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        riddles >> hard >> Distance Problem
        
(Message started by: Wonderer on Apr 28^th, 2007, 7:16am)

Title: Distance Problem
Post by Wonderer on Apr 28^th, 2007, 7:16am

Given a staight line with length L. Two points A and B are selected randomly on that straight line.
Question:
What is the probability that the distance between A and B (AB) is less than L/4?

Please provide walkthrough.

Title: Re: Distance Problem
Post by Grimbal on Apr 28^th, 2007, 7:31am

[hide] 7/16 [/hide]

http://florian.net/puzzle/pic/quarter.gif

Title: Re: Distance Problem
Post by Icarus on Apr 28^th, 2007, 7:40am

Switch the problem around a little bit: Consider the line to be infinite. Instead of choosing 2 points on a line segment, we choose 3 point-sets. Each set consists of points continuing indefinitely in each direction, all equally spaced at a distance L apart. One set is your point A, translated repeatedly by L. The second is your point B and its translates. The third, call it D, consists of the endpoints of your line segment, and all their translates. Since each endpoint is a translate of the other, this is the same as picking a single point and its translates, like A and B.

Why do this? Because it allows you to reorder the picks: Pick A first, B second, and D last. The probabilities are the same. If we now cut back to a single segment, we discover that the original problem is equivalent to:

Given a line segment of length L, choose two points B and D at random in it. What is the probability that B is within L/4 of an endpoint, and that D is not between B and the nearer endpoint? This is an easier question to approach.

[edit] the problem with being verbose. someone slips in with an easier solution while you are still putting yours together![/edit]