红糖取完时蓝绿都还剩的概率

Candies in a Jar

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

罐子里有 10 颗红糖、20 颗蓝糖、30 颗绿糖。你不放回地逐颗随机取出。

问：在你取完所有红糖的那一刻，罐子里仍至少剩 1 颗蓝糖且至少剩 1 颗绿糖的概率是多少？

There are 10 red candies, 20 blue candies, and 30 green candies in a jar. You draw candies one by one (no replacement). What is the probability that, by the time you have drawn all red candies, there is still at least one blue and at least one green left in the jar?

解析

答案是 $\frac{7}{12}$ 。

设 $T_r,T_b,T_g$ 分别为最后一颗红/蓝/绿糖被取出的时间（抽取序列中的位置）。要求“红先取完且蓝绿未取完”等价于

T_r<T_b\ \text{且}\ T_r<T_g.

对颜色顺序进行计数/对称性分析可得概率为 $7/12$ 。

Original Explanation

Let $T_r, T_b, T_g$ be the draw indices of the last red, blue, and green candy, respectively. We require $T_r < T_b$ and $T_r < T_g.$ One shows $P(T_r < T_b \,\cap\, T_r < T_g) = P(T_r < T_b < T_g) + P(T_r < T_g < T_b) = \frac{7}{12}.$