Will the two of them actually meet?

Question

Two traders agree to meet between 12:00 and 1:00. Each arrives at a uniformly random time in that hour, independently, and waits $15$ minutes for the other before leaving.

What is the probability that they actually meet?

Accepted Answer

Let x and y be the two arrival times in minutes, each uniform on [0, 60] and independent, so (x, y) is uniform over the 60 	imes 60 square (area 3600). They meet exactly when x - y \le 15. That is the band around the diagonal of the square. It is easier to measure the complement, the two corner triangles where they miss: - x - y 15: a right triangle with legs 60 - 15 = 45, area 	frac12(45)^2 = 1012.5. - y - x 15: the symmetric triangle, area 1012.5. Missing area = 2025, so the meeting area is 3600 - 2025 = 1575, and P(	ext{meet}) = \frac{1575}{3600} = \frac{7}{16} = 0.4375. Equivalently, P…

Will the two of them actually meet?

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