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Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #116

Bag A has3 red,5 black; B has4 red,4 black. Transfer2 uniform balls without replacement from A to B, then draw one uniformly from B. Given the drawn ball is red, find P(both transferred balls were red).

✅ Step-by-Step Solution

Final Answer: 18/133
  1. Let E be the observed event. The mutually exclusive exhaustive cases are RR, RB, BB. Their prior probabilities are P(RR)=3/28, P(RB)=15/28, P(BB)=5/14.
  2. The conditional observation probabilities are P(E|RR)=3/5, P(E|RB)=1/2, P(E|BB)=2/5.
  3. By total probability, P(E)=(3/28)(3/5) + (15/28)(1/2) + (5/14)(2/5)=19/40.
  4. The requested joint probability with E is 9/140=9/140.
  5. By Bayes’ theorem, the requested posterior is 9/140 / (19/40) = 18/133.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are RR, RB, BB. Their prior probabilities are P(RR)=3/28, P(RB)=15/28, P(BB)=5/14.
  • [1 mark] The conditional observation probabilities are P(E|RR)=3/5, P(E|RB)=1/2, P(E|BB)=2/5.
  • [1 mark] By total probability, P(E)=(3/28)(3/5) + (15/28)(1/2) + (5/14)(2/5)=19/40.
  • [1 mark] The requested joint probability with E is 9/140=9/140.
  • [1 mark] By Bayes’ theorem, the requested posterior is 9/140 / (19/40) = 18/133.

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