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Class 12 Probability · Long Answer3 Marksbayes_theorem

Class 12 Maths · Probability · Long Answer #61

A fair die is rolled. A witness answers the yes/no question “Was it a six?” correctly with probability 4/5, conditional on either actual yes or actual no. Given a “yes” report, find P(actual six).

✅ Step-by-Step Solution

Final Answer: 4/9
  1. Let E be the observed event. The mutually exclusive exhaustive cases are six, not six. Their prior probabilities are P(six)=1/6, P(not six)=5/6.
  2. The conditional observation probabilities are P(E|six)=4/5, P(E|not six)=1/5. By total probability, P(E)=(1/6)(4/5) + (5/6)(1/5)=3/10.
  3. The requested joint probability with E is 2/15=2/15.
  4. By Bayes’ theorem, the requested posterior is 2/15 / (3/10) = 4/9.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Let E be the observed event. The mutually exclusive exhaustive cases are six, not six. Their prior probabilities are P(six)=1/6, P(not six)=5/6.
  • [1 mark] The conditional observation probabilities are P(E|six)=4/5, P(E|not six)=1/5. By total probability, P(E)=(1/6)(4/5) + (5/6)(1/5)=3/10.
  • [1 mark] The requested joint probability with E is 2/15=2/15.
  • [0.5 marks] By Bayes’ theorem, the requested posterior is 2/15 / (3/10) = 4/9.

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