# Question

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.

(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.

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