# Question

Consider the following game. Player A flips a fair coin until a head appears. She pays player B 2n 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 2n for n = 1, 2, . . . and whose probability distribution is given by (1 2)n for n = 1, 2, . . . , that is, if X de-notes the payoff to player B,

The usual definition of a fair game between two players is for each player to have equal expectation for the amount to be won.

(a) How much should player B pay to player A so that this game will be fair?

(b) What is the variance of X?

(c) What is the probability of player B winning no more than $8 in one play of the game?

The usual definition of a fair game between two players is for each player to have equal expectation for the amount to be won.

(a) How much should player B pay to player A so that this game will be fair?

(b) What is the variance of X?

(c) What is the probability of player B winning no more than $8 in one play of the game?

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