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Class 12 Probability · Long Answer3 Marksbayes_theorem

Class 12 Maths · Probability · Long Answer #57

Choose bag I or II with equal probabilities, then draw one uniform ball. Bag I has 3 red,4 black; II has 5 red,6 black. Given red, find P(bag II).

✅ Step-by-Step Solution

Final Answer: 35/68
  1. Let E be the observed event. The mutually exclusive exhaustive cases are I, II. Their prior probabilities are P(I)=1/2, P(II)=1/2.
  2. The conditional observation probabilities are P(E|I)=3/7, P(E|II)=5/11. By total probability, P(E)=(1/2)(3/7) + (1/2)(5/11)=34/77.
  3. The requested joint probability with E is 5/22=5/22.
  4. By Bayes’ theorem, the requested posterior is 5/22 / (34/77) = 35/68.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Let E be the observed event. The mutually exclusive exhaustive cases are I, II. Their prior probabilities are P(I)=1/2, P(II)=1/2.
  • [1 mark] The conditional observation probabilities are P(E|I)=3/7, P(E|II)=5/11. By total probability, P(E)=(1/2)(3/7) + (1/2)(5/11)=34/77.
  • [1 mark] The requested joint probability with E is 5/22=5/22.
  • [0.5 marks] By Bayes’ theorem, the requested posterior is 5/22 / (34/77) = 35/68.

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