A factory produces items on two assembly lines, Line 1 (L1) and Line 2 (L2). 60% of items are produced on L1 and 40% on L2. The defect rate for L1 is 5%, and for L2 is 10%. If a randomly selected item is found to be defective (D), what is the probability that it came from Line 1?

Question

A factory produces items on two assembly lines, Line 1 (L1​) and Line 2 (L2​). 60% of items are produced on L1​ and 40% on L2​. The defect rate for L1​ is 5%, and for L2​ is 10%. If a randomly selected item is found to be defective (D), what is the probability that it came from Line 1?

BlackTom AI · Accepted Answer

The problem asks for P(L1 | D), the probability that a defect came from Line 1 given that the item is defective.
First, identify the needed probabilities: P(L1) = 0.60, P(L2) = 0.40, P(D | L1) = 0.05, P(D | L2) = 0.10.
Now apply Bayes’ theorem: P(L1 | D) = [P(D | L1) * P(L1)] / [P(D | L1) * P(L1) + P(D | L2) * P(L2)].
Compute the numerator: 0.05 * 0.60 = 0.03.
Compute the denominator: (0.05 * 0.60) + (0.10 * 0.40) = 0.03 + 0.04 = 0.07.
So P(L1 | D) = 0.03 / 0.07 ≈ 0.4286.
Now evaluate the options:
- Option a: 0.4286. This matches the calculated value, so this is correct.
- Option b: 0.7. This would imply that the defect is much more likely to come from L1 than supported by the given defect rates and production shares; the numbers do not align with Bayes’ calculation here.
- Option c: 0.2842. This is a miscalculation; it resembles misplacing the fractions or mixing up the numerator/denominator, not the correct Bayes expression.
- Option d: 0.01. This is far too small and not consistent with either the prior probabilities or the defect rates provided.
In summary, the correct reasoning yields 0.4286 for P(L1 | D), while the other choices arise from common slips such as incorrect formula application or arithmetic errors.

A factory produces items on two assembly lines, Line 1 (L1) and Line 2 (L2). 60% of items are produced on L1 and 40% on L2. The defect rate for L1 is 5%, and for L2 is 10%. If a randomly selected item is found to be defective (D), what is the probability that it came from Line 1?单项选择题

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