GATE CSE 2026 | Set 1 | Question: 1

Question

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

• A. $1 / 4$
• B. $1 / 3$
• C. $1 / 2$
• D. $2 / 3$

Comments

similar type of problem is asked in 2020 AMC 10B Problems

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@Jayvijay Chauhan I think there are a lot of questions directly taken from wiki. and other sites in discrete mathematics.

Answer 1

$\boxed{\text{Answer: B) } \frac{1}{3}}$

Tree Diagram

Answer 2

After $2$ draws we will have total of $4$ balls in the urn. In our case we want equal number of Red and Blue balls i.e. $2R \ \&\ 2B$
 

So one draw should be Blue and another should be Red, the 2 sequence can be $RB$ & $BR$

$\to P(RB) = P(first \ is \ R) \times P(second \ is \ B \ |\ first \ is \ R) = (\frac{1}{2})(\frac{1}{3}) = \frac{1}{6}$

$\to P(BR) = same\ case \ as \ P(RB) = \frac{1}{6}$

Total Probability  $P(Equal \ R \ \& \ B) = \frac{1}{6} + \frac{1}{6} = \frac{1}{3}$

Answer 3

.

Answer 4

Answer is 1/3 

As explained in below image