1/3 of golfers also play tennis, 3/5 of tennis players also surf. 4/7

1) gives ut that 2/3 of golfers surf
2) gives us that 3/5 of the other 1/3 of the golfers also surf

10/15 golfers surf
plus 3 of the 5 remaining golfers

13/15 golfers surf

4/7 surfers ski, so the number of surfers is a multiple of 7. It could be a total of 14 surfers or maybe a total of 140 surfers.

IF there was 14 surfers, then 13 of these would have been golfers, and 3 or 4 of these golfers would have skied. However, even with this known amount of surfers, we would not be able to answer the question.

I don't understand why this isn't solvable. I also don't see how this is not more a probability question than an overlapping sets question.

I'll use some real numbers to make my case clearer.

Say there are 105 golfers.

Using the prompt, we know 105 \* 1/3 = 35 of them play tennis (in fact, from (2), we know that these are all the tennis players). We can expect 35 \* 3/5 = 21 of them to surf. And then we can expect 21 \* 4/7 = 12 of them to ski.

The remainder of the golfers, 105 * 2/3 = 70 of them, don't play tennis, but from (1) we know they all surf. Using the prompt, we can expect 70 * 4/7 = 40 of them to ski.

This means we can expect 12 + 40 = 52 of the surfers to ski, and 52/105 < 1/2

As a quick summary: (1/3 * 3/5 * 4/7) + (2/3*4/7) = 52/105 < 1/2

What is wrong with my logic?

Edit: I now see what's wrong with my logic. Never mind. Will delete this in a moment. Just going to kick myself. Brb.

Could you please explain what IS wrong with this logic?

If 1/3 of golf players play tennis and it is given that all golfers who do not play tennis, ski, it means 2/3rd of all golf players ski. So the answer is supposed to be A. If I am wrong, can someone shed some light on where I went wrong?

can you please share your corrected approach?

Hello IanStewart, could you please help with this?

Sure - say we use both Statements, and there are 15 golfers. We know 5 golfers play tennis, and there are 5 tennis players in total, from Statement 2. From the "3/5" in the question stem, we know 3 of these golfer-tennis-players is also a surfer. We also know the remaining 10 golfers are also surfers, from Statement 1. So we have 13 tennis players who are surfers in total.

But that's really all we know, because there's no connection between the last fraction in the stem, the "4/7", and anything else we just learned. Maybe there are 7,000,000 surfers in total, most of whom don't golf at all, and 4,000,000 of them ski, so 3,000,000 of them don't ski. Our 13 tennis players who are surfers could all belong to that group of 3,000,000 non-skiers, or they could all belong to that group of 4,000,000 skiers. And we don't know anything about the other two golfers (the two who only play tennis) either. So the probability the question asks for could be anything between 0 and 1, and the answer is E.

It's a bit of a confusing question at first, and I imagine many people will just start multiplying the fractions as if it were a standard probability question -- that was the mistake in the solution posted above by physiltant (who realized their mistake but never corrected it). physiltant was assuming all of the people were golfers, and then was further assuming that when, say, "4/7 of surfers also ski", that you could then say that "from the group of tennis players who surf, 4/7 of them must ski", and we have no reason to think that's true. It's maybe a bit confusing to see in the context of this question, but the principle is the same in this example: if you knew in country X that 10% of people were unemployed, there'd be no reason to expect that exactly 10% of people with Engineering degrees were unemployed, say. The same is true: just because 4/7 of an entire group skis, that doesn't mean 4/7 of some specific subset of that group (the tennis players) will also ski.

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