JEE Advance - Mathematics (2024 - Paper 2 Online - No. 9)
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 $i^{\text {th }}$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P\left(W_1 \cap G_2 \cap B_3\right)=\frac{2}{5 N}$ and the conditional probability $P\left(B_3 \mid W_1 \cap G_2\right)=\frac{2}{9}$, then $N$ equals ________.
Answer
11
Explanation
3 White
6 Green
$$(\mathrm{N}-9)$$ Blue
$$\begin{aligned} & \text { Given } P\left(W_1 \cap G_2 \cap B_3\right)=\frac{2}{5 N} \\ & \text { and } P\left(B_3 \mid W_1 \cap G_2\right)=\frac{2}{9} \\ & \Rightarrow \frac{P\left(B_3 \cap W_1 \cap G_2\right)}{P\left(W_1 \cap G_2\right)}=\frac{2}{9} \\ & \Rightarrow \frac{2}{5 N} \times \frac{N \times(N-1)}{3 \times 6}=\frac{2}{9} \\ & \Rightarrow N=11 \end{aligned}$$
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