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JEE Advanced 2025 Paper 2 · Mathematics · Probability · medium

If the outcome is 1 or 2, two balls are drawn without replacement from urn A containing 3 red and 2 blue balls; otherwise two balls are drawn without…

Single correct option

A fair die is thrown. If the outcome is 11 or 22, two balls are drawn without replacement from urn AA containing 33 red and 22 blue balls; otherwise two balls are drawn without replacement from urn BB containing 11 red and 44 blue balls. Given that the two balls drawn are of the same colour, the probability that they were drawn from urn AA is
  1. A. 14\dfrac14
  2. B. 13\dfrac13
  3. C. 29\dfrac29
  4. D. 310\dfrac{3}{10}
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Answer: (A) 14\dfrac14

For urn AA: P(same)=(32)+(22)(52)=410=25P(\text{same})=\frac{\binom32+\binom22}{\binom52}=\frac{4}{10}=\frac25. For urn BB: P(same)=(12)+(42)(52)=610=35P(\text{same})=\frac{\binom12+\binom42}{\binom52}=\frac{6}{10}=\frac35. Priors: P(A)=13P(A)=\frac13, P(B)=23P(B)=\frac23. By Bayes' theorem, P(A∣same)=13⋅2513⋅25+23⋅35=2/152/15+6/15=14P(A\mid\text{same})=\dfrac{\frac13\cdot\frac25}{\frac13\cdot\frac25+\frac23\cdot\frac35}=\dfrac{2/15}{2/15+6/15}=\dfrac14. Using 12\tfrac12 for the die prior or drawing with replacement gives the other options.
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Original question written in the JEE Advanced 2025 Paper 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake