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 Failure to Communicate (Posted on 2011-02-04)
Alice and Bob need to discuss a business project for 5 minutes. However, they haven't coordinated when they are going to be in the office, and in fact they are going to follow strict schedules of being in the office for only 15 minutes each. Each will arrive at an independently chosen random time between 2 PM and 3 PM, and their 5 minute discussion must end by 3 PM.

Given the randomness and lack of coordination, what is the probability that Alice and Bob will get to complete their 5-minute discussion?

 Submitted by Charlie No Rating Solution: (Hide) 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.

 Subject Author Date graphic solution or Monte-Carlo simplified Ady TZIDON 2011-02-04 17:39:05 solution Justin 2011-02-04 14:21:46

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