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(C) Let $E$ be the event that the selected item is defective. Let $A, B, C$ be the events that the item was produced by machines $A, B, C$ respectivel

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$\frac{40}{71}$

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(C) Let $E$ be the event that the selected item is defective. Let $A, B, C$ be the events that the item was produced by machines $A, B, C$ respectively.Given probabilities:$P(A) = \frac{2500}{10000} = 0.25$$P(B) = \frac{3500}{10000} = 0.35$$P(C) = \frac{4000}{10000} = 0.40$Conditional probabilities of defect:$P(E|A) = \frac{2}{100} = 0.02$$P(E|B) = \frac{3}{100} = 0.03$$P(E|C) = \frac{5}{100} = 0.05$Using Bayes' theorem,the probability that the item was produced by machine $C$ given it is defective is:$P(C|E) = \frac{P(E|C) \cdot P(C)}{P(E|A) \cdot P(A) + P(E|B) \cdot P(B) + P(E|C) \cdot P(C)}$$P(C|E) = \frac{0.05 \cdot 0.40}{(0.02 \cdot 0.25) + (0.03 \cdot 0.35) + (0.05 \cdot 0.40)}$$P(C|E) = \frac{0.0200}{0.0050 + 0.0105 + 0.0200} = \frac{0.0200}{0.0355} = \frac{200}{355} = \frac{40}{71}$

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