A group of n friends often go out for dinner together. At their dinners, they play

Chapter 9, Problem 39

(choose chapter or problem)

A group of n friends often go out for dinner together. At their dinners, they play credit card roulette to decide who pays the bill. This means that at each dinner, one person is chosen uniformly at random to pay the entire bill (independently of what happens at the other dinners). (a) Find the probability that in k dinners, no one will have to pay the bill more than once (do not simplify for the case k n, but do simplify fully for the case k>n). (b) Find the expected number of dinners it takes in order for everyone to have paid at least once (you can leave your answer as a finite sum of simple-looking terms). (c) Alice and Bob are two of the friends. Find the covariance between how many times Alice pays and how many times Bob pays in k dinners (simplify fully)