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Probability & Cominatorics Test #10

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Re: Probability & Cominatorics Test #10 [#permalink] New post 15 Oct 2013, 16:11
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VeritasPrepKarishma wrote:
Why is 6/55 incorrect? The probability of achieving the 55 cases is not the same (It's similar to the throwing two dice situation if someone has come across those questions). The probability of achieving each of the favorable 6 un-arranged cases is the same i.e. you can get each of them in 2 ways. But the probability of achieving each of the total 55 un-arranged cases is not the same. You can get (20, 21) in 2 ways but (20, 20) in only one way in the un-arranged case. Hence you cannot add them up to get the total number of cases as 55. In total number of cases, the probability of achieving each case has to be the same. In case it isn't, we need to consider the order to get all the cases. And this is the reason I do not agree with GMAT TIGER's response.

Thoughts?

Dear Karishma,

Actually, this is why I think there is a fundamental flaw in the question, because something basic about the selection process is not specified. Here's what I mean:
Interpretation A
Is it correct to say, as you say, that there are two different "ways" to get, say, (20, 21) but only one "way" to get (20,20)? This leads to the 6/50 = 0.12 solution.
Interpretation B
Or are we completely ignoring the "ways" to select/produce the pairs, and just saying that all the resultant pairs are equally likely? In this case, we have 55 equally likely pairs, and the solution is 6/55.

The question does say: "The numbers are selected independently of each other"
The way I understood this: To say A and B are truly independent is to say --- If I tell you how A turns out, that gives you absolutely no information about how B turns out. (This is one of many ways to understand the idea of independence.)
Well, if we are told that one of the numbers selected is, say, 24, then in order for that to give us absolutely no information about what the other number is, it must mean that all numbers, including a second 24, are equally likely. This would occur only in Interpretation B, not Interpretation A. In interpretation A, if we are told one of the number is 24, then a second 24 would be half as likely as any other number.

If you think of Interpretation A, and the associated 10 x 10 square of possibilities, a cross-shape of 19 squares constitutes the space of "one number is 24". Within that cross-shape of 19 spaces, each other number has a probability of 2/19 of being paired with 24, but the second 24 has a probability of 1/19 of being paired with the first 24.
In Interpretation B, and the associated right triangle of 55 spaces, there's a rotated-L shape of 10 squares that contain at least one 24, and all the the numbers paired with 24, including the second 24, are equally likely. That's how I conceived of the independence in this situation.

Of course, there are other ways to frame the idea of independence, some of which I readily believe could be used to justify Interpretation A. That's precisely why I suspect there's a flaw in the question. What do you think?

Mike :-)
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Re: Probability & Cominatorics Test #10 [#permalink] New post 15 Oct 2013, 20:57
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mikemcgarry wrote:
VeritasPrepKarishma wrote:
Why is 6/55 incorrect? The probability of achieving the 55 cases is not the same (It's similar to the throwing two dice situation if someone has come across those questions). The probability of achieving each of the favorable 6 un-arranged cases is the same i.e. you can get each of them in 2 ways. But the probability of achieving each of the total 55 un-arranged cases is not the same. You can get (20, 21) in 2 ways but (20, 20) in only one way in the un-arranged case. Hence you cannot add them up to get the total number of cases as 55. In total number of cases, the probability of achieving each case has to be the same. In case it isn't, we need to consider the order to get all the cases. And this is the reason I do not agree with GMAT TIGER's response.

Thoughts?

Dear Karishma,

Actually, this is why I think there is a fundamental flaw in the question, because something basic about the selection process is not specified. Here's what I mean:
Interpretation A
Is it correct to say, as you say, that there are two different "ways" to get, say, (20, 21) but only one "way" to get (20,20)? This leads to the 6/50 = 0.12 solution.
Interpretation B
Or are we completely ignoring the "ways" to select/produce the pairs, and just saying that all the resultant pairs are equally likely? In this case, we have 55 equally likely pairs, and the solution is 6/55.

The question does say: "The numbers are selected independently of each other"
The way I understood this: To say A and B are truly independent is to say --- If I tell you how A turns out, that gives you absolutely no information about how B turns out. (This is one of many ways to understand the idea of independence.)
Well, if we are told that one of the numbers selected is, say, 24, then in order for that to give us absolutely no information about what the other number is, it must mean that all numbers, including a second 24, are equally likely. This would occur only in Interpretation B, not Interpretation A. In interpretation A, if we are told one of the number is 24, then a second 24 would be half as likely as any other number.

If you think of Interpretation A, and the associated 10 x 10 square of possibilities, a cross-shape of 19 squares constitutes the space of "one number is 24". Within that cross-shape of 19 spaces, each other number has a probability of 2/19 of being paired with 24, but the second 24 has a probability of 1/19 of being paired with the first 24.
In Interpretation B, and the associated right triangle of 55 spaces, there's a rotated-L shape of 10 squares that contain at least one 24, and all the the numbers paired with 24, including the second 24, are equally likely. That's how I conceived of the independence in this situation.

Of course, there are other ways to frame the idea of independence, some of which I readily believe could be used to justify Interpretation A. That's precisely why I suspect there's a flaw in the question. What do you think?

Mike :-)


Hey Mike,

The reason I would not dwell on the independence issue is that the question clarifies what they want us to take away from it.

"The numbers are selected independently of each other, i.e. they can be equal"

They tell us that the numbers are selected independently, that is they can be equal. They have given the implication of the previous part of the sentence - the numbers can be equal. So basically, we say that once you pick a number, you can pick the same number again. This gives the process a natural sequence and very often in P&C questions we rely on such inferences. It's like throwing two dice. What you get on one is independent of what you get on another but the probability of getting two 6s is half of the probability of getting a 1 and a 2.

To my mind, the intent of the question is clear. Of course, in case enough people find the interpretation suspect, it would be removed from the question bank in actual GMAT scenario since interpretation is a subjective issue.

Thanks for a great discussion!
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Re: Probability & Cominatorics Test #10   [#permalink] 15 Oct 2013, 20:57
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