Alice and Bob need to arrive within 10 minutes of one another, so on a graph of Alice's vs. Bob's arrival times in minutes past 2 PM, the bounds are y = x + 10 and y = x - 10. Two other bounds are the x-axis and the y-axis. The remaining two bounds are x = 55 and y = 55.
These are base-to-base trapezoids where the base of each is 55*sqrt(2). The shorter parallel sides of the trapezoids are (60-10-5)*sqrt(2) = 45*sqrt(2) each, and the slant height of each of the sides of the trapezoids is 10, so the height of each trapezoid is 10/sqrt(2).
The area of each trapezoid is therefore 50*sqrt(2)*10/sqrt(2) = 500, and the two together are 1000. The whole 60*60 square of potential arrivals has area 3600, so the probability is 1000/3600 = 5/18 = .277777....
From Enigma No. 1622, "Chances for a business meeting", by V.V.Bapeswara Rao, New Scientist, 20 November 2010, page 30.