communicated to me by co-blogger Steve Miller, which he's too busy to post. Two players, A and B, start out with $a and $b dollars, respectively -- where a and b are natural numbers. They flip a fair coin, and every time it comes up heads, A gives 1 dollar to B; for every tails, B gives 1 dollar to A. The game ends when one of the players has zero dollars (and the other one has a+b). What's the probability that player A wins?
I was able to guess the answer right away (Steve told this to me on the phone), but this is probably more luck than anything else, as these intuitions can often be misleading. In any case, an answer is worthless without a proof, which Steve tells me is not entirely trivial. Furthermore, we don't know what happens if the coin, instead of being fair, has bias p. Anyone out there know?