两个孩子问题

Two Children

一个家庭有两个孩子。已知至少有一个孩子是男孩，问两个孩子都是男孩的概率是多少？（假设男女各 1/2 且独立）

A family has two children. Given that one of the children is a boy, what is the probability that both children are boys?

解析

样本空间： $\{BB,BG,GB,GG\}$ 等可能。

条件“至少一个男孩”排除 $GG$ ，剩下 3 种等可能： $BB,BG,GB$ 。

因此

P(BB\mid \text{至少一个男孩})=\frac{1}{3}.

Let's list all equally likely combinations of two children:

Given that at least one child is a boy, we eliminate the GG case.

So, the remaining valid possibilities are:

Out of these three, only one (BB) has both children as boys.

Therefore, the conditional probability that both are boys, given that one is a boy, is:

P(\text{both boys} \mid \text{at least one is a boy}) = \frac{1}{3}

✅ Final Answer:

\boxed{\frac{1}{3}}

So, the probability that both children are boys is $\frac{1}{3}$ .