Like most major tennis events, the U.S. Open is a single elimination tournament

Question

Like most major tennis events, the U.S. Open is a single elimination tournament with 128 players entering the first round. A player is eliminated after losing a single match

(thus, 64 are eliminated after the first round, another 32 after the second, etc.). Now assume the

following: (i) in any match the probability of the better player winning is 0.8; (ii) the outcome of

every match is independent of every other; and (iii) how good a player is stays constant throughout

the tournament. What is the probability of the tournament being won by the best player of the 128?

Explanation / Answer

Number of matches to be won by best player to win the tournment.

After Round 1 number of players left =64 ;

After Round 2 number of players left = 32;

After Round 3 number of players left =16 ;

After Round 4 number of players left =8 ;

After Round 5 number of players left =4 ;

After Round 6 number of players left = 2 ;

Match 7 is the final game;

So the best player has to win 7 matches to win the tournment.

Probability of best player winning a match = 0.8

Probability of best player winning 7 matches = 0.8 x 0.8 x 0.8 x  0.8 x 0.8 x 0.8 x 0.8 = 0.87 = 0.209715

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