Two ships sharing a berth

Question

Two ships each arrive at a berth at a uniformly random time during a 24-hour day, independently. Each ship occupies the berth for 4 hours after it arrives.

What is the probability that their stays overlap (a scheduling conflict)?

Accepted Answer

Represent the two arrival times as a point (x, y) dropped uniformly into a 24-by-24 square, where all points are equally likely. Because the outcomes fill the square evenly, probability equals favorable area over total area. Two 4-hour stays overlap exactly when the arrivals are within 4 hours of each other: x - y \le 4. That is a diagonal band. Find the complement, the two corner triangles where the gap exceeds 4 hours. Each triangle has legs of length 24 - 4 = 20, so the two together have area 2 \cdot \frac{1}{2}(20)(20) = 20^2 = 400. The square has area 24^2 = 576. So the overlap probabili…

Two ships sharing a berth

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