JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Probability XII

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 ____.

Answer: 11

Solution

There are 3 white, 6 green, and

(N - 9)

blue balls.

Using the conditional probability formula:

P (B_{3} ∣ W_{1} \cap G_{2}) = \frac{P ( W _{1} \cap G _{2} \cap B _{3} )}{P ( W _{1} \cap G _{2} )}

Given:

\frac{2/ ( 5 N )}{P ( W _{1} \cap G _{2} )} = \frac{2}{9}

Now:

P (W_{1} \cap G_{2}) = \frac{3}{N} \cdot \frac{6}{N - 1}

And:

P (W_{1} \cap G_{2} \cap B_{3}) = \frac{3}{N} \cdot \frac{6}{N - 1} \cdot \frac{N - 9}{N - 2} = \frac{2}{5 N}

From the conditional probability:

\frac{N - 9}{N - 2} = \frac{2}{9}

Solving:

9 (N - 9) = 2 (N - 2) \Rightarrow 9 N - 81 = 2 N - 4 \Rightarrow 7 N = 77 \Rightarrow N = 11

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Integer answer

Practise more: all JEE Advanced PYQs on Probability XII →

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Probability XII

Solution

More JEE Advanced questions on Probability XII