Problem
The Problem
Two players, A and B, watch an infinite sequence of fair coin flips. Player A picks a triplet of heads/tails (e.g. HHH); player B, knowing A's choice, picks a different triplet. The first player whose triplet appears as three consecutive flips wins.
- If A picks HHH, what should B pick to maximise winning probability, and what is that probability?
- If A picks HHT, what should B pick?
- General principle: for any triplet A picks, can B always pick a triplet that beats it? This is Penney's game.
The non-transitivity of probabilities here is the bit that surprises people.