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Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #74

Bag I has 3 red,5 black; bag II has 4 red,3 black. Transfer one uniform ball from I to II, then draw one uniformly from II. Given the drawn ball is black, find P(the transferred ball was black).

✅ Step-by-Step Solution

Final Answer: 20/29
  1. Let E be the observed event. The mutually exclusive exhaustive cases are transfer black, transfer red. Their prior probabilities are P(transfer black)=5/8, P(transfer red)=3/8.
  2. The conditional observation probabilities are P(E|transfer black)=1/2, P(E|transfer red)=3/8.
  3. By total probability, P(E)=(5/8)(1/2) + (3/8)(3/8)=29/64.
  4. The requested joint probability with E is 5/16=5/16.
  5. By Bayes’ theorem, the requested posterior is 5/16 / (29/64) = 20/29.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are transfer black, transfer red. Their prior probabilities are P(transfer black)=5/8, P(transfer red)=3/8.
  • [1 mark] The conditional observation probabilities are P(E|transfer black)=1/2, P(E|transfer red)=3/8.
  • [1 mark] By total probability, P(E)=(5/8)(1/2) + (3/8)(3/8)=29/64.
  • [1 mark] The requested joint probability with E is 5/16=5/16.
  • [1 mark] By Bayes’ theorem, the requested posterior is 5/16 / (29/64) = 20/29.

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