My Notes Ask Your Teache I. -/6 points + Kim and Susan are playing a tennis matc

Question

My Notes Ask Your Teache I. -/6 points + Kim and Susan are playing a tennis match where the winner must win 2 sets in order to win the match. For each set, the probability that Kim wins the set is 0.53. The probability of Kim winning the set is not affected by who has won any previous sets Draw a tree to represent this tennis match and answer the following questions. (a) What is the expected value for the number of sets that will be played? (Hint: We only care about the number of sets played, not who wins.) (b) What is the expected value for the number of sets that Kim will win? (Hint: We only care about the number of sets Kim wins during the match, not if she ended up winning the match in the end.)

Explanation / Answer

Combinations: KK, SS, KSK, SKS, KSS,SKK

a) X 2 3
P(X=2) = P(KK) + P(SS) = 0.53^2 + 0.47^2 = 0.5018
P(X = 3) = P(KSK) + P(SKS) + P(KSS) + P(SKK) = 0.53*0.47*0.53 + 0.47*0.53*0.47 + 0.53*0.47*0.47 + 0.47*0.53*0.53 = 0.4982

E[X] = 2*0.5018 + 3*0.4982 = 2.982 sets

b) X 0 1 2
P(X = 0) = P(SS) = 0.47^2 = 0.2209
P(X = 1) = P(SKS) + P(KSS) = 2*0.53*0.47^2 = 0.234154
P(X = 2) = P(KK) + P(KSK) + P(SKK) = 0.53^2 + 2*0.47*0.53^2 = 0.544946

E[X] = 0*0.2209 + 1*0.234154 + 2*0.544946 = 1.3240 sets

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