Two signals close in time

Question

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?

Accepted Answer

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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