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Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #120

A fair die is rolled. A witness answers “Was it a4?” correctly with probability3/5 conditional on either true yes or true no. Given a “yes” response, find P(actual4).

✅ Step-by-Step Solution

Final Answer: 3/13
  1. Let E be the observed event. The mutually exclusive exhaustive cases are four, not four. Their prior probabilities are P(four)=1/6, P(not four)=5/6.
  2. The conditional observation probabilities are P(E|four)=3/5, P(E|not four)=2/5.
  3. By total probability, P(E)=(1/6)(3/5) + (5/6)(2/5)=13/30.
  4. The requested joint probability with E is 1/10=1/10.
  5. By Bayes’ theorem, the requested posterior is 1/10 / (13/30) = 3/13.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are four, not four. Their prior probabilities are P(four)=1/6, P(not four)=5/6.
  • [1 mark] The conditional observation probabilities are P(E|four)=3/5, P(E|not four)=2/5.
  • [1 mark] By total probability, P(E)=(1/6)(3/5) + (5/6)(2/5)=13/30.
  • [1 mark] The requested joint probability with E is 1/10=1/10.
  • [1 mark] By Bayes’ theorem, the requested posterior is 1/10 / (13/30) = 3/13.

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