/why-take-the-gmat-in-2021-when-many-schools-are-test-optional\" rel=\"noopener noreferrer\"> GMAC<\/a>, and<a href=\"https:\/\/www.businessbecause.com\/news\/gmat\/3263\/mba-admissions-gre-challenges-gmat\" rel=\"noopener noreferrer\"> Business Because<\/a>.\r\n\r\n\r\nIn his time at Magoosh, Chris has taught countless students how to tackle the<a href=\"https:\/\/www.amazon.com\/GRE-Prep-Magoosh\/dp\/1939418917\/\" rel=\"noopener noreferrer\"> GRE<\/a>, GMAT, SAT, ACT, MCAT (CARS), and LSAT exams with confidence. Some of his students have even gone on to get near-perfect scores. You can find Chris on <a href=\"https:\/\/www.youtube.com\/channel\/UCZ4sGYJ6l0cgnVNIHY8GY6w\">YouTube<\/a>,<a href=\"https:\/\/www.linkedin.com\/in\/chris-lele-095005a0\"> LinkedIn<\/a>, <a href=\"https:\/\/twitter.com\/chrismagoosh\">Twitter<\/a> and <a href=\"https:\/\/www.facebook.com\/ChrisLeleMagoosh1\/\">Facebook<\/a>!"}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/776","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=776"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/776\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=776"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=776"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=776"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=776"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}