Hemophilia is a hereditary disease. If a mother has it, then with probability 1/2, any of her sons independently will inherit it. Otherwise, none of the sons becomes hemophilic. Julie is the mother of...

Hemophilia is a hereditary disease. If a mother has it, then with probability 1/2, any of her sons independently will inherit it. Otherwise, none of the sons becomes hemophilic. Julie is the mother of two sons, and from her family’s medical history it is known that, with the probability 1/4, she is hemophilic. What is the probability that

(a) her first son is hemophilic;

(b) her second son is hemophilic;

(c) none of her sons are hemophilic? Hint: Let H, H₁, and H₂
denote the events that the mother, the first son, and the second son are hemophilic, respectively. Note that H₁
and H₂
are conditionally independent given H. That is, if we are given that the mother is hemophilic, knowledge about one son being hemophilic does not change the chance of the other son being hemophilic. However, H1 and H2 are not independent. This is because if we know that one son is hemophilic, the mother is hemophilic and therefore with probability 1/2 the other son is also hemophilic.