Guess each other's coin

Question

Alice and Bob are a team. They are put in separate rooms with no communication. Each flips a fair coin and sees only their own result. Then each must guess the other person's result. The team wins a prize of 3 dollars if both guesses are correct, nothing otherwise. Playing costs 1 dollar.

Guessing at random, each is right with probability 1/2, so the team wins with probability 1/4 and the game is a losing proposition.

Can Alice and Bob agree a strategy beforehand that makes the game worth playing? What is the best probability of winning?

Accepted Answer

Yes. If each guesses that the other's coin shows the same face as their own, they win with probability 1/2, and the game is worth 1.50 dollars against a cost of 1. The strategy. Alice guesses Bob's coin equals her own; Bob guesses Alice's coin equals his own. When does it win? Alice is right exactly when the two coins match, and Bob is right exactly when the two coins match. So both are right at the same time: whenever the two coins show the same face, which happens with probability 1/2. Both are wrong the other half of the time. Why this beats random guessing. Individually, each player is st…

Guess each other's coin

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