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

Date of Exam:

**Problem**

Smith and Jones, both 50% marksmen, decide to fight a duel in which they exchange alternate shots until one is hit. What are the odds in favor of the man who shoots first?

A. 1/3 | C. 2/3 |

B. 1/2 | D. 1/4 |

**Answer Key**

[ C ]

**Solution**

Smith shoots first.

$P = S + SJS + SJSJS + SJSJSJS + SJSJSJSJS + \ldots$Where

$S = 0.5$

$P = S + SJS + SJSJS + SJSJSJS + SJSJSJSJS + \ldots$

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

$S = 0.5$

$SJ = 0.5(0.5) = 0.25$

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

Sum of Infinite Geometric Progression

$P = \dfrac{a_1}{1 - r}$

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

$P = 2/3$