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A factory has three machines, M1, M2, and M3, that produce items (say, light bulbs). It is impossible to tell which machine produced a particular item, but some are defective. Here are the known numbers:

machine proportion of items made prob. any made item is defective
M1 0.2 0.001
M2 0.3 0.002
M3 0.5 0.003

You pick an item, test it, and find it is defective. What is the probability that it was made by M2?

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Probability of being defective:let D

Pr(Defective)=0.2x0.001+0.3x0.002+0.5x0.003=0.0024

P(M2/Defective)=P(Defective/M2)xP(M2)/P(Defective=0.002x 0.3/0.0024=0.25

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