100 点：99 只羊 1 只狼

99 sheep, 1 wolf on a circle

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

在圆周上 100 个等间距点上放置 99 只羊与 1 只狼。每个时间步狼以相等概率向顺/逆时针移动 1 格。若狼落到有羊的位置，该羊被吃掉。

问：最开始与狼“正对面”的那只羊成为最后幸存者的概率是多少？

We place 99 sheep, 1 wolf on these 100 points. Each time step, the wolf moves $\pm1$ (clockwise or counterclockwise) with equal probability. If it lands on a sheep, that sheep is eaten. We want the probability the sheep initially opposite the wolf is the last sheep remaining.

解析

由于对称性，在 99 只羊中每只羊成为最后幸存者的概率相同，均为 $\frac{1}{99}$ 。因此那只“正对面”的羊成为最后幸存者的概率是

\boxed{\frac{1}{99}}.

Original Explanation

By symmetry among all the sheep, each sheep is equally likely to be the final survivor. With 99 sheep, each has probability $\frac{1}{99}$ . So the chance that that particular sheep (the one opposite) is last is $\frac{1}{99}$ .