Putnam 1989 A4

If $\alpha$ is an irrational number, $0 < \alpha < 1$, is there a finite game with an honest coin such that the probability of one player winning the game is $\alpha$? (An honest coin is one for which the probability of heads and the probability of tails are both $\frac12$. A game is finite if with probability 1 it must end in a finite number of moves.)