Class 12 Probability · Very Long Answer5 Marksbayes_theorem

Class 12 Maths · Probability · Very Long Answer #72

Choose boxes I,II with equal probability. I contains 3 red,6 black; II contains 5 red,n black. Draw one uniform ball. Given it is red, P(II|red)=3/5. Find n.

✅ Step-by-Step Solution

Final Answer: n=5

Let E be red. The box priors are P(I)=P(II)=1/2.
The conditional red probabilities are P(E|I)=1/3 and P(E|II)=5/(5+n), with n a nonnegative integer.
Bayes gives P(II|E)=[5/(5+n)]/[1/3+5/(5+n)]=15/(20+n).
Set 15/(20+n)=3/5. Cross-multiplying gives 75=60+3n, so n=5.
With n=5 the posterior is 15/25=3/5. Therefore n=5 black balls.

📋 CBSE Stepwise Marking Scheme

[1 mark] Let E be red. The box priors are P(I)=P(II)=1/2.
[1 mark] The conditional red probabilities are P(E|I)=1/3 and P(E|II)=5/(5+n), with n a nonnegative integer.
[1 mark] Bayes gives P(II|E)=[5/(5+n)]/[1/3+5/(5+n)]=15/(20+n).
[1 mark] Set 15/(20+n)=3/5. Cross-multiplying gives 75=60+3n, so n=5.
[1 mark] With n=5 the posterior is 15/25=3/5. Therefore n=5 black balls.

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Class 12 Maths · Probability · Very Long Answer #72

✅ Step-by-Step Solution

📋 CBSE Stepwise Marking Scheme

Master Chapter 13: Probability

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