GATE CS 1995 Q2.14 — Probability: Random Variables and Distributions

To find the probability that one of the drawn balls is black and the other is white, we assume the standard case of drawing without replacement.

Given:

• Number of white balls = $10$
• Number of black balls = $15$
• Total number of balls = $10 + 15 = 25$

We can solve this using two standard approaches:

Method 1: Using Combinations

The total number of ways to choose any $2$ balls out of $25$ is:

The number of ways to choose exactly $1$ white ball (from $10$) and $1$ black ball (from $15$) is:

Thus, the required probability is:

Method 2: Sequential Probability

There are two mutually exclusive sequences to get one white and one black ball:

1. First ball is White, second is Black (W then B): $$P(W \text{ then } B) = \frac{10}{25} \times \frac{15}{24} = \frac{150}{600} = \frac{1}{4}$$
2. First ball is Black, second is White (B then W): $$P(B \text{ then } W) = \frac{15}{25} \times \frac{10}{24} = \frac{150}{600} = \frac{1}{4}$$

Adding these probabilities together:

Hence, the correct option is C.