A quick caller and a patient one

Question

Two colleagues each join a shared phone line at a uniformly random time during a $60$ -minute hour, independently. One holds the line for $10$ minutes, the other for $30$ minutes (neither past the end of the hour).

What is the probability their times on the line overlap?

Accepted Answer

Let a, b be the join times, each uniform on [0, 60]. They miss each other iff a - b 30 \quad	ext{or}\quad b - a 10 , triangles with legs 60 - 30 = 30 and 60 - 10 = 50 inside the 60 	imes 60 square: P(	ext{miss}) = \frac{	frac12(30)^2 + 	frac12(50)^2}{60^2} = \frac{450 + 1250}{3600} = \frac{1700}{3600} = \frac{17}{36} . Therefore P(	ext{overlap}) = 1 - 	frac{17}{36} = 	frac{19}{36} \approx 0.528. Takeaway. Longer patience on one side widens the meeting band asymmetrically. The formula 1 - \dfrac{(T - wA)^2 + (T - wB)^2}{2T^2} handles any pair of waits.

A quick caller and a patient one

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