Two friends with different waiting times

Question

Two friends agree to meet between noon and 1pm. Each arrives at a time uniform on the hour, independently. Friend A is impatient and waits only $10$ minutes; friend B waits $20$ minutes. They meet if either is still present when the other arrives.

What is the probability they meet?

Accepted Answer

Let X, Y \sim U(0, 60) be A's and B's arrival times. Their intervals [X, X+10] and [Y, Y+20] overlap iff X \le Y + 20 and Y \le X + 10, that is -20 \le X - Y \le 10. The band around the diagonal is now lopsided: it extends 10 on the side where A arrives after B, and 20 on the side where B arrives after A. The complement (no meeting) is two right triangles: - X - Y 10 (A arrives well after B): legs 60 - 10 = 50, area \frac{1}{2}(50)^2 = 1250. - Y - X 20 (B arrives well after A): legs 60 - 20 = 40, area \frac{1}{2}(40)^2 = 800. So P(	ext{no meeting}) = \frac{1250 + 800}{60^2} = \frac{2050}{360…

Two friends with different waiting times

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