Separate Rooms 1
Alice and Bob are each placed in separate, distant rooms. Alice will flip a coin, and Bob will also flip a coin. Alice must guess the outcome of Bob’s coin toss, while Bob must guess the outcome of Alice’s coin toss. They will win if both guesses are correct. Before the game, they are allowed to discuss and devise a strategy. What is the probability of their winning, assuming they use an optimal strategy?
Answer
The sample space for the pair \((A,B)\), where \(A\) is Alice’s outcome and \(B\) is Bob’s outcome, is
If both guess randomly, the probability that both guesses are correct is \(\tfrac{1}{4}\). An optimal strategy is for each to guess that the other’s coin flip is the same as their own. Then they are both correct whenever \(A = B\). Since there are 4 equally likely outcomes and 2 of them have \(A = B\), the probability of winning is \(\tfrac{2}{4} = \tfrac{1}{2}\).