4. A small factory produces window frames. Suppose there are three machines, A,

Question

4. A small factory produces window frames. Suppose there are three machines, A, B and C producing 20%, 30% and 50% of the window frames respectively. Also suppose that of the window frames produced by machine A 0.5% are defective, while 0.3% of those produces by machine B are defective and 0.2% of those produced by machine C are defective. A window frame is selected at random. (a) What is the probability that it is defective? (b) Given that it is defective, what is the probability that it was made by machine A?

Explanation / Answer

P(A) = 0.20, P(B) = 0.30, P( C) = 0.50

P( Defective | A) = 0.005, P( Defective | B) = 0.003, P( Defective | C) = 0.002

a)

P(Defective ) = P(A) * P( Defective | A) + P(B) * P( Defective | B) + P(C) * P( Defective | C)

= 0.2 * 0.005 + 0.3 * 0.003 + 0.5 * 0.002

= 0.0029

P(Defective ) = 0.0029

b)

P(A | Defective) = P(A) * P( Defective | A) / P(Defective)

= 0.2 * 0.005 / 0.0029

= 0.3448

P(A | Defective) = 0.3448

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