Class 12 Probability · Long Answer3 Marksbayes_theorem
Class 12 Maths · Probability · Long Answer #51
Choose uniformly a biased coin with head:tail odds 1:3 or a fair coin. Toss it once; given a head, find P(biased coin).
✅ Step-by-Step Solution
Final Answer: 1/3
- Let E be the observed event. The mutually exclusive exhaustive cases are biased, fair. Their prior probabilities are P(biased)=1/2, P(fair)=1/2.
- The conditional observation probabilities are P(E|biased)=1/4, P(E|fair)=1/2. By total probability, P(E)=(1/2)(1/4) + (1/2)(1/2)=3/8.
- The requested joint probability with E is 1/8=1/8.
- By Bayes’ theorem, the requested posterior is 1/8 / (3/8) = 1/3.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let E be the observed event. The mutually exclusive exhaustive cases are biased, fair. Their prior probabilities are P(biased)=1/2, P(fair)=1/2.
- [1 mark] The conditional observation probabilities are P(E|biased)=1/4, P(E|fair)=1/2. By total probability, P(E)=(1/2)(1/4) + (1/2)(1/2)=3/8.
- [1 mark] The requested joint probability with E is 1/8=1/8.
- [0.5 marks] By Bayes’ theorem, the requested posterior is 1/8 / (3/8) = 1/3.
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