2. Let A, B, C, D denote four events.

(a) (4 marks) Suppose P(C D) > 0, and prove that P(A B|C D) = P(ABC|D) P(C|D) .

(b) (9 marks) Suppose that A, B, C are independent given the event D, i.e., all these equalities hold:

P(A B C|D) = P(A|D)P(B|D)P(C|D),

P(A B|D) = P(A|D)P(B|D),

P(A C|D) = P(A|D)P(C|D),

P(C B|D) = P(C|D)P(B|D).

Define AB = (A Bc ) (Ac B). Is the following conditional independence true?

P((AB) C|D) = P((AB)|D)P(C|D)

Prove your answer.