![\"Manhattan](\"http:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/03\/3-16-17-social-1.png\")
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Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We\u2019re not kidding! <\/i><\/b>Check out our upcoming courses here<\/i><\/b><\/a>.<\/i><\/b><\/p>\n <\/i><\/b>There\u2019s a classic brain teaser called the Monty Hall problem. It\u2019s named after the host of an old-timey TV game show, who used it to confound contestants. He\u2019d present each contestant with three closed doors. Behind one door was a new car, and behind the other two doors were goats. <\/span><\/p>\n Monty invited the player to pick one of the three doors. Whichever door the player chose, Monty would then open a <\/span>different<\/span><\/i> one, revealing a goat, not the car. Then, he would offer the player a choice. If the player wanted, he could <\/span>switch<\/span><\/i> doors, picking the other unopened door. Or, he could stick with the door he picked in the first place. Whichever decision he made, he would win the prize behind the door he chose. <\/span><\/p>\n The question is, <\/span>does switching doors give the contestant a higher probability of picking the car<\/span><\/i>? It should seem obvious that it doesn\u2019t. The car is equally likely to be behind any of the doors. So, it seems like once one of the three doors is open, you have 50\/50 odds of picking the car, regardless of whether you switch or stay. <\/span><\/p>\n However, <\/span>that isn\u2019t true<\/span><\/i>. Believe it or not, switching actually increases the probability of picking the car to 2\/3, while staying on the same door means that you only have a 1 in 3 chance of picking it. If you didn\u2019t figure that out on your own, you\u2019re in the company of many very smart people, including a number of famous mathematicians.<\/span><\/p>\n From a GMAT perspective, here\u2019s what to take away from this little anecdote. GMAT probability and combinatorics are counterintuitive for <\/span>everyone<\/span><\/i>. I majored in math in college \u2013 I even took a couple of courses on probability \u2013 and I still can\u2019t solve these problems by intuiting which formulas to use. It seems crazy to me that switching doors after the fact would make you more likely to pick the car. So, what do I do \u2013 and what should you do? I think about almost all GMAT probability and combinatorics problems in terms of <\/span>counting out possibilities<\/span><\/i>. <\/span><\/p>\n Let\u2019s give it a try with the Monty Hall problem. Imagine that you play three games with Monty. Additionally, imagine we\u2019re in probability land, where everything happens exactly according to its probability. Since there are three possible scenarios, and they\u2019re all equally likely, here\u2019s what your three games will look like. <\/span><\/p>\n Now, Monty opens one of the two doors you didn\u2019t pick. Remember that Monty always opens a door with a goat behind it. So, in two out of the three games, Monty is stuck: there\u2019s only one door he can choose. <\/span><\/p>\n In both of these games, you started with a goat. If you switch, you\u2019ll win! <\/span><\/p>\n In game 3, Monty could pick either door, since they both have goats. Let\u2019s say he flips a coin to decide which one to pick. If he gets heads, he\u2019ll open the first mystery door. If he gets tails, he\u2019ll open the second one. This scenario only comes up when you picked the car in the first place \u2013 so, Monty only needs to worry about it 1\/3 of the time.<\/span><\/p>\n Interestingly, it doesn\u2019t matter what Monty flips. No matter what, if Monty flips a coin at all, and you switch, you\u2019ll lose. If you always switch, you\u2019ll win 2 out of your 3 games, and lose 1 of them. That means that switching gives you a 2\/3 probability of winning. <\/span><\/p>\n The reason the Monty Hall problem is so counterintuitive, is that people don\u2019t remember that you\u2019re more likely to start with a goat than with a car. Since there are two possibilities \u2013 goat or car \u2013 your brain wants to think that they\u2019re equally likely. They aren\u2019t. If you start with a goat, switching is always good, and if you start with a car, switching is always bad. You\u2019re more likely to start with a goat than with a car, so without any more information, you should assume that switching is good. <\/span><\/p>\n How do you apply this to the GMAT? There are a few ways. First, recognize that probability is counterintuitive. Not just for you \u2013 for many people who are far better at math than you and I, as well. If you see a super tough GMAT probability and combinatorics problem, feel free to guess, since everyone else is probably getting it wrong too. And <\/span>don\u2019t<\/span><\/i> make the \u201cobvious\u201d guess, since it\u2019s probably a trap. Second, the best way to work through counterintuitive GMAT probability and combinatorics problems is to write out, or think out, the different possibilities in an organized way. This is even possible when you\u2019re doing problems with very large numbers, as you\u2019ll learn from the next article in this series<\/a>. ?<\/span><\/p>\n Want more guidance from our GRE gurus? You can attend the first session of any of our online or in-person GRE courses absolutely free! We\u2019re not kidding. <\/i><\/b>Check out our upcoming courses here<\/i><\/b><\/a>.<\/i><\/b><\/p>\n Chelsey Cooley<\/a> Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We\u2019re not kidding! Check out our upcoming courses here. There\u2019s a classic brain teaser called the Monty Hall problem. It\u2019s named after the host of an old-timey TV game show, who used it to […]<\/p>\n","protected":false},"author":127,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13,873,929,52871,930,2,8,52807],"tags":[53000,53001],"yst_prominent_words":[],"class_list":["post-13314","post","type-post","status-publish","format-standard","hentry","category-challenge-problem","category-for-current-studiers","category-gmat-prep","category-gmat-strategies","category-gmat-study-guide","category-how-to-study","category-quant-on-gmat","category-word-problems","tag-gmat-probability-and-combinatorics","tag-probability-and-combinatorics"],"_links":{"self":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/13314","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/users\/127"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/comments?post=13314"}],"version-history":[{"count":6,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/13314\/revisions"}],"predecessor-version":[{"id":13514,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/13314\/revisions\/13514"}],"wp:attachment":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/media?parent=13314"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/categories?post=13314"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/tags?post=13314"},{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/yst_prominent_words?post=13314"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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\n![\"Chelsey](\"https:\/\/cdn2.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2015\/11\/chelsey-cooley-150x150.jpg\" "\\"Chelsey")
<\/a> is a Manhattan Prep instructor based in Seattle, Washington.<\/strong>\u00a0<\/em><\/i><\/b>Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master\u2019s degree in linguistics, a 790 on the GMAT, and a perfect 170\/170 on the GRE.\u00a0<\/em><\/i>Check out Chelsey\u2019s upcoming GRE prep offerings here<\/a>.<\/em><\/i><\/p>\n","protected":false},"excerpt":{"rendered":"