Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #118

Machines A,B,C make25%,35%,40% of bolts, with defect rates5%,4%,2%. A uniform bolt is defective. Find P(it was produced by A or C).

✅ Step-by-Step Solution

Final Answer: 41/69

Let E be the observed event. The mutually exclusive exhaustive cases are A, B, C. Their prior probabilities are P(A)=1/4, P(B)=7/20, P(C)=2/5.
The conditional observation probabilities are P(E|A)=1/20, P(E|B)=1/25, P(E|C)=1/50.
By total probability, P(E)=(1/4)(1/20) + (7/20)(1/25) + (2/5)(1/50)=69/2000.
The requested joint probability with E is 1/80 + 1/125=41/2000.
By Bayes’ theorem, the requested posterior is 41/2000 / (69/2000) = 41/69.

[1 mark] Let E be the observed event. The mutually exclusive exhaustive cases are A, B, C. Their prior probabilities are P(A)=1/4, P(B)=7/20, P(C)=2/5.
[1 mark] The conditional observation probabilities are P(E|A)=1/20, P(E|B)=1/25, P(E|C)=1/50.
[1 mark] By total probability, P(E)=(1/4)(1/20) + (7/20)(1/25) + (2/5)(1/50)=69/2000.
[1 mark] The requested joint probability with E is 1/80 + 1/125=41/2000.
[1 mark] By Bayes’ theorem, the requested posterior is 41/2000 / (69/2000) = 41/69.

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