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

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tony richards

Posts: 164
Registered: 12/8/04
Re: Random Triangle Problem
Posted: Jul 29, 1997 4:04 AM
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The reason I persist with this problem, is that I remain unconvinced by
handwaving arguments and special pleading involving 'common sense' assumptions which turn out
to be unwarranted. Several contributions to this subject have stumbled because of this.

I have spotted the one remaining fault with my previous effort, which no one spotted,
but which I now correct here.

I obtained the following expression for the probability for an OBTUSE
triangle

P(OBTUSE)=[ 1 - abs(x)*2*L/(4*L^2) +(pi*abs(x)^2/4)/(4*L^2)]*P(x)

where the first point is located at the origin (or vice versa) of an infinite plane,
side 2*L in X and Y, the second point is located a distance x from the first and the third
is located in (a) the area to the left of X=0, or in (b) the area to the right of X=x,
or in (c) the circle diameter x centred at X=x/2 .

I then made the erroneous assumption that the probability/weighting function for
the second point to be at distance x from the first point is P(x)=dx/(2*L)
when further thought shows that it must actually be P(x)=2*pi*x*dx/(pi*L^2).
That is,the probability that the second point is distance x away from the first
is actually proportional
to the area of an annulus, radius x, thickness dx, centred at the first point, divded by
the total area available (pi*L^2) (orienting the axes after the location of the second point
so that the second point is along X does not mean that the spacing is only along the x axis)

With this correct weighting/probability function, the result for the probability integrated
over x, taken over the range 0<x<L, is

P(OBTUSE)= 1- (1/3) + pi/32 = 0.76484.

This is close to, but not identical to the 3/4 everyone seems to expect.

My final question is , 'why persist in believing that the probabilities for each
vertex being obtuse must be the same', when it is clear that, given the first two points,
the probabilities for the third point falling within the three different areas of
the infinite plane which will result in the triangle being OBTUSE are clearly different?'.


--
Tony Richards 'I think, therefore I think I am right this time'
Rutherford Appleton Lab '
UK '







Date Subject Author
7/16/97
Read Random Triangle Problem
Mike Housky
7/21/97
Read Re: Random Triangle Problem
Bill Taylor
7/22/97
Read Re: Random Triangle Problem
tony richards
7/24/97
Read Re: Random Triangle Problem
Brian M. Scott
7/23/97
Read Re: Random Triangle Problem
tony richards
7/23/97
Read Re: Random Triangle Problem
T. Sheridan
7/24/97
Read Re: Random Triangle Problem
Bill Taylor
7/24/97
Read Re: Random Triangle Problem
Bill Taylor
7/25/97
Read Re: Random Triangle Problem
Ilias Kastanas
7/23/97
Read Re: Random Triangle Problem
Robert Hill
7/23/97
Read Re: Random Triangle Problem
tony richards
7/27/97
Read Re: Random Triangle Problem
Bill Taylor
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/28/97
Read Re: Random Triangle Problem
tony richards
7/30/97
Read Re: Random Triangle Problem
Bill Taylor
7/30/97
Read Re: Random Triangle Problem
tony richards
8/1/97
Read Re: Random Triangle Problem
Bill Taylor
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/25/97
Read Re: Random Triangle Problem
Robert Hill
7/30/97
Read Re: Random Triangle Problem
Bill Taylor
8/1/97
Read Re: Random Triangle Problem
Charles H. Giffen
8/1/97
Read Re: Random Triangle Problem
John Rickard
8/1/97
Read Re: Random Triangle Problem
Chris Thompson
8/1/97
Read Re: Random Triangle Problem
John Rickard
8/4/97
Read Re: Random Triangle Problem
Bill Taylor
8/5/97
Read Re: Random Triangle Problem
John Rickard
7/25/97
Read Re: Random Triangle Problem
Charles H. Giffen
7/25/97
Read Re: Random Triangle Problem
Charles H. Giffen
7/28/97
Read Re: Random Triangle Problem
Hauke Reddmann
7/28/97
Read Re: Random Triangle Problem
Robert Hill
7/28/97
Read Re: Random Triangle Problem
Robert Hill
7/28/97
Read Re: Random Triangle Problem
Robert Hill
7/29/97
Read Re: Random Triangle Problem
tony richards
7/30/97
Read Re: Random Triangle Problem
Keith Ramsay
7/30/97
Read Re: Random Triangle Problem
tony richards
8/2/97
Read Re: Random Triangle Problem
Keith Ramsay
7/29/97
Read Re: Random Triangle Problem
tony richards
8/4/97
Read Re: Random Triangle Problem
Bill Taylor
8/5/97
Read Re: Random Triangle Problem
Charles H. Giffen
8/6/97
Read Re: Random Triangle Problem
Terry Moore
8/7/97
Read Re: Random Triangle Problem (LONG summary)
Terry Moore
8/16/97
Read Re: Random Triangle Problem (LONG summary)
Kevin Brown
8/17/97
Read Re: Random Triangle Problem (LONG summary)
Kevin Brown
7/30/97
Read Re: Random Triangle Problem
Muad'dib
7/30/97
Read Re: Random Triangle Problem
Robert Hill
7/31/97
Read Re: Random Triangle Problem
tony richards
8/6/97
Read Re: Random Triangle Problem
Terry Moore
7/31/97
Read Re: Random Triangle Problem
John Rickard
7/30/97
Read Re: Random Triangle Problem
Robert Hill
7/31/97
Read Re: Random Triangle Problem
Robert Hill
7/31/97
Read Re: Random Triangle Problem
Robert Hill
8/1/97
Read Re: Random Triangle Problem
R J Morris
8/4/97
Read Re: Random Triangle Problem
Robert Hill
8/4/97
Read Re: Random Triangle Problem
Robert Hill
8/5/97
Read Re: Random Triangle Problem
Charles H. Giffen
8/6/97
Read Re: Random Triangle Problem
Robert Hill

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