Question

Studies indicate that there are 16% maintenance-related shutdowns of the underground train system in a particular country. If 100 underground trains are randomly chosen,                       

(a) Find the probability that more than fifteen shutdowns will occur.

(b) Find the probability that exactly twelve shutdowns will occur.   

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 100 * 0.16 ) = 16
Variance = n * P * Q = ( 100 * 0.16 * 0.84 ) = 13.44
Standard deviation = √(variance) = √(13.44) = 3.6661

Part a)

P ( X > 15 )
Using continuity correction
P ( X > n + 0.5 ) = P ( X > 15 + 0.5 ) = P ( X > 15.5 )

X ~ N ( µ = 16 , σ = 3.6661 )
P ( X > 15.5 ) = 1 - P ( X < 15.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 15.5 - 16 ) / 3.6661
Z = -0.14
P ( ( X - µ ) / σ ) > ( 15.5 - 16 ) / 3.6661 )
P ( Z > -0.14 )
P ( X > 15.5 ) = 1 - P ( Z < -0.14 )
P ( X > 15.5 ) = 1 - 0.4443
P ( X > 15.5 ) = 0.5557

Part b)

P ( X = 12 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 12 - 0.5 < X < 12 + 0.5 ) = P ( 11.5 < X < 12.5 )

X ~ N ( µ = 16 , σ = 3.6661 )
P ( 11.5 < X < 12.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 11.5 - 16 ) / 3.6661
Z = -1.23
Z = ( 12.5 - 16 ) / 3.6661
Z = -0.95
P ( -1.23 < Z < -0.95 )
P ( 11.5 < X < 12.5 ) = P ( Z < -0.95 ) - P ( Z < -1.23 )
P ( 11.5 < X < 12.5 ) = 0.1711 - 0.1093
P ( 11.5 < X < 12.5 ) = 0.0617