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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.2010-07-25
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    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.2010-07-25
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    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](http://www.amazon.com/Probability-Jim-Pitman/dp/0387979743), a standard textbook at Berkeley.2010-07-26

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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.

plot of the inequalities

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.2010-07-25
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    Nice. It's funny we both used Mathematica and then uploaded to imgur for the region plot.2010-07-25
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    @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.2010-07-25
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    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!2010-07-25
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    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.2010-07-25
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    @Tom,Ben: Without Mathematica, you can get WolframAlpha to do some graphs like this, but it sometimes takes more fiddling around to figure out how to get Alpha to use its underlying Mathematica engine the way you want.2010-07-25
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    A good diagram is always worth +1.2010-10-18