Problem #3 (15 points) - An Interesting Coin Game

Suppose that two people A and B are playing a game with a single coin

which has probability p of coming up heads and q = 1 p of coming up

tails. The game begins with A flipping the coin and then B flipping the coin

and then A flipping the coin and then B flipping, and so on, until the coin

comes up heads. The winner of the game is the one that flips a heads on

the coin.

a.) (10 points) The game is in no person's favor if the probability that A wins

is the same as the probability that B wins, both being 1/2. Show that the

game is always in A's favor (i.e., P(A) > 1/2 and P(B) < 1/2) for any

0 < p < 1.

b.) (5 points) Suppose that it cost A $a (which goes to the Casino) to play

the game and it cost B $b (which goes to the Casino) to play the game and

suppose that the winner of the game gets $c (c > a and c > b) from the

Casino. The game is called fair to a given player if the average winnings

(per play) for that player is $0. Determine (in terms of c and p) the values

of a and b if the game is to be fair to both players.