Q. 12. In a bolt factory machines A, B and C manufacture respectively 25, 30 and 40 percent of the total. Out of their total outputs 5, 4 and 2 percent are defective. A bolt is drawn from the produce at random and is found to be defective. What is the probability that it is manufactured by (i) factory A (ii) factory C? (June 99)

Solution. This problem is based on Bayes theorem.
Let A = an event that the output is defective
B₁ = an event that bolt is manufactured by machine A
B₂ = an event that bolt is manufactured by machine B
B₃ = an event that bolt is manufactured by machine C

Therefore, P(B₁) = 25/100, P(B₂) = 35/100, P(B₃) = 40/100
P(A | B₁) = 5/100, P(A | B₂) = 4/100, P(A | B₃) = 2/100

or P(B₁ | A) = 25/69

or P(B₃ | A) = 16/69

Q. 13. In a bolt factory machines A, B and C manufacture respectively 30, 35 and 35 percent of the total. Out of their total outputs 3, 4 and 3 percent are defective. A bolt is drawn at random and is found to be defective. What is the probability that it is manufactured by (i) factory A (ii) factory B? (June 2002)

Solution.
Let X, Y and Z denote the event that a bolt is manufactured by the machine A, B and C respectively.
Let S denotes the event that the bolt is drawn now.

P(X Ç S) = (30/100) X (3/100) = 90/10000
P(Y Ç S) = (35/100) X (4/100) = 140/10000
P(Z Ç S) = (35/100) X (3/100) = 105/10000

P(S) = [(90/10000) + (140/10000) + (105/10000)]
or P(S) = 335/10000

or P(X | S) = 18/67

or P(Y | S) = 28/67