Question

A pitcher pitches a strike ( a good pitch) an average of 75% of the time. What is the probability that in a given game, at least 77 pitches of 110 thrown were strikes? Represent this graphically as well as numerically.

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 110 * 0.75 ) = 82.5
Variance = n * P * Q = ( 110 * 0.75 * 0.25 ) = 20.625
Standard deviation = √(variance) = √(20.625) = 4.5415

P ( X >= 77 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 77 - 0.5 ) = P ( X > 76.5 )

X ~ N ( µ = 82.5 , σ = 4.5415 )
P ( X > 76.5 ) = 1 - P ( X < 76.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 76.5 - 82.5 ) / 4.5415
Z = -1.32
P ( ( X - µ ) / σ ) > ( 76.5 - 82.5 ) / 4.5415 )
P ( Z > -1.32 )
P ( X > 76.5 ) = 1 - P ( Z < -1.32 )
P ( X > 76.5 ) = 1 - 0.0934
P ( X > 76.5 ) = 0.9066