Ann and Bruce play a game of alternating turns in which each turn consists of a single roll

of a fair six-sided die. Ann rolls first. The first person to roll a number that is greater than

two times the number that was last rolled (by their opponent) wins the game, and the game

ends then.

(a) What is the probability that Ann wins the game?

(b) What is the expected value of N, the number of turns until the game ends?

(c) What is the moment-generating function of N, m(t) = Ee¹?

(d) Ann suspects that Bruce's die is on average rolling 6 more often than hers. They decide

to perform an experiment to test this against the null hypothesis that Bruce's die rolls

6 no more often than hers. Write down a formula for an appropriate test statistic and

its approximate distribution under the null hypothesis.

In 130 rolls of Ann's die, 19 rolls were 6. In 90 rolls of Bruce's die, 23 rolls were 6.

What is the value of the test statistic and the p-value? Is this strong evidence that

Bruce's die rolls 6 more often than Ann's?