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Topic: Random Triangle Problem
Replies: 57   Last Post: Aug 17, 1997 10:51 PM

 Messages: [ Previous | Next ]
 Brian M. Scott Posts: 863 Registered: 12/6/04
Re: Random Triangle Problem
Posted: Jul 24, 1997 2:09 PM

On 22 Jul 1997 15:56:59 GMT, tony richards <tony.richards@rl.ac.uk>
wrote:

>mathwft@math.canterbury.ac.nz (Bill Taylor) wrote:
>>Mike Housky <mike@webworldinc.com> writes:

>>|> > Derive a method of finding the chances of 3 random points placed on
>>|> > a plane that will yield an obtuse triangle.

>without loss of generality, we can assume point 1 is at the origin with probability =1
>so P1(anywhere)=1.0.

>Without loss of generality, point 2 can be on the x axis at either x>0 (probability 0.5)
>or at x<0 (probabilty = 0.5).
> SO P2(x>0)=0.5 and P2(x<0)=0.5

>Considering only the x-co-ordinate of point 3, x3

>You get an ACUTE triangle if x3 is BETWEEN x1 and x2

Only if the point is not within the circle of which the segment P1P2
is a diameter.

>I propose that it is equally likely that x3>x1 as x3<x1 (probability = 0.5 each)
>and also that it is equally likely that x3>x2 as x3<x2 (probability=0.5 each)
>So, the probability that x3 lies between x1 and x2 will be the
>product of the probabilities 0.5 and 0.5, one factor drawn from each of the above pairs
> giving P=0.25 for x3 between x1 and x2.

Even assuming for the sake of argument that there's a reasonable
probability measure floating around here somewhere, this calculation
seems unjustifiable: the events in question aren't independent. But
the whole approach seems unworkable. If Pr(x3 < x1) = Pr(x3 < x2) =
1/2, then (assuming that x1 < x2) Pr(x1 < x3 < x2) = 0. The plane is
just too big.

Brian M. Scott

Date Subject Author
7/16/97 Mike Housky
7/21/97 Bill Taylor
7/22/97 tony richards
7/24/97 Brian M. Scott
7/23/97 tony richards
7/23/97 T. Sheridan
7/24/97 Bill Taylor
7/24/97 Bill Taylor
7/25/97 Ilias Kastanas
7/23/97 Robert Hill
7/23/97 tony richards
7/27/97 Bill Taylor
7/24/97 Robert Hill
7/28/97 tony richards
7/30/97 Bill Taylor
7/30/97 tony richards
8/1/97 Bill Taylor
7/24/97 Robert Hill
7/24/97 Robert Hill
7/24/97 Robert Hill
7/25/97 Robert Hill
7/30/97 Bill Taylor
8/1/97 Charles H. Giffen
8/1/97 John Rickard
8/1/97 Chris Thompson
8/1/97 John Rickard
8/4/97 Bill Taylor
8/5/97 John Rickard
7/25/97 Charles H. Giffen
7/25/97 Charles H. Giffen
7/28/97 Hauke Reddmann
7/28/97 Robert Hill
7/28/97 Robert Hill
7/28/97 Robert Hill
7/29/97 tony richards
7/30/97 Keith Ramsay
7/30/97 tony richards
8/2/97 Keith Ramsay
7/29/97 tony richards
8/4/97 Bill Taylor
8/5/97 Charles H. Giffen
8/6/97 Terry Moore
8/7/97 Terry Moore
8/16/97 Kevin Brown
8/17/97 Kevin Brown
7/30/97 Robert Hill
7/31/97 tony richards
8/6/97 Terry Moore
7/31/97 John Rickard
7/30/97 Robert Hill
7/31/97 Robert Hill
7/31/97 Robert Hill
8/1/97 R J Morris
8/4/97 Robert Hill
8/4/97 Robert Hill
8/5/97 Charles H. Giffen
8/6/97 Robert Hill