It belongs to the family of non-transitive games in which choosing second is not a disadvantage, but rather a way of responding to the rival sequence with a better one.
Penney's game
Riddle statement
Two players take turns choosing a sequence of three coin outcomes.
Then a fair coin is tossed repeatedly until one of the two sequences appears for the first time as a consecutive block.
Whoever chose that sequence wins.
The first player has chosen:
heads, heads, heads
You choose later.
Which sequence should you choose to maximize your chances of winning?
Show solution
Solution
The optimal response to heads-heads-heads is to choose tails-heads-heads.
The first player can only win if the game starts with exactly three consecutive heads from the first toss. The probability of that start is barely 1/8.
In any other case—that is, as soon as a tail appears before completing that initial streak—the sequence tails-heads-heads is structurally better positioned to close before heads-heads-heads.
The result: tails-heads-heads wins with probability 7/8.