Discussion
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
Answer: B
The probability is 1/3.
After two steps the urn holds 4 balls. Equal numbers means 2 red and 2 blue, which needs one red draw and one blue draw.
\[ \begin{aligned} P(R \text{ then } B) &= \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \\ P(B \text{ then } R) &= \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \end{aligned} \]
After the first draw the urn holds 3 balls, two of the drawn colour, so the other colour has probability 1/3. The total is 1/6 + 1/6 = 1/3.
1/2 treats the second draw as if the urn were unchanged, which ignores the reinforcement. The other outcomes, 3 red and 1 blue or the reverse, have probability 1/3 each, which is a useful check: 1/3 + 1/3 + 1/3 = 1.
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