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Problem of the Week 1234
Chess in Black and White
Alice will play a 5-game chess match against Bob in which they alternate colors. She wins the match if she wins two games in a row.
Suppose that the probability of Alice winning when playing White is greater than when she has Black. Alice can choose which color to start the match with.
What should she choose?
The two probabilities can be anything and need not sum to 1 because the game can have draws, or Alice can have very high win probabilities in both cases. The condition applies only to Alice: if Bob wins two games in a row, that has no impact on whether Alice wins the match.
Extra Credit: There are obvious extensions.
Assume the match is n games long, not 5. The case of
Let the probabilities be p and 1 − p as opposed to the more general p and q in the stated problem. I am not sure this matters, but perhaps it changes things.
Assume Alice needs to win Z in a row, not 2 in a row.
Assume that patterns other than the alternating pattern are allowed.
Source: Based on a problem in the very nice new problem book Half a Century of Pythagoras Magazine, eds A. van den Brandhof, J. Guichelaar, and A. Jaspers, MAA, 2015.