ITPEC FE Morning October 2018 Question 3
A factory manufactures the same products in two production lines A and B, where 60% of the products are manufactured in A, and the remaining in B. Defective rates of the products manufactured in A and B are 2% and 1%, respectively. When a product was randomly chosen for inspection, it was found to be defective. What is the percent probability that the product was manufactured in 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.