GATE 2026 CS (CS1) – Question 11
An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is put back in the urn. The probability that there are equal number of red and blue balls after two steps is
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Correct answer: (B) $\frac{1}{3}$
Explanation
Initially, the urn has 1 Red and 1 Blue ball (total 2 balls).
Step 1: Pick a ball.
- Red is picked with probability $\frac{1}{2}$. It is returned with another Red ball. The urn now has 2 Red and 1 Blue ball (total 3 balls).
- Blue is picked with probability $\frac{1}{2}$. It is returned with another Blue ball. The urn now has 1 Red and 2 Blue balls (total 3 balls).
Step 2: To have an equal number of red and blue balls after two steps, the total number of balls will be 4, so there must be exactly 2 Red and 2 Blue balls.
- From state (2R, 1B): we must pick a Blue ball, which has probability $\frac{1}{3}$. This gives (2R, 2B).
- From state (1R, 2B): we must pick a Red ball, which has probability $\frac{1}{3}$. This gives (2R, 2B).
By the law of total probability:
$$P(\text{equal R and B}) = \left(\frac{1}{2} \times \frac{1}{3}\right) + \left(\frac{1}{2} \times \frac{1}{3}\right) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$$
Therefore, option (B) is correct.