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   Author  Topic: Distance Problem  (Read 773 times)
Wonderer
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Distance Problem  
« on: Apr 28th, 2007, 7:16am »
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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.
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Grimbal
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Re: Distance Problem  
« Reply #1 on: Apr 28th, 2007, 7:31am »
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7/16
 
« Last Edit: Apr 28th, 2007, 11:47am by Grimbal » IP Logged
Icarus
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Re: Distance Problem  
« Reply #2 on: Apr 28th, 2007, 7:40am »
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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]
« Last Edit: Apr 28th, 2007, 7:42am by Icarus » IP Logged

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