MathProblem 52 Two children problem

A woman is chosen at random among all women that have two children. She is asked do you have at least one boy, and she answers 'yes.' What is the probability her other child is a boy? Assume every pregnancy has a 50/50 chance to be a boy or a girl.

Solution

询问得到已知的结果：至少有一个男生。现在问另一个孩子也是男生的概率。

设事件 $A$: 两个孩子都是男生，$B$: 两个孩子中至少有一个孩子是男生

$$P(A|B)=\frac{P(AB)}{P(B)}=\frac{1/4}{3/4}=\frac{1}{3}$$