20070414, 07:09  #1 
Dec 2005
3·5·13 Posts 
4 Points in a Plane
Choose 4 points at random in a plane.
What is the probability that the convex hull of these 4 points is a triangle ? 
20070414, 08:23  #2 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2×3×5×193 Posts 
Is the plane infinite?
What is the type of random distribution? 
20070414, 08:51  #3 
"Lucan"
Dec 2006
England
6,451 Posts 
Is this the same as asking whether they can constitute
the vertices of a quadrilateral whose interior angles are all < 180 degrees? 
20070414, 08:51  #4 
Dec 2005
3·5·13 Posts 
the plane is infinite, the distribution is that every point in the plane is as likely to be selected

20070414, 08:56  #5  
Dec 2005
3×5×13 Posts 
Quote:


20070414, 09:57  #6 
"Lucan"
Dec 2006
England
1933_{16} Posts 
I say it's evens.
Take three points at random. Produce the three sides of the triangle in both directions. This divides the plane into seven regions: the triangle itself, 3 unbounded regions adjoining an edge of the triangle, and 3 unbounded regions with one vertex. The two types of infinite region each span 180 degrees. The fourth point has equal probability of lying in each type of infinite region. (The hull is a triangle iff the fourth point lies in a "vertex" type region or the triangle itself) (I shudder to think what Silverstone will make of this ) David Last fiddled with by davieddy on 20070414 at 10:04 
20070414, 10:03  #7 
Dec 2005
3·5·13 Posts 
My solution is essentially the same and I too fear that it is not very rigourous

20070414, 10:46  #8 
"Lucan"
Dec 2006
England
1933_{16} Posts 
If we consider a sphere rather than a plane, does this
clarify or obfuscate the problem? David 
20070414, 11:19  #9 
"Lucan"
Dec 2006
England
6,451 Posts 

20070414, 14:57  #10 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2·3·5·193 Posts 

20070414, 21:33  #11 
"Lucan"
Dec 2006
England
1100100110011_{2} Posts 

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