Question

2-172. An inspector working for a manufacturing com- pany has a 99% chance of correctly identifying defective items and a 0.5% chance of incorrectly classifying a good item as defective. The company has evidence that 0.9% of the items its line produces are nonconforming. (a) What is the probability that an item selected for inspection is classified as defective? (b) If an item selected at random is classified as nondefective, what is the probability that it is indeed good?

Answer 1

Let us denote the events :

A : the event that a randomly selected item is defective

B : the event that the an item selected for inspection is classified as defective

Given

"Let us denote the events: A: the event that a randomly selected item is defective B: the event that the an item selected for inspection is classified as defective Given P(B

a) The probability that an item selected for inspection is classified as defective

$=P(B)$

$=P(B|A)*P(A)+P(B|A^c)*P(A^c)$

$=(0.99*0.009)+(0.005*(1-0.009))$

b) If an item selected at random is classified as nondefective, the probability that it is indeed good

$=P(A|B)$

$=\frac{P(A\cap B)}{P(B)}$

$=\frac{P(B|A)*P(A)}{P(B)}$

$=\frac{(0.99*0.009)}{0.0138650}=0.642625$