JEE Advanced 2024 — Paper 2 — MATH-III solution

JEE Advanced2024Paper 2MATH-III

A bag contains $N$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $i = 1, 2, 3$ , let $W_{i}$ , $G_{i}$ , and $B_{i}$ denote the events that the ball drawn in the $i^{th}$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P (W_{1} \cap G_{2} \cap B_{3}) = \frac{2}{5 N}$ and the conditional probability $P (B_{3} ∣ W_{1} \cap G_{2}) = \frac{2}{9}$ , then $N$ equals ______.

Your answer

Solution

The bag contains 3 white, 6 green, and $N - 9$ blue balls. The conditional probability gives the blue fraction after one white and one green are removed:

$P (B_{3} ∣ W_{1} \cap G_{2}) = \frac{N - 9}{N - 2} = \frac{2}{9} .$

Cross-multiplying: $9 (N - 9) = 2 (N - 2) \Rightarrow 7 N = 77 \Rightarrow N = 11.$

Verify with the joint probability: $P (W_{1} \cap G_{2} \cap B_{3}) = \frac{3}{11} \cdot \frac{6}{10} \cdot \frac{2}{9} = \frac{36}{990} = \frac{2}{55} = \frac{2}{5 \cdot 11} .$ Consistent.

$∴ N = 11.$

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