(a) We say two events A and B are conditionally independent given an event C iff P(A B|C) = P(A|C)P(B|C). Verify this is equivalent to saying P(A|B C) = P(A|C). Also, verify it is equivalent to saying P(B|A C) = P(B|C). You may assume P(A) > 0, P(B) > 0 and P(C) > 0.

(b) We have 5 coins in a box. Four of these coins comes up heads 80% of the time, the last comes up heads 20% of the time. One coin is picked at random and independently tossed twice. Let Ai be the event you get heads on the i-th toss. Show that A1 and A2 are not independent.

(c) Consider two independent fair coin tosses, in which all four possible outcomes are equally likely. Let H1 be the event the first shows a head, H2 the second shows a head, and D the event the two tosses have different results. Verify that H1 and H2 are independent, but are not conditionally independent given D