### Coin Sequence Game

Source: Plus Magazine Issue 55 Problem: Let's witness a game. The game involves flipping a coin, which you assume has equal probability of coming up heads or tails. The game is played by two players, A and B, who each select a sequence of three flips. For example, assume that Player A selected "heads-heads-heads" (HHH) and Player B has selected "tails-heads-heads" (THH). Then the coin is flipped repeatedly, resulting in a sequence like the following: HTHTHHHHTHHHTTTTHTHH... The player whose sequence showed up first (HHH for Player A or THH for Player B) is declared the winner. A chooses first and you have to help B. Show that given what A has chosen, B can always choose a sequence such that B is in a better position than A. For example: If A chooses HHH, B chooses THH to make sure that now it has 7/8 probability to win. Update (27/07/10): Solution: Posted in comments by me!