A robot wrestling tournament with 8 participants is taking place. The defending

Question

A robot wrestling tournament with 8 participants is taking place. The defending champion is expected to win a match with the probability of 0.87 regardless of the opponent, and matches outcomes are assumed to be independent. (a) The single elimination tournament requires 3 consecutive match wins to win the tournament. What is the probability that the defending champion wins the tournament? Round your answer to three decimal places (e.g. 98.765) (b) The defending champion won the tournament again and now accepts open challenges. What is the expected number of matches until this robot is defeated by a challenger? Round your answer to two decmal places (e.g. 98.76) (c) After the first defeat, the robot's joints are replaced to more flexible ones, increasing the winning probability to 0.96. What is the probability that this robot's first loss is the fifth challenge? Round your answer to three decimal places (e.g. 98.765)

Explanation / Answer

The defending champion is expected to win a match with probability p = 0.87

a) Here the single elimination tournament required 3 consecutive match wins to win the tournament.

So the probability of the defending champion will win all the 3 matches = p^3 = 0.87^3 = 0.659

b) Here p = 0.87

So the defending champion is expected to loss a match with probability 1- p = 1 - 0.87 = 0.13

So the expected number of matches until this robot defeated by the

chalengers = 1/(1 - p) = 1/0.13 = 7.69

c) After the replacement of joint parts of the robot , his winning chances increases upto 0.96.

so that the probability of he is loss a match is 1 - 0.96 = 0.04

So the probability that this robot's first loss is the 5th challenge = p^4 * ( 1 - p ) = 0.96^4 * 0.04 = 0034

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