Question

A manufacturer obtains​ clock-radios from three different​ subcontractors, 20​% from Upper B 1​, 30​% from Upper B 2​, and 50​% from Upper B 3. The defective rates for​ clock-radios from these subcontractors are 1​%, 4 %​, and 5​% respectively. If a defective​ clock-radio is returned by a​ customer, what is the probability that a defective​ clock-radio came from subcontractor Upper B 1​?

Answer 1

Let B1 be the event that denotes the clock-radio is obtained from contractor 1.

Let B2 be the event that denotes the clock-radio is obtained from contractor 2.

Let B3 be the event that denotes the clock-radio is obtained from contractor 3.

Let D be the event that denotes the clock-radio is defective.

Given, P(B1) = 0.20 ; P(B2) = 0.30 ; P(B3) = 0.50 ; P(D | B1) = 0.01 ; P(D | B2) = 0.04 ; P(D | B3) = 0.05

The required probability is

P(B1 | D) = (P(D | B1) * P(B1)) / P(D | Bi) * P(Bi)      .............(using the Bayes' theorem)

                = (P(D | B1) * P(B1)) / ((P(D | B1) * P(B1)) + (P(D | B2) * P(B2)) + (P(D | B3) * P(B3)))

                = (0.01 * 0.20) / ((0.01 * 0.20) + (0.04 * 0.30) + (0.05 * 0.50))

                = 2 / 39

                = 0.0513

Therefore, if a defective clock-radio is returned by a customer, the probability that a defective clock-radio came from subcontractor B 1 is 0.0513