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Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #79

Machines A,B produce 60%,40% of a factory output. Their defect rates are 2%,1%. A uniform item is found defective. Find P(machine B).

✅ Step-by-Step Solution

Final Answer: 1/4
  1. Let E be the observed event. The mutually exclusive exhaustive cases are A, B. Their prior probabilities are P(A)=3/5, P(B)=2/5.
  2. The conditional observation probabilities are P(E|A)=1/50, P(E|B)=1/100.
  3. By total probability, P(E)=(3/5)(1/50) + (2/5)(1/100)=2/125.
  4. The requested joint probability with E is 1/250=1/250.
  5. By Bayes’ theorem, the requested posterior is 1/250 / (2/125) = 1/4.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are A, B. Their prior probabilities are P(A)=3/5, P(B)=2/5.
  • [1 mark] The conditional observation probabilities are P(E|A)=1/50, P(E|B)=1/100.
  • [1 mark] By total probability, P(E)=(3/5)(1/50) + (2/5)(1/100)=2/125.
  • [1 mark] The requested joint probability with E is 1/250=1/250.
  • [1 mark] By Bayes’ theorem, the requested posterior is 1/250 / (2/125) = 1/4.

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