A puzzle about an imaginary game show has revealed a counterintuitive strategy that doubles the chance of winning from 25% to 50%. The puzzle, set by Alex Bellos on his Monday puzzle column, involves two contestants each flipping a fair coin and then guessing the other's result. If both guess correctly, they win a prize.
The Game Show Setup
In the show, two people are chosen and placed in separate booths. Each flips a fair coin, visible only to the audience. They must then guess what the other person flipped—heads or tails. If both guess correctly, they receive a prize. Naively, each person has a 50% chance of guessing correctly, so the probability of both being correct is 25%.
The Winning Strategy
However, if the two contestants agree on a strategy beforehand, they can increase their odds. The solution: each person announces the result of their own coin flip. This raises the probability of winning to 50%. The four equally likely outcomes of two flips are HH, TH, HT, and TT. When both flip the same value (HH or TT), which happens 50% of the time, each correctly predicts the other's flip (since they announce their own), and they win. Alternatively, they could agree to guess the opposite of their own flip, yielding the same odds.
Explanation and Impact
The puzzle was contributed by Henk Tijms, Emeritus Professor of Operations Research at VU Amsterdam and author of several probability books. Bellos has been setting puzzles on alternate Mondays since 2015 and encourages readers to submit their own puzzles.
