If 5 of toys that roll off the assembly line are defective w

If 5% of toys that roll off the assembly line are defective, what is the probability that in a shipment of 200 toys, there are fewer than 5 defective toys?

Solution

If 5% of toys that roll off the assembly line are defective, what is the probability that in a shipment of 200 toys, there are fewer than 5 defective toys?

p=0.05

n=200

np=200*0.05 =10 and n(1-p)=190 both > 5, so we can use normal approximation.

Normal approximation to binomial used.

Expectation = np = 10

Variance = np(1 - p) = 9.5

Standard deviation = 3.0822

P( x <5) = P( z <(4.5-10)/3.0822 =p( z < -1.78) = 0.0375

The required P=0.0375

Direct Method

If we use binomial probabilities we get p( x <5) = 0.0264 ( from software.)