ITPEC FE Morning October 2020 Question 3
A factory manufactures the same products on two production lines, A and B. Line A manufactures 60% of all products, and B manufactures the remaining 40%. Defect rates of products manufactured in A and in B are 2% and 1%, respectively. When a product was randomly chosen for inspection among the products manufactured in this factory, it was found to be defective. What is the percentage of probability that the product was manufactured in Line A?
Explanation
75% — of all defective products, three quarters come from Line A, because A both makes more and fails more often.
Weight each line's defect rate by its share of production:
- •Defective from A: 0.60 × 0.02 = 0.012
- •Defective from B: 0.40 × 0.01 = 0.004
- •All defective: 0.012 + 0.004 = 0.016
- •From A given defective: 0.012 ÷ 0.016 = 0.75, that is 75%
Answer (d)
Why not others:
- •(a) 40 — Line B's share of production; the chance that a defective product came from B is only 25%
- •(b) 50 — would hold only if both lines produced the same number of defects, but A produces three times as many (0.012 against 0.004)
- •(c) 60 — Line A's share of all products, the prior, ignoring that A's defect rate is twice B's
Key rule: Bayes' theorem: multiply each source's share by its likelihood of the observed outcome, then divide one source's product by the sum over all sources.
Basic Theory · AI-written, may contain errors
This question comes from an official ITPEC past paper. ITPEC Practice is an independent study tool and is not affiliated with ITPEC. See the official FE past-paper collection or Report an issue.