A probability puzzle from Henk Tijms, Emeritus Professor of Operations Research at VU Amsterdam, imagines a TV game show where two contestants each flip a fair coin and must guess the other's result. If both guess correctly, they win a prize. Individually, each has a 50% chance, so the chance of both guessing correctly is 25%. But can you and a friend devise a strategy that improves those odds?
The Game Show Scenario
The compere announces that at the end of the show, two people will be chosen and placed in separate booths. Each flips a fair coin, out of sight of the other but visible to the audience. Then each must guess what the other flipped – heads or tails. If both guess correctly, they receive a prize.
As you watch from the studio, you and your friend are called up. While walking to the stage, you whisper a strategy to your companion that gives you a better than 25% chance of winning. What is this strategy, and what is the probability of winning?
Understanding the Problem
The naive probability of both guessing correctly is indeed 1/2 × 1/2 = 1/4, or 25%. However, the contestants can coordinate their guesses based on their own coin flip. Since each sees their own coin result, they can use that information to inform their guess about the other's coin. The key is that the guesses are not independent; they can be correlated.
The Optimal Strategy
The winning strategy is for each person to guess that the other's coin flip matches their own. In other words, if you flip heads, you guess the other flipped heads; if you flip tails, you guess tails. This ensures that both guess correctly exactly when both coins show the same face (both heads or both tails). The probability of both coins matching is 50% (since there are four equally likely outcomes: HH, HT, TH, TT, and two of them have matching results).
So, by using this strategy, the chance of winning increases from 25% to 50%.
Why It Works
The strategy works because it aligns the guesses with the actual outcomes. Each person's guess is determined by their own coin flip, so the pair guesses correctly exactly when the two flips are identical. This is a classic example of how coordination can improve probability in a cooperative game.
Henk Tijms, author of several books on probability, notes that this puzzle illustrates the power of correlated strategies. The solution is simple yet counterintuitive to those who assume independence.
Discussion
This puzzle was originally posted as part of Alex Bellos's Monday puzzle series. Readers are encouraged to discuss their favorite game shows without spoiling the answer. The strategy described above is the optimal one, giving a 50% chance of winning.
