You choose the pattern HHH and I then choose THH. A fair coin is flipped until one of the two patterns appears. What is my probability of winning, and why does choosing second help?

You choose the pattern HHH and I then choose THH. A fair coin is flipped until one of the two patterns appears. What is my probability of winning, and why does choosing second help?

Approach: Think about what must happen before HHH can appear at all. Condition on the first three flips and use the fact that any tail occurring before a run of three heads puts my pattern ahead permanently.

7/8. HHH wins only when the first three flips are HHH, which has probability 1/8. On every other sequence a tail appears before the first run of three heads, and once a tail has appeared, any later HHH must be preceded by that T, so THH completes first. Hence THH wins with probability 7/8. The game is non-transitive, so there is no best pattern to name first: for any pattern of length three the second player prepends the correct symbol and takes an edge of at least 2 to 1. The second mover advantage comes from pattern waiting time being about precedence between two patterns rather than about each pattern's own frequency.

Follow-up: Which pattern of length four maximises the second player's winning probability against HHTT?

Key concepts: non-transitive game, pattern waiting time, second mover advantage, first three flips.