Will they meet within a two-hour window?

Question

Two traders agree to be online sometime in a two-hour window ( $120$ minutes). Each logs in at a time uniform on the window, independently, and stays exactly $20$ minutes before logging off.

What is the probability that their sessions overlap?

Accepted Answer

Let X, Y \sim U(0, 120) (minutes into the window) be the login times. Their sessions overlap iff X - Y \le 20. In the 120 	imes 120 square, the complement (no overlap) is two congruent right triangles where one logs in more than 20 minutes after the other, each with legs 120 - 20 = 100: P(	ext{no overlap}) = \frac{2 \cdot \frac{1}{2}(100)^2}{120^2} = \frac{100^2}{120^2} = \left(\frac{5}{6}ight)^2 = \frac{25}{36}. Therefore P(	ext{overlap}) = 1 - \frac{25}{36} = \frac{11}{36} \approx 0.3056. General form. With stay time w out of a window T, P(	ext{overlap}) = 1 - \left(1 - \frac{w}{T}i…

Will they meet within a two-hour window?

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