Question

A large firm uses three different types (A, B, and C) of raw materials to manufacture its product. Previous records show that 25% of the manufactured products are produced using material A, 50% using material B, and 25% using material C. If it is known that 5% of the product made with material A are defective, 2% made with material B are defective, and 5% made with C are defective.

a. What is the probability that a product selected at random from this firm will be defective?

b. If a product selected at random from this firm is found to be defective, what is the probability that it was made from material A.

Answer 1

We are given here that:

P(A) = 0.25,
P(B) = 0.50,
P(C) = 0.25

P(defective | A) = 0.05
P(defective | B) = 0.02
P(defective | C) = 0.05

a) Using law of total probability, we get here:

P(defective ) = P(defective | A)P(A) + P(defective | B)P(B) + P(defective | C)P(C)
P(defective ) = 0.25*0.05 + 0.5*0.02 + 0.25*0.05 = 0.035

Therefore 0.035 is the required probability here.

b) Given that product is defective, probability that it was from A is computed using bayes theorem as:

P( A | defective ) =   P(defective | A)P(A) / P( defective ) = 0.25*0.05 / 0.035 = 0.3571

Therefore 0.3571 is the required probability here.