Romeo and Juliet try to meet each other everyday at a certain place between 12:00 and 13:00. They arrive at the place at a random time between 12:00 and 13:00 and they wait 15 minutes (but never after 13:00). If nobody shows up they come back home. Every day, what is the probability that they meet each other?
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1 Answer
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Let $R$ and $J$ be uniform $[0,1]$ random variables to represent the arrival times of Romeo and Juliet, respectively. The star-crossed lovers meet if $|R-J|\le 1/4$.
The desired probability is the area of the shaded region here, and this area is
$(1-0)^2 - (3/4)^2 = 7/16$.
