The Puzzle Setup
Earlier today, I presented a puzzle involving a fictional game show.
At the conclusion of the show, two contestants are selected and each is placed in a separate booth.
Inside their booths, each contestant flips a fair coin, unseen by the other but visible to the audience.
Following the coin flips, each contestant must guess the result of the other person's coin—either heads or tails. If both guess correctly, they win a prize.
Since each guess has a 50% chance of being correct, the probability that both guess correctly is 25%.
A Surprising Strategy
Imagine you are in the studio watching the show, and unexpectedly, you and a friend are called to participate. As you approach the stage, you quietly share a strategy with your companion that increases your chances of winning beyond 25%. What is this strategy, and what is the new probability of winning?
Remarkably, if each contestant announces the result of their own coin flip, the probability of winning rises to 50%.
The four equally likely outcomes of the two coin flips are HH, TH, HT, and TT.
In half of these cases (HH or TT), both contestants will have flipped the same result, allowing each to correctly predict the other's coin flip and thus win the prize.
Alternatively, the contestants could agree to guess that the other person's coin flip is the opposite of their own. This approach yields the same probability of success.
Closing Remarks
I hope you found today’s puzzle engaging. I will return with another puzzle in two weeks.
Thanks to Henk Tijms for providing this puzzle. Henk is Emeritus Professor of Operations Research at VU Amsterdam and the author of several books on probability.
Since 2015, I have been posting a puzzle every alternate Monday and am always seeking excellent puzzles. If you would like to suggest one, please email me.
“Earlier today I set you this puzzle about an imaginary game show.
At the end of the show two people will be chosen and each placed in a separate booth.
In the booth, each of them will flip a fair coin, out of sight of the other person but visible to the audience.
Then each of them must guess what the other person flipped—heads or tails. If they both guess correctly, they receive a prize.
The chance of each person guessing correctly is 50 per cent. So the chance of both guessing is 25 per cent.
You’re in the studio watching the show, and to your surprise, you and your friend are called up to play the game. As you walk up to the stage, you whisper a strategy to your companion that gives you a better than 25% chance of winning the prize. What is this strategy, and what is the probability of winning?
Remarkably, if each of you announces the result of your own coin flip you will raise your probability of winning to 50 per cent!
The four equally likely outcomes of two flips are HH, TH, HT and TT.
You will both flip the same value (HH, or TT) 50 per cent of the time, and when you do you will both have predicted the other person’s flip, and therefore win the prize.
Alternatively, you could each agree to guess that the other has flipped the opposite of your own flip. The odds work out the same.
I hope you enjoyed today’s puzzle. I’ll be back in two weeks.
Thanks to Henk Tijms for today’s puzzle. Henk is Emeritus Professor of Operations Research at VU Amsterdam and the author of several books on probability.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
