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 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] = 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] = 1 − (1 − p)n+1/(n + 1)p
when B is a binomial random variable with parameters n and p.