Three companies make brake linings for GM and supply 20 30 a

Three companies make brake linings for GM and supply 20%, 30% and 50% of the linings respectively. Their defect rate is 1%, 0.5% and 2% respectively. Find the chance that a defective brake lining came from the third company.

Solution

Let

1,2,3 = the component came from these companies
D = defective

Thus, as

P(3|D) = P(3) P(D|3) / P(D)

and

P(D) = P(1) P(D|1) + P(2) P(D|2) + P(3) P(D|3) = 0.20*0.01 + 0.30*0.005 + 0.50*0.02 = 0.0135

Thus,

P(3|D) = 0.50*0.02/0.0135 = 0.740740741 [answer]