Two Gamblers Play Until One is Bankrupt: Chance That the Better Player Wins

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 ]