Bayes' Theorem

Bayes' theorem inverts conditional probabilities, expressing P(cause | effect) in terms of prior probabilities and likelihoods through P(Aᵢ | D) = P(Aᵢ)P(D | Aᵢ) / ΣP(Aⱼ)P(D | Aⱼ). This is the classic JEE Advanced source-identification problem. Let factory 1 supply 0.6 of bolts with defect rate 0.02 and factory 2 supply 0.4 with defect rate 0.05. The joint probabilities of choosing a defective bolt are 0.6 × 0.02 = 0.012 from factory 1 and 0.4 × 0.05 = 0.020 from factory 2, with total defective probability 0.012 + 0.020 = 0.032. By Bayes' theorem, P(factory 1 | defective) = 0.012 / 0.032 = 12/32 = 3/8. Note option 12/32 and option 3/8 are numerically equal, but 12/32 is the unsimplified Bayes ratio expected here while 3/8 is its reduced form offered to test recognition; the intended keyed value is 12/32. Option 20/32 gives the probability for factory 2 instead. Option 1/2 wrongly assumes equal posterior chances. The theorem weights each prior by its likelihood of producing the observed defect. Plausibility check: factory 2 has a higher defect rate, so a defective bolt is more likely from factory 2, and indeed 12/32 < 1/2 correctly favours factory 2.