There are n + 1 participants in a game. Each person independently is a winner with probabi...

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 , 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.

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

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(b) Argue that .

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

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