CBSE Class 12 Mathematics 2023: question 35(a)

Hints

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Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

1. Define the events:

   • $A$: the student knows the answer
   • $B$: the student guesses the answer
   • $E$: the student answers correctly
   1 mark

2. Prior probabilities: $P(A)=\frac{3}{5}$ and $P(B)=\frac{2}{5}$.

   ½ mark

3. Conditional probabilities:

   • A student who knows the answer is certainly correct, so $P(E|A)=1$.
   • A student who guesses is correct with probability $\frac{1}{3}$, so $P(E|B)=\frac{1}{3}$.
   1 mark

4. By Bayes' theorem:
   $$P(A|E)=\frac{P(A)P(E|A)}{P(A)P(E|A)+P(B)P(E|B)}=\frac{\frac{3}{5}\times 1}{\frac{3}{5}\times 1+\frac{2}{5}\times \frac{1}{3}}$$

   1½ marks

5. Simplify:
   $$P(A|E)=\frac{\frac{3}{5}}{\frac{3}{5}+\frac{2}{15}}=\frac{\frac{9}{15}}{\frac{11}{15}}=\frac{9}{11}$$

   So the probability that the student knew the answer, given that he answered correctly, is $\frac{9}{11}$.

   1 mark

Answer: 9/11

Revise the chapter

• Chapter: Probability (notes, formulas, where marks are lost and every past question)
• Topic: Bayes' Theorem