A manufacturing plant has two facilities, A and B which produce 30% and 70% of output respectively. On average, twelve out of every thousand components produced in facility A are defective while eight out of every thousand from facility B are defective. If a component is drawn at random from the plant’s total output, use Bayes’s rule to determine the probability that the component is from facility A.

(Hint: let the two events be A = from facility A and C = from total plant output, i.e., from either facility A or B. Find P (A|C). Remember that P (C|A) is the probability of C occurring, given the probability of A, hence P (C|A) = 0.012; also P (T) = P (B) = 0.7.)

(Answer: P (A|C) = 9/23 = 0.3913.)