A bag has 2 red and 3 white balls. A and B take turns to draw one ball each, without replacement, with A going first; whoever first draws a red ball wins. What is the probability that A wins?

Explanation

1. 01

   A wins on the first draw with probability $\frac{2}{5}$, or on the third, after two white balls, with probability $\frac{3}{5} \times \frac{2}{4} \times \frac{2}{3} = \frac{1}{5}$.

2. 02

   With only 3 white balls, a red one appears by the fourth draw, so A gets no fifth draw: $\frac{2}{5} + \frac{1}{5} = \frac{3}{5}$.