GATE CS 2026 Set 1 Q.11: Probability and Statistics

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

$1/4$

$1/3$

$1/2$

$2/3$

Official answer and solution

Official answer: B
From: Official GATE 2026 answer key

Solution

Compute the probability of picking one red ball and one blue ball in either order across the two steps.

Initially the urn has $1$ red ( $R$ ) and $1$ blue ( $B$ ) ball ( $2$ balls total). After two steps, the urn has $4$ balls, so having an equal number of red and blue balls means $2$ red and $2$ blue (one $R$ and one $B$ picked in the two steps):

Step 1 picks $R$ and Step 2 picks $B$ : $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$
Step 1 picks $B$ and Step 2 picks $R$ : $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$

Adding the two mutually exclusive cases gives $\frac{1}{6} + \frac{1}{6} = \frac{1}{3}$ .

Final answer

Option (B)

Common trap

Assuming independent steps with probability $1/2$ at the second step.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 1 (CS1)
Question: 11 of 65
Section: Computer Science and Information Technology

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