A short explainer on Penney's Game, the non-transitive coin-flip game where the second player can always pick a three-flip sequence that beats the first player's — sometimes 7 to 1. Includes the optimal-response rule and the full odds table.
A short tour of a non-transitive coin game where the smart move is to let your opponent choose first.
TL;DR
Penney's Game is a two-player coin game. Each player picks a length-three sequence of heads and tails, like HHT. You then flip a fair coin over and over until one of the two chosen sequences shows up as three flips in a row — that player wins. The catch is that the game is non-transitive, just like rock-paper-scissors: whatever the first player picks, the second player can always pick something that beats it, sometimes by a mile. The whole trick is to never pick first.
The rules
- Player 1 announces a sequence of three, say
HHH. - Player 2, having heard it, announces their own, say
THH. - Flip a fair coin, tracking the running stream, until one player's pattern appears as three consecutive flips.
- That player wins.
It feels perfectly symmetric. It is not.
Try the race yourself. Pick two patterns, flip one coin at a time, or run a thousand fresh games. The theoretical bar is calculated; the observed bar records your trials.
Loading probability experiment…
Why going second wins
Take HHH vs THH. For HHH to win, the very first three flips must all be heads — probability 1/8. If they aren't, then a tails has appeared somewhere before the run of heads finished, and the instant that tails is followed by two heads, THH has already won. So out of the gate, THH wins 7 times out of 8.
The same logic, more gently, gives THH the edge over HHT at 3/4. Your sequence is essentially lying in wait to swallow the opponent's.
The second player's recipe
There's a one-line rule for the optimal response. Given Player 1's sequence ABC, Player 2 picks:
the opposite of the middle letter, then the first two letters →
(not B) A B.
So HTH → middle is T, flip to H, prepend to HT → Player 2 plays HHT.
The full table of best responses and the odds they give the second player:
| Player 1 | Player 2 | Odds for P2 |
|---|---|---|
HHH | THH | 7 : 1 |
HHT | THH | 3 : 1 |
HTH | HHT | 2 : 1 |
HTT | HHT | 2 : 1 |
THH | TTH | 2 : 1 |
THT | TTH | 2 : 1 |
TTH | HTT | 3 : 1 |
TTT | HTT | 7 : 1 |
Every row is a winning bet for the player who moved second.
The surprising part
There is no best sequence. Notice the loop: HHT beats HTH, HTH beats HTT... and THH beats HHT, which would beat something that beats THH. Pick any sequence and another one beats it — exactly the rock-paper-scissors structure, except here the imbalance can be a brutal 7-to-1 rather than a fair third.
Our intuition assumes that "more likely to win than X" is a transitive ranking. Penney's Game is the cleanest possible demonstration that it doesn't have to be. So the next time someone offers to let you pick your three coin flips first, decline — and make them go first instead.