If a father is type O, what is the probability of the child being his son if he's type A and the mom is type B.

Nil.

Explanation:

In the given question the parent blood type, father is O blood type and mother blood type is B. The genotype of blood type of father is OO and genotype of mother's blood will be either BB or BO. If mother is BB then all the offspring will have blood group B with genotype BO. If mother is BO, then half of the offspring will have O blood type and half will have B blood type. In no case the son will have blood group A. Thus, the probability of Blood group of child A is nil.