Suposse that in a bolt manufacturing plant machines A B and

Suposse that in a bolt manufacturing plant, machines A, B, and C manufacture respectively 45%, 25% and 30% of the bolts. It is observed that three machines produce, respectvely, 5%, 2%, and 3% defective bolts. If the bolts produced by these machines are mixed in a well-mixed lot and then a randomly selected bolt is found to be defecetive, find the probability that the defective bolt is produced by (a) machine A, (b) machine B, and (c) machine C.

Solution

Let

D = defective

Thus,

P(D) = P(A)P(D|A) + P(B)P(D|B) + P(C)P(D|C)

= 0.45*0.05 + 0.25*0.02 + 0.30*0.03

P(D) = 0.0365

Thus,

a)

P(A|D) = P(A) P(D|A) / P(D)

= 0.45*0.05/0.0365

= 0.616438356 [ANSWER]

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b)

P(B|D) = P(B) P(D|B) / P(D)

= 0.25*0.02/0.0365

= 0.136986301 [ANSWER]

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c)

P(C|D) = P(C) P(D|C) / P(D)

= 0.30*0.03/0.0365

= 0.246575342 [ANSWER]