Q.62 In a factory, two machines M1 and M2 manufacture 60% and 40% of the autocomponents
respectively. Out of the total production, 2% of M1 and 3% of M2 are found to be defective. If a
randomly drawn autocomponent from the combined lot is found defective, what is the probability
that it was manufactured by M2?
(A) 0.35 (B) 0.45 (C) 0.5 (D) 0.4
Bayes Theorem Factory Defective Parts: Probability M2 Given Defective
This classic Bayes’ theorem problem involves two machines producing autocomponents with different defect rates. When a defective part is found, the probability it came from M2 (40% production, 3% defective) is exactly 0.4, making option (D) correct.
Problem Setup
Machine M1 produces 60% of parts with 2% defect rate; M2 produces 40% with 3% defect rate. Find P(M2|Defective) using Bayes’ theorem.
Define events:
-
P(M1) = 0.6, P(D|M1) = 0.02
-
P(M2) = 0.4, P(D|M2) = 0.03
Correct Solution: Bayes Theorem
P(M2|D) = [P(D|M2) × P(M2)] / P(D total)
Step 1: Total defect probability
P(D) = (0.02 × 0.6) + (0.03 × 0.4) = 0.012 + 0.012 = 0.024
Step 2: Posterior probability
P(M2|D) = (0.03 × 0.4) / 0.024 = 0.012 / 0.024 = 0.4 ✓
Complete Calculation Table
| Description | Probability | Calculation |
|---|---|---|
| M1 production | 60% | 0.6 |
| M1 defectives | 1.2% of total | 0.6 × 0.02 = 0.012 |
| M2 production | 40% | 0.4 |
| M2 defectives | 1.2% of total | 0.4 × 0.03 = 0.012 |
| Total defectives | 2.4% | 0.024 |
| P(M2|D) | 40% | 0.012/0.024 = 0.4 |
Why Other Options Wrong
-
(A) 0.35: Underestimates M2 contribution (ignores equal defective contributions)
-
(B) 0.45: Slightly overestimates (simple 40% production ratio wrong)
-
(C) 0.5: Assumes equal defective contribution despite different rates/production
Key insight: Equal defective amounts (0.012 each) from both machines means 50-50 split among defectives, but M2’s higher defect rate exactly balances lower production.
Real-World Application
Used in quality control, medical testing (false positives), spam filtering. Always calculate total probability of evidence first.
