Question

A factory tests all its products. The proportion of defective items is 0.02. The probability that the test will catch a defective product is 0.95. The test will also reject nondefective products with probability 0.01. (a) Given that a product passes the test, what is the probability that it is defective? (b) Given that the product does not pass the test, what is the probability that the product is defective?

Answer 1

Solution:

probability of defective items = 0.02

probability of finding nondefective item = 1 - 0.02 = 0.98

probability of a test to catch a defective product = 0.95

probability of a test failing to catch a defective product = 1 - 0.95 = 0.05

probability of rejecting nondefective products = 0.01

probability of allowing a nondefective product = 1 - 0.01 = 0.99

a)probability of an item being defective given that it has passed the test =

probability of an item passing the test = p(non defective and passes) + P(defective and passes)

= (0.98 * 0.99) + (0.02* 0.05) = 0.9712

(probability of item being defective times probability of the item passing the test) / probability of it having passed the test

= 0.02 * (0.05) /0.9712 = 0.001

b)probability of an item being defective given that it didn't pass the test.

probability of an item not passing the test = 1 - probability of an item to pass the test

= 1 - 0.9712 = 0.0288zz

= probabilitu of defective and probability of not passing )/ total probablility of not passing

= (0.02* 0.95) / 0.0288 = 0.658