JEE Advanced · Mathematics · 32. Probability
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 ________.
- A 11
- B 22
- C 33
- D 44
Answer & Solution
Correct Answer
(A) 11
Step-by-step Solution
Detailed explanation
Given \(\mathrm{P}\left(\mathrm{W}_1 \cap \mathrm{G}_2 \cap \mathrm{B}_3\right)=\frac{2}{5 \mathrm{~N}}\)
and \(\mathrm{P}\left(\mathrm{B}_3 \mid \mathrm{W}_1 \cap \mathrm{G}_2\right)=\frac{2}{9}\)
\(\begin{aligned} & \Rightarrow \frac{\mathrm{P}\left(\mathrm{B}_3 \cap \mathrm{W}_1 \cap \mathrm{G}_2\right)}{\mathrm{P}\left(\mathrm{W}_1 \cap \mathrm{G}_2\right)}=\frac{2}{9} \\ & \Rightarrow \frac{2}{5 \mathrm{~N}} \times \frac{\mathrm{N} \times(\mathrm{N}-1)}{3 \times 6}=\frac{2}{9} \\ & \Rightarrow \mathrm{N}=11\end{aligned}\)
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