In the context of the random experiment of tossing a coin twice, derive the probability of event A = {(HT), (TH)} given event B = {(HH), (HT)}. Explain why events A and B are independent.

25*. For two events A and B in S, use Venn diagrams to evaluate

(a) the smallest possible value of P(A ∪ B) and

(b) the greatest possible value of P(A ∩ B), under the following two scenarios: (i) P(A) = .4, P(B) = .6; (ii) P(A) = .3, P(B) = .5. Hint: Consider all possible relationships between A and B within S.