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Sets R and S each contain three distinct positive integers.

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Sets R and S each contain three distinct positive integers.

by BTGmoderatorDC » Sat Jun 29, 2019 6:36 pm

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Sets R and S each contain three distinct positive integers. If integer r is randomly selected from R and integer s is randomly selected from S, what is the probability that rs = r?

(1) The probability that rs = s is 1/3

(2) The probability that r + s = 2 is 1/9

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Source: Princeton Review
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Source: — Data Sufficiency |

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by Ian Stewart » Mon Jul 01, 2019 6:20 am

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Since r is positive (so since it cannot be zero), we can divide by r in the question "What is the probability that rs = r?" to get the simpler question "What is the probability that s = 1?"

So it doesn't matter what is in set R, since we only care if we pick a "1" from set S. Since S contains 3 distinct elements, the answer will either be 1/3 if "1" is in set S, or will be 0 if "1" is not in set S.

Statement 1 tells us that the probability that rs = s, so the probability that r = 1 (we can divide by s since it is nonzero) is 1/3. We don't care at all about values in set R, so this Statement is useless information.

Statement 2 tells us that the probability r+s = 2 is 1/9. The only way, if r and s are positive integers, for r+s = 2 to be true is if r = 1 and s = 1. Since Statement 2 ensures that it is possible s = 1, the value "1" must be in set S, and Statement 2 is sufficient.
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