Suppose we were to play a game with the following rules. We pay 8 dollars to play the game. We then flip a (possibly biased, probability of heads p) coin 21 times. Our payoff is equal to three dollars plus an amount of dollars equal to half the number of heads obtained. Since we don't know how many heads will result, the payoff is a random variable. Call that random variable A (for amount of payoff). a) Find the moment generating function, M_X(t), of the random variable X. b) Differentiate to obtain M_X(t) to obtain E(X). c) Use the result of part b to find the expected winnings from playing this game. What value of p (if any) makes this a fair game?