Probability that two people see each other at the coffee shop

probability

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)?


Solution 1:

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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