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JEE Main 2025 · Mathematics · Probability · medium

Bag I has 3 red and 2 black balls; Bag II has 2 red and 4 black balls. A fair die is thrown: if 1 or 2 appears, a ball is drawn from Bag I, otherwise…

Single correct option

Bag I has 3 red and 2 black balls; Bag II has 2 red and 4 black balls. A fair die is thrown: if 1 or 2 appears, a ball is drawn from Bag I, otherwise from Bag II. Given that the ball drawn is red, the probability that it came from Bag I is
  1. A. 919\tfrac{9}{19}
  2. B. 35\tfrac{3}{5}
  3. C. 1019\tfrac{10}{19}
  4. D. 13\tfrac{1}{3}
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Answer: (A) 919\tfrac{9}{19}

P(I)=13P(\text{I})=\tfrac13, P(II)=23P(\text{II})=\tfrac23, P(R∣I)=35P(R|\text{I})=\tfrac35, P(R∣II)=13P(R|\text{II})=\tfrac13. By Bayes' theorem, P(I∣R)=13⋅3513⋅35+23⋅13=1/51/5+2/9=9/4519/45=919P(\text{I}|R)=\dfrac{\frac13\cdot\frac35}{\frac13\cdot\frac35+\frac23\cdot\frac13}=\dfrac{1/5}{1/5+2/9}=\dfrac{9/45}{19/45}=\dfrac{9}{19}.
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