Mathematics Solutions and Resources Online

Maths probability problem?

Mike and Dougal are going to the casino. They agree to meet at a coffee shop near by. Each arrives randomly at a time between 8.00pm and 8.30pm. They have agreed to wait for 5 minutes for the other to arrive. If after 5 minutes the other person has not arrived they leave and go to the casino alone. What is the probability that Mike and Dougal meet?

This is maybe the most well known geometric probability problem, cited in many sources (‘Meeting Problem’). Let x and y are the moments past 8:00 of arrivals of both persons:
0 ≤ x ≤ 30, 0 ≤ y ≤ 30
All possible cases can be illustrated by all points with coordinates (x, y) in a square with side 30:

http://farm4.static.flickr.com/3639/3672223992_ec992a35bd_o.gif

The meeting will take place if both moments differ with 5 at most:
|x – y| ≤ 5, so the favorable cases are the above points in the stripe
-5 ≤ x – y ≤ 5 (green points on the picture)
Now the required probability is
Area(green hexagon) / Area(square) =
= 1 – Area(both gray triangles) / Area(square) =
= 1 – 25² / 30² = 1 – 625/900 = 275/900 = 11/36

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