Question

A certain tennis player makes a successful first serve 71% of the time. Suppose the tennis player serves 110 times in a match. What's the probability that she makes at least 89 first serves? The probability she makes at least 89 first serves is . ? (Use the answers from part a to find this answer. Round to four decimal places as needed.)

Answer 1

X ~ Bin ( n , p)

Where n = 110 , p = 0.71

Using Normal Approximation to Binomial
Mean = n * P = ( 110 * 0.71 ) = 78.1
Variance = n * P * Q = ( 110 * 0.71 * 0.29 ) = 22.649
Standard deviation = √(variance) = √(22.649) = 4.7591

P ( X >= 89 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 89 - 0.5 ) =P ( X > 88.5 )

Using normal approximation,

P ( X < x) = P ( (Z < X - µ ) / σ )
P ( X > 88.5 ) = P(Z > (88.5 - 78.1 ) / 4.7591 )
= P ( Z > 2.19 )
= 1 - P ( Z < 2.19 )
= 1 - 0.9857
= 0.0143