A puzzle about an imaginary game show has been solved, revealing a strategy that doubles the chance of winning from 25% to 50%. The puzzle, set earlier today by Alex Bellos, involved two contestants each flipping a fair coin and then guessing the other's result. If both guess correctly, they win a prize.
The puzzle setup
In the game show, two people are placed in separate booths. Each flips a fair coin, visible only to the audience. Then each must guess whether the other flipped heads or tails. Individually, each has a 50% chance of guessing correctly, so the probability of both guessing correctly is 25%. However, a strategy exists to improve the odds.
The winning strategy
The solution, as explained by Bellos, is for each contestant to announce 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 occurs 50% of the time, each will have predicted the other's flip correctly, thus winning the prize. Alternatively, each can agree to guess that the other flipped the opposite of their own flip, yielding the same odds.
Source and background
Today's puzzle was contributed by Henk Tijms, Emeritus Professor of Operations Research at VU Amsterdam and author of several books on probability. Bellos has been setting puzzles on alternate Mondays since 2015 and invites readers to suggest future puzzles via email.
