56. 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 receive anything.) Let *A* denote a specified one of the players, and let *X* denote the amount that is received by *A*.

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

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

(c) Compute *E[X]* by conditioning on whether *A* is a winner, and conclude that

$$E[(1 + B)^{-1}] = \frac{1 - (1-p)^{n+1}}{(n+1)p}$$

when *B* is a binomial random variable with parameters *n* and *p*.