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

Let $R$ denote a red ball and $B$ denote a blue ball.

Initial state: Urn contains $1R + 1B$ (Total = $2$ balls).

Step 1:

• With probability $\frac{1}{2}$, an $R$ is drawn. It is replaced along with an additional red ball. The urn now has $2R + 1B$ (Total = $3$ balls).

• With probability $\frac{1}{2}$, a $B$ is drawn. It is replaced along with an additional blue ball. The urn now has $1R + 2B$ (Total = $3$ balls).

Step 2:

• If urn state is $(2R, 1B)$:

  • To get an equal number of red and blue balls after step 2 (i.e., $2R + 2B$), a blue ball must be drawn in this step.

  • Probability of drawing $B = \frac{1}{3}$.

  • Joint probability for path $R \to B = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$.

• If urn state is $(1R, 2B)$:

  • To get an equal number of red and blue balls after step 2 (i.e., $2R + 2B$), a red ball must be drawn in this step.

  • Probability of drawing $R = \frac{1}{3}$.

  • Joint probability for path $B \to R = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$.

Total Probability:

Hence, the correct option is (B).