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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, 16:09
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Difficulty:

75% (hard)

Question Stats:

47% (01:28) correct 53% (01:38) wrong based on 204 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, 17:35
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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, 17:38
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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.

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

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06 Feb 2012, 20:06
thank you guys for the response.
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Re: On a game show, a contestant is given three keys, each of [#permalink]

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21 Oct 2014, 14:56
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Re: On a game show, a contestant is given three keys, each of [#permalink]

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31 Oct 2015, 11:51
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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.

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

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16 Dec 2016, 00:24
Hi mikemcgarry

Had the question been, What is the probability of winning a Prize of 1000 or less?

Best
RzS

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

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21 Dec 2016, 08:44
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

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

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21 May 2017, 15:09
TheRzS wrote:
Hi mikemcgarry

Had the question been, What is the probability of winning a Prize of 1000 or less?

Best
RzS

It will be 1-1/6 = 5/6.

Thanks.

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Re: On a game show, a contestant is given three keys, each of   [#permalink] 21 May 2017, 15:09
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