Probability that two people see each other at the coffee shop

Two mathematicians each come into a coffee shop at a random time between 8:00 a.m. and 9:00 a.m. each day. Each orders a cup of coffee then sits at a table, reading a newspaper for 20 minutes before leaving to go to work.

On any day, what is the probability that both mathematicians are at the coffee shop at the same time (that is, their arrival times are within 20 minutes of each other)?

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This is a common homework question. I'd like to see more effort on the part of the asker than this. –  Larry Wang Jul 25 '10 at 17:35
I don't know which classes you're taking where this is a common homework question, but I just really like the graphical solution for this problem and so I thought I'd ask it so that someone could put up the solution. –  Ben Alpert Jul 25 '10 at 18:23
I agree with your goals, I'm not implying that you got assigned this question for homework. I just think that this looks a lot like a student retyping a question from his textbook and getting a complete answer, which is a bad precedent to set. For instance, this is exactly problem 5.1.6 out of Pitman's Probability, a standard textbook at Berkeley. –  Larry Wang Jul 26 '10 at 8:35

1 Answer

Working in hours and letting 8:00 a.m. be t=0, each mathematician's arrival time is a number between 0 and 1. The sample space can be represented by the unit square in the coordinate plane with one professor's arrival time as x and the other's as y, where regions with equal areas are equally likely. We want x - 1/3 < y < x + 1/3 -- that is, the second professor arrives earlier than the first by no more than 1/3 of an hour or later than the first by no more than 1/3 of an hour.

The area of the desired region is 5/9.

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This is a rather elegant answers =) I do find graphical proofs and solutions rather captivating because (in my mind) because they seem so much more...universal. –  Justin L. Jul 25 '10 at 7:42
Nice. It's funny we both used Mathematica and then uploaded to imgur for the region plot. –  Ben Alpert Jul 25 '10 at 15:44
@Ben: No coincidence there--I liked the way it looked in your answer to my question and I'd been looking for a quick-and-easy image host for images in my answers here, so checked where you'd put that image up. –  Isaac Jul 25 '10 at 15:56
These images are phenomenal. I am going to figure out what you guys are doing and then do the same for future questions/answers. Thanks! –  Tom Stephens Jul 25 '10 at 18:27
Tom Stephens: In Mathematica, you can generate the above plot by running RegionPlot[x - 1/3 < y < x + 1/3, {x, 0, 1}, {y, 0, 1}, FrameTicks -> {{0, 1/3, 2/3, 1}, {0, 1/3, 2/3, 1}}] then selecting Save Graphic As… from the right-click menu to make a PNG. If you don't have Mathematica, you'll need to find some other tool. –  Ben Alpert Jul 25 '10 at 21:55