Penney's Game: How Aunt Linda Wins the Washing-Up Coin Toss 2 Times in 3
The short version: any particular three tosses of a fair coin are equally likely to be heads-heads-heads or tails-heads-heads (1 in 8 each). But in a race to see which pattern appears first in a run of tosses, they're not equal. Against heads-heads-heads, tails-heads-heads wins 7 times in 8. Whatever pattern you choose, the second player can pick one that beats it at least 2 times in 3. The trick: take the opponent's first two tosses, put them at the end, and start with the opposite of their middle toss.
Uncle Gary and Aunt Linda settle the washing-up with a coin. Each picks a pattern of three tosses. They toss until one of the patterns appears. Whoever's pattern comes first doesn't wash up.
Gary always picks first, because he's the guest of honour in his own kitchen. He picks heads-heads-heads, because it's "strong". Linda picks tails-heads-heads. Gary has washed up on eleven of the last twelve Sundays. He has started wearing the rubber gloves to dinner, to save time.
Every pattern is equally likely, isn't it?
If you toss a fair coin three times, every pattern of three is equally likely: HHH, HHT, HTH and the rest each have a 1 in 8 probability. Each toss is independent of the others. So a contest between two patterns ought to be fifty-fifty.
It isn't, because the contest isn't "which pattern comes up in three tosses". It's "which pattern turns up first in a long run of tosses", and patterns get in each other's way.
Why tails-heads-heads beats heads-heads-heads
Think about how HHH can ever appear. The only way Gary wins is if the very first three tosses are all heads: a 1 in 8 chance. If a tail turns up at any point before HHH, then the next time two heads in a row arrive, they come straight after a tail, and that's T, H, H. Linda's pattern. She gets there before the third head can.
So Linda wins whenever the first three tosses aren't all heads: 7 times in 8. Gary's "strong" pattern is a sitting duck.
There's always a better pattern
Gary eventually switched patterns. It didn't help, because Linda always chose second. Against each of his choices, her answer won:
- Gary HHH, Linda THH: Linda wins 7 in 8.
- Gary HHT, Linda THH: Linda wins 3 in 4.
- Gary HTH, Linda HHT: Linda wins 2 in 3.
- Gary TTT, Linda HTT: Linda wins 7 in 8.
There's no best pattern. Like rock, paper, scissors, every choice has another that beats it. The rule Linda uses: take Gary's first two tosses and put them at the end of hers, then start with the opposite of his middle toss. Against HHT, his first two are HH and his middle is H, so she picks T, H, H. It wins at least 2 times in 3, every time. This is called Penney's game, after the mathematician Walter Penney, who described it in 1969.
How often the second player's pattern comes first
How to win the washing-up
1. Always choose second. Be gracious. Offer your opponent the first pick.
2. Apply the rule. Opposite of their middle toss, then their first two tosses. It works for every pattern of three.
3. Remember the lesson. "Equally likely" for a fixed set of tosses is not the same as "equally likely to come first". The question you ask changes the answer.
Where it costs you money
When someone offers you a game, a bet or a deal where they choose after seeing your choice, the order may be worth more than the rules. In markets, whoever sees your order before setting their price has a version of Linda's advantage. And our intuitions about random sequences are unreliable enough that "it should be fifty-fifty" deserves a check before money depends on it.
The moral
Linda explained the trick. Gary listened carefully and insisted on choosing second from now on. Linda agreed, and chose first: HTH. Gary, applying the rule, picked HHT and won 2 times in 3 for a month. Then Linda pointed out that the rule had been invented by a mathematician, that the washing-up rota had been invented by her, and that from now on they'd take turns. Gary has kept the gloves.
Probabilities are exact for a fair coin; we checked each match-up with 200,000 simulated games. Educational content, not financial advice.