Correct Answer: Option A
Explanation:
By Bayes: P(M1|Def) = [P(Def|M1)P(M1)] / [P(Def|M1)P(M1) + P(Def|M2)P(M2)] = (0.02×0.6) / (0.02×0.6 + 0.05×0.4) = 0.012 / (0.012 + 0.02) = 0.012/0.032 = 0.375. Wait, recalculate: 0.02*0.6=0.012, 0.05*0.4=0.02, total=0.032, 0.012/0.032=0.375. But option B is 0.48. Adjust numbers: Let Machine 1: 70% production, 1% defect; Machine 2: 30%, 4% defect. Then P(M1|Def)=(0.01*0.7)/(0.01*0.7 + 0.04*0.3)=0.007/(0.007+0.012)=0.007/0.019≈0.368. Still not matching. To get 0.48: Suppose M1: 50% prod, 2% defect; M2: 50%, 5% defect. Then P(M1|Def)=(0.02*0.5)/(0.02*0.5+0.05*0.5)=0.01/(0.01+0.025)=0.01/0.035≈0.2857. Not 0.48. Let me solve: Want P(M1|Def)=0.48. Set P(M1)=p, defect rates d1,d2. Then (d1 p)/(d1 p + d2 (1-p)) = 0.48. Assume d1=0.02, d2=0.05. Then (0.02p)/(0.02p + 0.05(1-p)) = 0.48 → 0.02p = 0.48(0.02p + 0.05 - 0.05p) → 0.02p = 0.48(0.05 - 0.03p) → 0.02p = 0.024 - 0.0144p → 0.02p + 0.0144p = 0.024 → 0.0344p=0.024 → p≈0.6977. So if Machine 1 produces ~70%, with defect rates 2% and 5%, P(M1|Def)≈0.48. I'll adjust the question text accordingly in final output. For accuracy, ensure numbers yield correct answer.
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