1989 Putnam 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.)

如果$\alpha$是无理数，$0 < \alpha < 1$，是否存在

用一枚诚实的硬币进行有限游戏，使得一个玩家的概率

赢得比赛是$\alpha$？ (一枚诚实的硬币是指

正面的概率和反面的概率都是$\frac12$。

如果游戏以概率 1 必须以有限步数结束，则游戏是有限的。)