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# On a game show, a contestant is given three keys, each of

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On a game show, a contestant is given three keys, each of [#permalink]

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05 Feb 2012, 17:09
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Question Stats:

49% (02:31) correct 51% (01:38) wrong based on 163 sessions

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On a game show, a contestant is given three keys, each of which opens exactly one of three identical boxes. The first box contains $1, the second$100, and the third $1000. The contestant assigns each key to one of the boxes and wins the amount of money contained in any box that is opened by the key assigned to it. What is the probability that a contestant will win more than$1000?

(A) 1/9
(B) 1/8
(C) 1/6
(D) 1/3
(E) 1/2
[Reveal] Spoiler: OA

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05 Feb 2012, 18:35
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Hi, there. I'm happy to help with this.

This is a cool question!

Let's call the boxes that contain $1,$100, and $1000, respectively, Box A, Box B, Box C. These are opened, respectively, by Key A, Key B, and Key C. We want to know the probability of winning more than$1000. Notice that if the distribution of keys is:

Box A = Key B
Box B = Key A
Box C = Key C

then the contestant wins exactly $1000, not more than$1000. The only configuration that leads to winning more than $1000 is: Box A = Key A Box B = Key B Box C = Key C i.e., getting all three keys correct. That's the only way to be more than$1000. So, really, the question can be rephrased: what is the probability of guessing the order of keys so that each key matches the correct box?

Well, for a set of three items, the number of possible permutations is 3! = 3*2*1 = 6.

Of those 6 possible permutations, only one of them leads to all three keys being paired with the right box. So, the answer is

[Reveal] Spoiler:
Probability = 1/6, answer = C

Does that make sense?

Here's another free practice question involving permutations:

http://gmat.magoosh.com/questions/857

The question at that link should be followed by a video solution.

Please let me know if you have any other questions.

Mike
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05 Feb 2012, 18:38
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calreg11 wrote:
On a game show, a contestant is given three keys, each of which opens exactly one of three identical boxes. The first box contains $1, the second$100, and the third $1000. The contestant assigns each key to one of the boxes and wins the amount of money contained in any box that is opened by the key assigned to it. What is the probability that a contestant will win more than$1000?

(A) 1/9
(B) 1/8
(C) 1/6
(D) 1/3
(E) 1/2

Total # of assignment of keys to boxes is 3!=6. Notice that it's impossible to assign only two keys correctly, so to win more than $1000 a contestant must assign all keys to the right boxes. Probability of that is 1/6. Answer: C. _________________ Manager Joined: 27 Oct 2011 Posts: 190 Location: United States Concentration: Finance, Strategy GMAT 1: Q V GPA: 3.7 WE: Account Management (Consumer Products) Followers: 5 Kudos [?]: 158 [0], given: 4 Re: On a game show, a contestant is given three keys, each of [#permalink] ### Show Tags 06 Feb 2012, 21:06 thank you guys for the response. _________________ DETERMINED TO BREAK 700!!! GMAT Club Legend Joined: 09 Sep 2013 Posts: 14367 Followers: 601 Kudos [?]: 174 [0], given: 0 Re: On a game show, a contestant is given three keys, each of [#permalink] ### Show Tags 21 Oct 2014, 15:56 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ SVP Joined: 17 Jul 2014 Posts: 2371 Location: United States (IL) Concentration: Finance, Economics Schools: Stanford '19 (D) GMAT 1: 650 Q49 V30 GPA: 3.92 WE: General Management (Transportation) Followers: 25 Kudos [?]: 287 [0], given: 147 Re: On a game show, a contestant is given three keys, each of [#permalink] ### Show Tags 31 Oct 2015, 12:51 2 This post was BOOKMARKED probability of winning 1000 is 1/3. But winning 1000 does not satisfy the condition. The contestant needs to open all boxes correctly so that to win more than 1000. The probability of winning is thus 1/3 * 1/2, since 1/2 is the probability of opening the remaining 2 boxes correctly. This results in 1/6. Intern Joined: 28 Jun 2015 Posts: 40 Location: Australia Followers: 0 Kudos [?]: 6 [0], given: 98 On a game show, a contestant is given three keys, each of [#permalink] ### Show Tags 16 Dec 2016, 01:24 Hi mikemcgarry Had the question been, What is the probability of winning a Prize of 1000 or less? Would the answer be 1/2? Best RzS Manager Joined: 13 Dec 2013 Posts: 55 Location: United States Concentration: Nonprofit, International Business GPA: 4 WE: Consulting (Consulting) Followers: 2 Kudos [?]: 11 [0], given: 86 Re: On a game show, a contestant is given three keys, each of [#permalink] ### Show Tags 21 Dec 2016, 09:44 For the contestant to win more that$1000, they must unlock the \$1000 box and one of the remaining boxes. However, if two boxes are unlocked it is impossible not to unlock the third box. The probability of opening the first two boxes correctly is: (1/3)*(1/2)=1/6
Re: On a game show, a contestant is given three keys, each of   [#permalink] 21 Dec 2016, 09:44
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