Problem 2 (20 points) A factory uses 3 types of machine A, B and C for construct

Question

Problem 2 (20 points) A factory uses 3 types of machine A, B and C for constructing products. It uses the machine A with probability of 0.4, the machine B with 0.25 and 0.35 for C The defect rate for the machine A is 0.3, 0.2 for B and 0.15 for C. A product was observed randomly 1. What is the probability that this product is defective and made by machine A? Defective and made by machine B? Defective and made by machine C? 2. If the product is a defective product, what is the probability that this is a C-product?

Explanation / Answer

Consider P(A) = 0.4

P(B) = 0.25

P(C) = 0.35

Similarly consider defect rate P(X|A) = 0.3

P(X|B) = 0.2

P(X|C) = 0.15

Q1 Solution:-

Defective product made by machine A = 0.4 * 0.3 = 0.12

Defective product made by machine B = 0.25 * 0.2 = 0.05

Defective product made by machine C = 0.35 * 0.15 = 0.0525

Total defective products from all three machines = 0.12 + 0.05 + 0.0525

= 0.2225

If part is found to be defective then,

Probability that it is made by machine A = 0.12/0.2225

= 0.53932

Probability that it is made by machine B = 0.05/0.2225

= 0.22471

Probability that it is made by machine C = 0.0525/0.2225

= 0.23595

Q2 Solution;-

Here we require to find the probability that the product selected is produced by machine C given that the product is defective i.e P(C|X). By Baye's theorem we find that,

P(C|X) = (P(X|C)*P(C))/ P(A)*P(X|A) + P(B)*P(X|B) + P(C)*P(X|C)

=(0.15*0.35)/0.4*0.3 + 0.25*0.2 + 0.35*0.15

=0.0525/0.2225

= 0.23595

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