Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #117

Machines A,B,C produce30%,50%,20% of bolts and have defect rates3%,4%,1%. A uniform bolt is defective. Find P(it was not produced by B).

✅ Step-by-Step Solution

Final Answer: 11/31

Let E be the observed event. The mutually exclusive exhaustive cases are A, B, C. Their prior probabilities are P(A)=3/10, P(B)=1/2, P(C)=1/5.
The conditional observation probabilities are P(E|A)=3/100, P(E|B)=1/25, P(E|C)=1/100.
By total probability, P(E)=(3/10)(3/100) + (1/2)(1/25) + (1/5)(1/100)=31/1000.
The requested joint probability with E is 9/1000 + 1/500=11/1000.
By Bayes’ theorem, the requested posterior is 11/1000 / (31/1000) = 11/31.

[1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are A, B, C. Their prior probabilities are P(A)=3/10, P(B)=1/2, P(C)=1/5.
[1 mark] The conditional observation probabilities are P(E|A)=3/100, P(E|B)=1/25, P(E|C)=1/100.
[1 mark] By total probability, P(E)=(3/10)(3/100) + (1/2)(1/25) + (1/5)(1/100)=31/1000.
[1 mark] The requested joint probability with E is 9/1000 + 1/500=11/1000.
[1 mark] By Bayes’ theorem, the requested posterior is 11/1000 / (31/1000) = 11/31.

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