(i) Show that if P(A) = 0 or P(A) = 1, then A is independent of every other event. Show that if A is independent of itself then P(A) is either O or 1. (ii) Suppose k events form a partition of the...

Extracted text: (i) Show that if P(A) = 0 or P(A) = 1, then A is independent of every other event. Show that if A is independent of itself then P(A) is either O or 1. (ii) Suppose k events form a partition of the sample space N, i.e., they are disjoint and UL, A; = N. Assume that P(B) > 0. Prove that if P(A1|B) < p(a1),="" then="" p(a;|b)=""> P(A;) for some i = 2, ..., k.