Duel of Two 50% Marksmen: Odds in favor of the man who shoots first

Smith shoots first.
$P = S + SJS + SJSJS + SJSJSJS + SJSJSJSJS + \ldots$

$P = S + (SJ)S + (SJ)^2 S + (SJ)^3 S + (SJ)^4 S + \ldots$
 

Where
$S = 0.5$

$SJ = 0.5(0.5) = 0.25$

 

$P = 0.5 + (0.25)(0.5) + (0.25^2)(0.5) + (0.25^3)(0.5) + (0.25^4)(0.5) + \ldots$
 

Sum of Infinite Geometric Progression
$P = \dfrac{a_1}{1 - r}$

$P = \dfrac{0.5}{1 - 0.25}$

$P = 2/3$