We are given a coin for which the probability of heads is p (0

(a) Let Y be the number of flips up to and including the flip on which the first head occurs. Write down the PMF of Y .

(b) Let X be the number of heads that occur on any particular flip. Write down E[X] and var(X).

(c) Let K be the number of heads that occur on the first n flips of the coin. Determine the PMF, mean, and variance of K.

(d) Given that a total of exactly six heads resulted from the first nine flips, what is the conditional probability that both the first and seventh flips were tails?

(e) Let H be the number of heads that occur on the first twenty flips. Let C be the event that a total of exactly ten heads resulted from the first eighteen flips. Find E[H | C] and the conditional variance var(H | C).