转移后从 B 抽到绿球的概率

What is the probability that this ball is green?

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

There are 1000 green balls and 3000 red balls in container $A$ , and 3000 green balls and 1000 red balls in container $B$ . You take half of the balls from $A$ at random and transfer them to $B$ . Then you take one ball from $B$ at random. What is the probability that this ball is green?

解析

把 A 的 4000 个球随机取一半（2000 个）转移到 B。

转移的绿球期望数为

2000\cdot\frac{1000}{4000}=500.

因此 B 中绿球期望数为 $3000+500=3500$ ，B 总球数为 $4000+2000=6000$ 。

随机从 B 抽一球为绿的概率等于绿球比例的期望（由对称性/线性期望可严格证明），因此

\boxed{\frac{3500}{6000}=\frac{7}{12}}.