Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #76

Choose uniformly among a two-headed coin, a coin with P(head)=3/4, and a fair coin. Toss once and observe a head. Find P(two-headed coin).

✅ Step-by-Step Solution

Final Answer: 4/9

Let E be the observed event. The mutually exclusive exhaustive cases are two-headed, biased, fair. Their prior probabilities are P(two-headed)=1/3, P(biased)=1/3, P(fair)=1/3.
The conditional observation probabilities are P(E|two-headed)=1, P(E|biased)=3/4, P(E|fair)=1/2.
By total probability, P(E)=(1/3)(1) + (1/3)(3/4) + (1/3)(1/2)=3/4.
The requested joint probability with E is 1/3=1/3.
By Bayes’ theorem, the requested posterior is 1/3 / (3/4) = 4/9.

[1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are two-headed, biased, fair. Their prior probabilities are P(two-headed)=1/3, P(biased)=1/3, P(fair)=1/3.
[1 mark] The conditional observation probabilities are P(E|two-headed)=1, P(E|biased)=3/4, P(E|fair)=1/2.
[1 mark] By total probability, P(E)=(1/3)(1) + (1/3)(3/4) + (1/3)(1/2)=3/4.
[1 mark] The requested joint probability with E is 1/3=1/3.
[1 mark] By Bayes’ theorem, the requested posterior is 1/3 / (3/4) = 4/9.

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