🟢 Repeat: Bayes' Theorem — Final Boss
Do this one cold. No formula lookup.
"A factory has two machines. Machine A produces 60% of items and has a 2% defect rate. Machine B produces 40% and has a 5% defect rate. A randomly chosen item is defective. What's the probability it came from Machine A?"
Work it out, then check.
Answer:
P(A|Defective) = P(Def|A) × P(A) / [P(Def|A) × P(A) + P(Def|B) × P(B)]
= (0.02 × 0.60) / (0.02 × 0.60 + 0.05 × 0.40)
= 0.012 / (0.012 + 0.020)
= 0.012 / 0.032
= 0.375 → 37.5%Even though Machine A makes more items, its lower defect rate means a defective item is more likely from Machine B (62.5%).