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Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #80

Bag I has 5 red,4 white; II has 3 red,3 white. Transfer two uniform balls without replacement from I to II; then draw one uniformly from II. Given it is red, find P(one red and one white were transferred).
Transfer composition and the red-draw observation
Transfer composition and the red-draw observation

✅ Step-by-Step Solution

Final Answer: 20/37
  1. Let E be the observed event. The mutually exclusive exhaustive cases are RR, WW, RW. Their prior probabilities are P(RR)=5/18, P(WW)=1/6, P(RW)=5/9.
  2. The conditional observation probabilities are P(E|RR)=5/8, P(E|WW)=3/8, P(E|RW)=1/2.
  3. By total probability, P(E)=(5/18)(5/8) + (1/6)(3/8) + (5/9)(1/2)=37/72.
  4. The requested joint probability with E is 5/18=5/18.
  5. By Bayes’ theorem, the requested posterior is 5/18 / (37/72) = 20/37.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are RR, WW, RW. Their prior probabilities are P(RR)=5/18, P(WW)=1/6, P(RW)=5/9.
  • [1 mark] The conditional observation probabilities are P(E|RR)=5/8, P(E|WW)=3/8, P(E|RW)=1/2.
  • [1 mark] By total probability, P(E)=(5/18)(5/8) + (1/6)(3/8) + (5/9)(1/2)=37/72.
  • [1 mark] The requested joint probability with E is 5/18=5/18.
  • [1 mark] By Bayes’ theorem, the requested posterior is 5/18 / (37/72) = 20/37.

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