Question

Assume that a factory has two machines ??A_1 and ??A_2 . Past records shows that machine ??A_1 produces 60% of the items of output and machine ??A_2 produces 40% of the items. Further, 2% of the items produced by machine ??A_1 were defective and only 1% produced by machine ??A_2 were defective.

If a detective item is drawn at random,

what is the probability that it was produced by machine ??A_1 ?

Answer 1

Let A be the event that denotes the items are produced by machine A1.

Let B be the event that denotes the items are produced by machine A2.

Let D be the event that denotes the item produced is defective.

Given, P(A) = 0.60 ; P(B) = 0.40 ; P(D | A) = 0.02 ; P(D | B) = 0.01

P(A | D) = (P(A) * P(D | A)) / (P(A) * P(D | A) + P(B) * P(D | B))       ............(by Bayes' theorem)

              = (0.60 * 0.02) / (0.60 * 0.02 + 0.40 * 0.01)

              = 0.012 / 0.016

              = 0.75

Therefore, if a defective item is drawn at random, then the probability that it was produced by machine A1 is 0.75