MathProblem 56 Babies in the nursery problem

A baby is added to a hospital nursery. Before the baby was added there were two boys in the nursery and an uncounted number of girls. After the new baby is added a baby is selected at random among all the babys. The selected baby is a boy. What is the probability that the added baby was a girl?

Solution

设事件 \(A:\) 新增的宝宝是女的， \(B:\) 选中的宝宝是男性

\[P(A|B)=\frac{P(AB)}{P(B)}=\frac{2/(x+3)}{\sum P(B|C)P(C)}=\frac{2/(x+3)\times 1/2}{\frac{1}{2}\frac{2}{x+3}+\frac{1}{2}\frac{3}{x+3}}=2/5 \]