W = win
L = lose
Player M wins:
$P = WW + WLWW + WLWLWW + WLWLWLWW + \ldots$
$P = W^2 + (WL)W^2 + (WL)^2 W^2 + (WL)^3 W^2 + \ldots$
$P = \left( \frac{2}{3} \right)^2 + \left( \frac{2}{3} \cdot \frac{1}{3} \right) \left( \frac{2}{3} \right)^2 + \left( \frac{2}{3} \cdot \frac{1}{3} \right)^2 \left( \frac{2}{3} \right)^2 + \left( \frac{2}{3} \cdot \frac{1}{3} \right)^3 \left( \frac{2}{3} \right)^2 + \ldots$
Sum of Infinite Geometric Progression
$P = \dfrac{a_1}{1 - r} = \dfrac{\left( \frac{2}{3} \right)^2}{1 - \left( \frac{2}{3} \cdot \frac{1}{3} \right)}$
$P = 4/7$ ← Answer: [ C ]
