1.(a) Let. A, B, C are mutually independent. Prove that A. is conditionally independent of B given C.

(b) Assume that A, B are both independent and conditionally independent given C. Is it necessary that A, B, C are mutually independent?

2. 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 will become heniophilic. Julie is a mother of two sons and from her medical history it is known that, with probability 1/4, she is hemophilic. That is the probability that

(a) her first son is hemophilic;

(h) her second son is hemophilic;

(c) both sons are hemophilic?

Cards are drawn at random from an ordinary deck of 52 cards, one- by-one without replacement. What is the probability that no heart is drawn before the ace of spades is drawn?