3. In a semiconductor company three machines, A, B, and C, fabricate integrated circuits (ICs). Machine...

3. In a semiconductor company three machines, A, B, and C, fabricate integrated circuits (ICs). Machine A produces 20% of the ICs, machine B produces 30% of the ICs, and machine C produces 50% of the ICs. The ICs are assumed to be identical. It is known that 3 in every 1000 ICs made by machine A are defective, 1 in every 125 ICs made by machine B is defective, and 1 in every 250 ICs made by machine C is defective.

(a) [6 points] A quality control inspector randomly selects an IC without knowing which machine fabricated it. What is the probability that it is defective?

(b) [6 points] The inspector draws a defective IC from a pile of rejects. Determine the probabilities that the defective IC was fabricated by machines A, B, and C, respectively. Do these probabilities add up to one?

(c) [8 points] The inspector draws 8 defective ICs at random from a pile of rejects. What is the probability that 2 were fabricated by machine A, 3 by machine B, and 3 by machine C?

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