3. Three players, Alan, Bob, and Cindy, throw a perfect die in turn independently in the order A, B,C, A, . . . until one wins by throwing a 5 or a 6. Show that the probability generating function F(s) for the random variable X which takes the value r if the game ends on the r th throw can be written as F(s) =

9s 27 − 8s3 +

6s2 27 − 8s3 +

4s3 27 − 8s3

.

Hence find the probabilities of winning for Alan, Bob, and Cindy. Find the mean duration of the game. (Oxford 1973M)

4.5 Problems 59