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Class 12 Probability · Long Answer3 Marksbayes_theorem

Class 12 Maths · Probability · Long Answer #60

A bag has four balls. Before drawing, its number W of white balls is uniformly distributed over {0,1,2,3,4}; other balls are nonwhite. Two balls drawn uniformly without replacement are white. Find P(W=4).

✅ Step-by-Step Solution

Final Answer: 3/5
  1. Let E be the observed event. The mutually exclusive exhaustive cases are W=0, W=1, W=2, W=3, W=4. Their prior probabilities are P(W=0)=1/5, P(W=1)=1/5, P(W=2)=1/5, P(W=3)=1/5, P(W=4)=1/5.
  2. The conditional observation probabilities are P(E|W=0)=0, P(E|W=1)=0, P(E|W=2)=1/6, P(E|W=3)=1/2, P(E|W=4)=1. By total probability, P(E)=(1/5)(0) + (1/5)(0) + (1/5)(1/6) + (1/5)(1/2) + (1/5)(1)=1/3.
  3. The requested joint probability with E is 1/5=1/5.
  4. By Bayes’ theorem, the requested posterior is 1/5 / (1/3) = 3/5.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Let E be the observed event. The mutually exclusive exhaustive cases are W=0, W=1, W=2, W=3, W=4. Their prior probabilities are P(W=0)=1/5, P(W=1)=1/5, P(W=2)=1/5, P(W=3)=1/5, P(W=4)=1/5.
  • [1 mark] The conditional observation probabilities are P(E|W=0)=0, P(E|W=1)=0, P(E|W=2)=1/6, P(E|W=3)=1/2, P(E|W=4)=1. By total probability, P(E)=(1/5)(0) + (1/5)(0) + (1/5)(1/6) + (1/5)(1/2) + (1/5)(1)=1/3.
  • [1 mark] The requested joint probability with E is 1/5=1/5.
  • [0.5 marks] By Bayes’ theorem, the requested posterior is 1/5 / (1/3) = 3/5.

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