Defective Bulbs from Two Machines
A bulb factory operates two machines, M1 and M2. M1 produces 60% of the day's output and M2 produces 40%. Past inspection records show that 2% of M1's bulbs are defective and 5% of M2's bulbs are defective. At the end of a day, an inspector picks one bulb at random from the combined output and tests it. The production manager says, 'M1 makes most of our bulbs, so if the bulb is defective it most likely came from M1.' The maintenance team can overhaul only one machine this week. The factory also wants to know whether the machine that made a bulb and whether the bulb is defective are independent events, because the quality report must state whether defect status depends on the machine. The inspector finds that the chosen bulb is defective. The factory has no other information about the day's production mix.
1. Find the probability that a randomly chosen bulb is defective and the probability that a defective bulb came from M1. Is the manager right?
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Use the law of total probability: P(D) = P(M1)P(D|M1) + P(M2)P(D|M2) = 0.60(0.02) + 0.40(0.05) = 0.012 + 0.020 = 0.032. So 3.2% of all bulbs are defective. Given D, P(M1|D) = P(M1 ∩ D)/P(D) = 0.012/0.032 = 0.375. Thus only 37.5% of defective bulbs come from M1; 62.5% come from M2. The manager's statement is wrong: M1 makes more bulbs, but M2's higher defect rate puts most defective bulbs in M2's smaller share.
2. Which machine should be overhauled first? Justify using probabilities, not just the defect rates.
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M2 should be serviced first. Its defect rate is 5% versus 2% for M1, and it accounts for 0.020/0.032 = 62.5% of all defective bulbs. M1 contributes 0.012 of the 0.032 total, i.e. 37.5%. If servicing reduces defects proportionally, the larger immediate reduction comes from M2. Cost and downtime could change the final decision, but on defect evidence M2 is the priority.
3. Are the machine and defect status independent? Show a check.
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Check P(D|M1) = 0.02 against P(D) = 0.032. They are not equal, so defect status and machine are not independent. Equivalently, P(M1|D) = 0.375 ≠ P(M1) = 0.60. Knowing the machine changes the probability of a defect. The quality report should state dependence, not independence.