Penney's Game

The problem
YouHHHFriendTHHvsflip a fair coin repeatedlyfirst pattern to appear wins

You and a friend play a game. You each pick a sequence of three coin flips:

  • You: Heads, Heads, Heads (HHH)
  • Friend: Tails, Heads, Heads (THH)

A fair coin is flipped repeatedly. The first sequence to appear in order, as consecutive flips wins. Who wins more often?

Tempting (but wrong)
any random 3 flips → each pattern has prob 1/8HHH1/8THH1/8→ "fair game" (wrong question)

The instinct: on any random three consecutive flips, HHH and THH each occur with probability 1/8. Same odds, no edge. So it's a fair game.

This argument is right about individual three-flip windows but wrong about which sequence appears first in an ongoing flip stream. The two are not the same question.