# Hurried Duelers

## Problem

Duels in the town of Discretion are rarely fatal. There, each contestant comes at a random moment between $$5$$ AM and $$6$$ AM on the appointed day and leaves exactly $$5$$ minutes later, honor served, unless his opponent arrives within the time interval and then they fight. What fraction of duels lead to violence?

## Solution

Let $$X$$ be the arrival time of the first contestant, and $$Y$$ that of the second. For simplicity, we will scale the duration between $$5$$ AM and $$6$$ AM to the value $$1$$, so that $$\frac{1}{12}$$ corresponds to $$5$$ minutes. Using this notation, the fraction of duels that will to violence can be stated as:

$P\left(|X - Y| < \frac{1}{12}\right) = P\left(-\frac{1}{12} < X - Y < \frac{1}{12}\right)$

Graphing this inequality gives the following shaded region representing duels that lead to violence:

The relevant region is a unit square (since $$0 \leq X \leq 1$$ and $$0 \leq Y \leq 1$$), and $$X$$ and $$Y$$ are both uniformly distributed, so we can calculate the probability of violence as the total area of the square, minus the areas that do not lead to violence, which gives $$1 - \left(\frac{11}{12}\right)^2 = \frac{23}{144} \approx \frac{1}{6}$$.