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Answer and discussion

Posted By: Timothy Chow
Date: Wednesday, 21 August 2024, at 4:29 a.m.

In Response To: My best probability puzzle (Timothy Chow)

The answer is that Andrew is more likely to win. I like this puzzle a lot because the result is so surprising. It is somehow even more surprising after you analyze the case of a single prize, and find that Andrew and Barbara are equally likely to win in that case.

That Andrew is more likely to win can be verified just by going through all 105 possibilities. This doesn't actually take that long to do because there are shortcuts; for example, if A does not contain a prize but B does, then of the 13 possible placements of the remaining prize, 12 favor Andrew and the last case is a tie. However, this brute-force approach is not very enlightening. I have a fairly general argument that can be used to analyze arbitrarily large rectangles, but it is rather complicated. I'm still looking for a nice argument that lets you see that Andrew has an advantage, without having to do a complicated case analysis, but I haven't found it yet. Bob Koca's analysis makes a good start, but I don't think it's quite there yet.

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