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This doesn’t require any programming. By drawing a graph we can determine that the time when it is possible to have a successful meeting is when 0 < x < 55/60, 0 < y < 55/60, |x-y| < 10/60, which is the diagonal hexagonal area in the graph.

This has area (55/60)² – 2 × 1/2 x 45/60 × 45/60 (the area of the whole square less the two triangles on the NW and SE corners).

Solution:The chances of business being transacted today is 5/18.The following code runs a simulation of the problem. It runs a million random trials by default (in around 780ms) and gets an answer of around 0.278, close to the expected answer of 0.2777…