# Question: In a certain contest the players are of equal skill

In a certain contest, the players are of equal skill and the probability is 1/2 that a specified one of the two contestants will be the victor. In a group of 2n players, the players are paired off against each other at random. The 2n−1 winners are again paired off randomly, and so on, until a single winner remains. Consider two specified contestants, A and B, and define the events Ai, i ≤ n, E by

Ai: A plays in exactly i contests:

E: A and B never play each other.

(a) Find P(Ai), i = 1, . . . , n.

(b) Find P(E).

(c) Let Pn = P(E). Show that

and use this formula to check the answer you obtained in part (b).

Find P(E) by conditioning on which of the events Ai, i = 1, . . . , n occur. In simplifying your answer, use the algebraic identity

For another approach to solving this problem, note that there are a total of 2n − 1 games played.

(d) Explain why 2n − 1 games are played. Number these games, and let Bi denote the event that A and B play each other in game i, i = 1, . . . , 2n − 1.

(e) What is P(Bi)?

(f) Use part (e) to find P(E).

Ai: A plays in exactly i contests:

E: A and B never play each other.

(a) Find P(Ai), i = 1, . . . , n.

(b) Find P(E).

(c) Let Pn = P(E). Show that

and use this formula to check the answer you obtained in part (b).

Find P(E) by conditioning on which of the events Ai, i = 1, . . . , n occur. In simplifying your answer, use the algebraic identity

For another approach to solving this problem, note that there are a total of 2n − 1 games played.

(d) Explain why 2n − 1 games are played. Number these games, and let Bi denote the event that A and B play each other in game i, i = 1, . . . , 2n − 1.

(e) What is P(Bi)?

(f) Use part (e) to find P(E).

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