Chapter 7, Problem 7.62P

There are $n+1$ participants in a game. Each person independently is a winner with probability p. The winners share a total prize of 1 unit. (For instance, if 4 people win, then each of them receives $\frac{1}{4}$ whereas if there are no winners, then none of the participants receives anything.) Let A denote a specified one of the players, and let X denote the amount that is received by

a. Compute the expected total prize shared by the players.

b. Argue that $E\left[X\right]=\frac{1-{\left(1-p\right)}^{n+1}}{n+1}$

c. Compute $E\left[X\right]$ by conditioning on whether A is a winner, and conclude that $E\left[{\left(1+B\right)}^{-1}\right]=\frac{1-{\left(1-p\right)}^{n+1}}{\left(n+1\right)p}$ when B is a binomial random variable with parameters n and p.