Jack and Diane take turns tossing the same coin. Diane wins if she gets a head before Jack gets a tail. Jack wins if he gets a tail before Diane gets a head. Diane goes first.
(a) If the coin is fair, show that the probability that Diane wins is greater than one-half.
Suppose that the probability of getting a head on any one toss is p.
(b) Find the value of p for which Jack & Diane have the same
probability of winning.
(c) Find the value of p that maximizes the expected length
of the game (i.e., maximizes the expected number of tosses).
(d) Find the value of p that maximizes the variance of
the length of the game.
(e) Find the moment generating function for Y= total # of tosses
in the game.