# Conditional probability in family

Assume that a family has four children. The probability of having a boy is p, and the genders of children in a family is assumed to be independent. Now, if we know that the eldest is a boy, the conditional probability of having two children of each gender is $3p(1-p)^2$. But what is the conditional probability of two children of each gender if we instead know that there is at least one boy?

This is an exercise in Blom. Thanks for your help!

Two boys and two girls: $6p^2(1-p)^2$. No boy: $(1-p)^4$. At least one boy: $1-(1-p)^4$. Two boys and two girls conditionally on at least one boy: $\dfrac{6p^2(1-p)^2}{1-(1-p)^4}$.