Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #122

Roll a fair die. On5 or6 toss a fair coin3 times; on1,2,3,4 toss it2 times, independently. Given exactly2 heads, find P(die was1,2,3,4).

✅ Step-by-Step Solution

Final Answer: 4/7

Let E be the observed event. The mutually exclusive exhaustive cases are die5,6, die1,2,3,4. Their prior probabilities are P(die5,6)=1/3, P(die1,2,3,4)=2/3.
The conditional observation probabilities are P(E|die5,6)=3/8, P(E|die1,2,3,4)=1/4.
By total probability, P(E)=(1/3)(3/8) + (2/3)(1/4)=7/24.
The requested joint probability with E is 1/6=1/6.
By Bayes’ theorem, the requested posterior is 1/6 / (7/24) = 4/7.

[1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are die5,6, die1,2,3,4. Their prior probabilities are P(die5,6)=1/3, P(die1,2,3,4)=2/3.
[1 mark] The conditional observation probabilities are P(E|die5,6)=3/8, P(E|die1,2,3,4)=1/4.
[1 mark] By total probability, P(E)=(1/3)(3/8) + (2/3)(1/4)=7/24.
[1 mark] The requested joint probability with E is 1/6=1/6.
[1 mark] By Bayes’ theorem, the requested posterior is 1/6 / (7/24) = 4/7.

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