I am playing in a tennis tournament, and I am up against a player I have watched

Question

I am playing in a tennis tournament, and I am up against a player I have watched but never played before. Based on what I have seen, I consider three possible models for our relative strengths:

Model A: We are evenly matched, so that each of us is equally likely to win each game.

Model B: I am slightly better, so that I win each game independently with probability 0:6.

Model C: My opponent is slightly better and wins each game independently with probability 0:6.

Before we play, I consider each of these possibilities to be equally likely. In our match, we play until one player wins three games. I win the second game, but my opponent wins the rst, third, and fourth games. After the match, what is the posterior probability of model C (i.e., that my opponent is slightly better than me)?

Explanation / Answer

p(A) = p(B) =p(C) = 1/3,

Pr(G) is games probability ,
so ,
P( G|C) = 0.6*0.4*0.6*0.6*1/3 = 0.0288
P( G|B) = 0.4*0.6*0.4*0.4*1/3 = 0.0128
P( G|A) = 0.5*0.5*0.5*0.5*1/3 = 0.020833

so ,
By Bayes theorem ,
P(C|G) = P(G|C)/( P(G|C) + P(G|A)+P(G|B)

so ,
P(C|G) = 0.0288/( 0.0288+0.0128+0.020833)

P(C|G) =0.4613

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