Problem 144 Answer

The answer is a bit hard to explain and appears paradoxical on the surface. Consider the fact that whenever a bet is resolved the probability of the same player winning the next one is 7/13. So in this game there will be longer strings of the same player winning in a row than in a game where every trial is independent of the last, like flipping a coin. Player B can expect winnings even closer together than player A because the probability of B winning the roll after a win is just 1/6, as opposed to the probability of 1/36 of A winning one roll after the last win. So with Bs wins being more clumped together picking a flip at random is more likely to preceed an A win, since As wins are more spaced apart.

Thanks to Extra Stuff: Gambling Rambling by Peter Griffin for this problem. See chapter 6.

Michael Shackleford, ASA, August 19 1999

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