Two arrivals, unequal patience

Question

Alice and Bob each arrive at a cafe at a time chosen uniformly at random during a $60$ -minute window, independently. Alice waits $10$ minutes for Bob before leaving; Bob waits $20$ minutes for Alice. Neither stays past the end of the window.

What is the probability that they meet?

Accepted Answer

Let a and b be their arrival minutes, each uniform on [0, 60], plotted as a point in a 60 	imes 60 square (all points equally likely). Alice is present during [a, a+10] and Bob during [b, b+20]. Their visits overlap unless one arrives too long after the other, that is, they miss iff a - b 20 \quad	ext{or}\quad b - a 10 . Each miss region is a right triangle in the square: - a - b 20: a triangle with legs 60 - 20 = 40, area 	frac12 (40)^2 = 800. - b - a 10: a triangle with legs 60 - 10 = 50, area 	frac12 (50)^2 = 1250. The full square has area 3600, so P(	ext{miss}) = \frac{800 + 1250}{36…

Two arrivals, unequal patience

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