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Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #75

A student knows an answer with probability 3/5 and otherwise guesses with probability 2/5. Knowing guarantees correctness; a guess is correct with probability 1/3. Given a correct answer, find P(the student knew it).

✅ Step-by-Step Solution

Final Answer: 9/11
  1. Let E be the observed event. The mutually exclusive exhaustive cases are knows, guesses. Their prior probabilities are P(knows)=3/5, P(guesses)=2/5.
  2. The conditional observation probabilities are P(E|knows)=1, P(E|guesses)=1/3.
  3. By total probability, P(E)=(3/5)(1) + (2/5)(1/3)=11/15.
  4. The requested joint probability with E is 3/5=3/5.
  5. By Bayes’ theorem, the requested posterior is 3/5 / (11/15) = 9/11.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are knows, guesses. Their prior probabilities are P(knows)=3/5, P(guesses)=2/5.
  • [1 mark] The conditional observation probabilities are P(E|knows)=1, P(E|guesses)=1/3.
  • [1 mark] By total probability, P(E)=(3/5)(1) + (2/5)(1/3)=11/15.
  • [1 mark] The requested joint probability with E is 3/5=3/5.
  • [1 mark] By Bayes’ theorem, the requested posterior is 3/5 / (11/15) = 9/11.

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