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Class 12 Probability · Long Answer3 Marksbayes_theorem

Class 12 Maths · Probability · Long Answer #58

In a hypothetical population with equal numbers of men and women, 5% of men and 0.25% of women have grey hair. Given a uniformly chosen person has grey hair, find P(man).

✅ Step-by-Step Solution

Final Answer: 20/21
  1. Let E be the observed event. The mutually exclusive exhaustive cases are man, woman. Their prior probabilities are P(man)=1/2, P(woman)=1/2.
  2. The conditional observation probabilities are P(E|man)=1/20, P(E|woman)=1/400. By total probability, P(E)=(1/2)(1/20) + (1/2)(1/400)=21/800.
  3. The requested joint probability with E is 1/40=1/40.
  4. By Bayes’ theorem, the requested posterior is 1/40 / (21/800) = 20/21.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Let E be the observed event. The mutually exclusive exhaustive cases are man, woman. Their prior probabilities are P(man)=1/2, P(woman)=1/2.
  • [1 mark] The conditional observation probabilities are P(E|man)=1/20, P(E|woman)=1/400. By total probability, P(E)=(1/2)(1/20) + (1/2)(1/400)=21/800.
  • [1 mark] The requested joint probability with E is 1/40=1/40.
  • [0.5 marks] By Bayes’ theorem, the requested posterior is 1/40 / (21/800) = 20/21.

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