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JEE Advanced 2021 Paper 2 · Mathematics · Probability · hard

A fair die is rolled. If the result is 1 or 2, two balls are drawn without replacement from bag A containing 3 red and 2 blue balls; otherwise two…

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A fair die is rolled. If the result is 11 or 22, two balls are drawn without replacement from bag AA containing 33 red and 22 blue balls; otherwise two balls are drawn without replacement from bag BB containing 22 red and 44 blue balls. Let R1R_1 and R2R_2 denote the events that the first and the second ball drawn are red, respectively. Which of the following statements is/are correct?
  1. A. P(R1∩R2)=1390P(R_1\cap R_2)=\dfrac{13}{90}
  2. B. P(bag A was used∣R1∩R2)=913P(\text{bag }A\text{ was used}\mid R_1\cap R_2)=\dfrac{9}{13}
  3. C. P(R2∣R1)=1338P(R_2\mid R_1)=\dfrac{13}{38}
  4. D. R1R_1 and R2R_2 are independent events
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Answer: A, B, C

Condition on the bag. P(R1∩R2)=13⋅3⋅25⋅4+23⋅2⋅16⋅5=110+245=1390P(R_1\cap R_2)=\frac13\cdot\frac{3\cdot2}{5\cdot4}+\frac23\cdot\frac{2\cdot1}{6\cdot5}=\frac1{10}+\frac2{45}=\frac{13}{90}, so (A) is true. By Bayes, P(A∣R1R2)=1/1013/90=913P(A\mid R_1R_2)=\frac{1/10}{13/90}=\frac9{13}, so (B) is true. Also P(R1)=13⋅35+23⋅13=1945P(R_1)=\frac13\cdot\frac35+\frac23\cdot\frac13=\frac{19}{45}, giving P(R2∣R1)=13/9019/45=1338P(R_2\mid R_1)=\frac{13/90}{19/45}=\frac{13}{38}, so (C) is true. For (D), by symmetry P(R2)=P(R1)=1945P(R_2)=P(R_1)=\frac{19}{45}, and P(R1)P(R2)=3612025≠1390P(R_1)P(R_2)=\frac{361}{2025}\neq\frac{13}{90}, so the events are dependent and (D) is false. Answer: A, B and C (checked with exact fractions).
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Original question written in the JEE Advanced 2021 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