aez-notes

Duels in the town of Discretion are rarely fatal. There, each contestant comes at a random moment between 5am and 6pm 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?

This is the same as asking what is the probability that two random variables, uniformly distributed on the unit interval, differ by less than \(5/60\). This is a simple integral, we just need to distinguish between the cases where one of the variables is less than \(5/60\), when it is greater than \(1 - 5/60\) and when it is between these values.

intervalLength : 5 / 60;

result : integrate(y + intervalLength, y, 0, intervalLength) +
  integrate(1/6, y, intervalLength, 1 - intervalLength) +
  integrate(1 - y + intervalLength, y, 1 - intervalLength, 1);

print(result);
print(float(result));

The answer is \(23/144\approx 0.160\).