Penney’s Game: The Coin Flip Bet That Is Not Fair
Penney’s Game looks like the fairest bet in the world. Two players each pick a pattern of three coin flips, such as heads-heads-tails, and a coin is flipped until one of the patterns appears. Every pattern of three is equally likely, so it should be 50/50. It is not. Whatever the first player picks, the second player can pick a pattern that wins at least two times out of three.
How to Play
- Player 1 picks a pattern of three flips, such as HTH (heads, tails, heads), and says it out loud.
- Player 2 picks a different pattern of three, after hearing player 1’s choice.
- Flip a coin again and again and write down each result in a row: H, T, T, H...
- The first pattern to appear wins. As soon as the last three flips match one player’s pattern, that player wins the round.
You need one coin, or our Coin Flip tool, and a piece of paper to write the flips on.
The Trick: How Player 2 Always Wins
Player 2 uses a simple rule. Take player 1’s first two flips, put them at the end, and in front of them put the opposite of player 1’s middle flip.
For example, player 1 picks HHT. The first two are HH. The middle flip is H, so its opposite is T. Player 2 picks THH, and wins 3 times out of 4.
Here is the best reply to every pattern, with player 2’s exact chance of winning:
| If player 1 picks | Player 2 should pick (and wins) |
|---|---|
| HHH | THH (7 in 8) |
| HHT | THH (3 in 4) |
| HTH | HHT (2 in 3) |
| HTT | HHT (2 in 3) |
| THH | TTH (2 in 3) |
| THT | TTH (2 in 3) |
| TTH | HTT (3 in 4) |
| TTT | HTT (7 in 8) |
We checked every row exactly, and with 200,000 simulated games for a few of them. The worst player 2 can do with this rule is win 2 games in 3.
Why It Works
Patterns are equally likely in any three fixed flips, but in a race they overlap. Look at HHT against THH. The only way for HHT to win is if the very first two flips are HH: from then on, the coin is waiting for a T, and the HH run keeps going until it comes. If the first two flips are anything else, a T has already appeared before any HH, and THH will turn up before HHT can. The first two flips are HH only 1 time in 4, so THH wins the other 3.
Player 2’s pattern is built to sneak in just before player 1’s: its last two flips are player 1’s first two. Whenever player 1’s pattern is about to begin, player 2’s pattern has usually just finished.
A Game with No Best Pattern
Penney’s Game is non-transitive, like Rock Paper Scissors: THH beats HHT, HHT beats HTT, HTT beats TTH, and TTH beats THH. There is no single best pattern. Whoever chooses second can always win the advantage, so the fair way to play is to take turns choosing first.
A Second Surprise: Waiting Times
Here is another one that catches people out. If you flip a coin and count how long it takes for a pattern to appear, some patterns take much longer than others:
| Pattern | Average flips until it first appears |
|---|---|
| HHT, HTT, THH, TTH | 8 |
| HTH, THT | 10 |
| HHH, TTT | 14 |
HTH takes 10 flips on average but HTT only 8, although both are just as likely in any three flips. When HTH fails at its last flip, you get HTT, and have to start over from that final T. When HTT fails at its last flip, you get HTH, and that final H is already the start of a new try.
None of this means a coin has a memory. Each single flip is still exactly 50/50, as our guide to coin flip odds explains, and a run of heads never makes tails due; that is the gambler’s fallacy. The surprises come only from how patterns overlap.
Where the Game Comes From
The game is named after Walter Penney, who described it in a recreational mathematics journal in 1969. The mathematician John Conway later found a neat way to work out the odds between any two patterns, and Martin Gardner made the game famous in his puzzle columns.
Play It in Class or at a Party
- Let a volunteer pick first, then pick second yourself and win again and again. Ask the class to work out your secret.
- Play with the whole class: split into two teams, and flip the coin on a big screen with the Coin Flip tool.
- Try four-flip patterns once everyone knows the trick. For more ideas, see our coin flip games.
FAQ
What is Penney’s Game?
A coin game in which two players each pick a pattern of three flips, and a coin is flipped until one of the patterns appears. Although every pattern is equally likely in any three flips, the second player can always choose a pattern that wins more often.
How do you always win Penney’s Game?
Choose second. Take the first player’s first two flips, put them at the end of your pattern, and start it with the opposite of their middle flip. Against HHT, choose THH.
What are the odds in Penney’s Game?
With the best reply, the second player wins at least 2 games in 3. Against HHH or TTT, the right reply wins 7 games in 8.
Is there a best pattern in Penney’s Game?
No. Every pattern can be beaten by another, like in Rock Paper Scissors, so the player who chooses second has the advantage.
Why does HTH take longer to appear than HTT?
On average HTH takes 10 flips and HTT 8. When an attempt at HTH fails, you have to start again, but a failed HTT leaves an H that already begins a new try.