Two signals close in time

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Two signals each arrive at a uniformly random time within one hour, independently of each other. What is the probability that the two signals arrive within 12 minutes of each other?

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Work in fractions of the hour so both arrival times are uniform on 0 to 1, and 12 minutes is 12/60 = 0.2 of the hour. Drop the pair (x, y) uniformly into the unit square, where probability equals area. "Within 12 minutes" means x - y \le 0.2. This is a diagonal band. The complement is the two corner triangles where the gap exceeds 0.2, each with legs 1 - 0.2 = 0.8. Their combined area is 2 \cdot \frac{1}{2}(0.8)(0.8) = 0.8^2 = 0.64. The unit square has area 1, so P(x - y \le 0.2) = 1 - 0.64 = 0.36. Rescaling to fractions of the window is often the cleanest move: it turns any such problem into…

Two signals close in time

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