Serena Williams made a successful first serve 67 of the time

Serena Williams made a successful first serve 67% of the time in a Wimbledon finals match against her sister Venus. If she continues to serve at the same rate the next time they play and serve 6 times in the first game, determine the following probabilities.(assume that each serve is independent of the others.)

a. All 6 first serves will be in.

b. Exactly 4 first serves will be in.

c. At least 4 first serves will be in.

Solution

Given X follows Binomial distribution with n=6 and p=0.67

P(X=x)=6Cx*(0.67^x)*((1-0.67)^(6-x)) for x=0,1,2,...,6

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a. All 6 first serves will be in.

P(X=6) =6C6*(0.67^6)*((1-0.67)^(6-6)) =0.09045838

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b. Exactly 4 first serves will be in.

P(X=4) =6C4*(0.67^4)*((1-0.67)^(6-4)) =0.3291686

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c. At least 4 first serves will be in.

P(X>=4) =1-P(X=0)-P(X=1)-P(X=2)-P(X=3)

=0.6869517