Answer Happy

5. (2 points) Consider the following game. Player A flips a fair coin until a head appears. She pays player B 2” dollars, where n is the number of tosses required until a head appears. For example, if a head appears on the first trial, player A pays player B $2. If the game results in 4 tails followed by a head, player A pays player B 25 = $32. Therefore, the payoff to player B is a random variable that takes on the values 2" for n = 1,2, ... and whose probability distribution is given by (42)” for n = 1,2, ..., that is, if X denotes the payoff to player B, = (4)" P(X = = 2") n = 1,2,3,... (a) What is the probability of player B winning no more than $8 in one play of the game? (b) What is the expected value of the payoff to player B?