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 Wi, Gi and Bi denote the events that the ball drawn in the ith draw is a white ball, green ball, and blue ball, respectively. If the probability and the conditional probability , then N equals ______

Engineering

Mathematics

Addition Theorem on Probability

Conditional Probability

Question

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 _______ .

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3 White

6 Green

(N – 9) Blue

Given $P (W_{1} \cap G_{2} \cap B_{3}) = \frac{2}{5 N}$

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

$\Rightarrow \frac{P (B_{3} \cap W_{1} \cap G_{2})}{P (W_{1} \cap G_{2})} = \frac{2}{9}$

$\Rightarrow \frac{2}{5 N} \times \frac{N \times (N - 1)}{3 \times 6} = \frac{2}{9}$

⇒ N = 11