81. Let S = {1, 2, ..., n} and suppose that A and B are, independently, equally likely to be any of the 2ⁿ subsets (including the null set and S itself) of S.

(a) Show that

$$P(A \subset B) = \binom{n}{i}$$.

HINT: Let N(B) denote the number of elements in B. Use

$$P(A \subset B) = \sum_{i=0}^{n} P(A \subset B | N(B) = i)P(N(B) = i)$$

(b) Show that P(AB = Ø) = $$\binom{n}{i}$$.