Gaussian triangle picking studies triangles whose three vertices have independent coordinates drawn from the standard
normal distribution in 
dimensions. Finch (2010) gives an overview of known results
for these random triangles. For integer 
,
the probability that such a triangle
is obtuse is
 |  | ![(3Gamma(n))/([Gamma(1/2n)]^2)int_0^(1/3)(x^((n-2)/2))/((1+x)^n)dx](/images/equations/GaussianTrianglePicking/Inline5.svg) |
(1)
|
 |  | ![(3Gamma(n))/([Gamma(1/2n)]^22^(n-1))int_0^(pi/3)sin^(n-1)thetadtheta](/images/equations/GaussianTrianglePicking/Inline8.svg) |
(2)
|
 |  | ![(6Gamma(n)_2F_1(1/2n,n;1+1/2n;-1/3))/(3^(n/2)n[Gamma(1/2n)]^2)](/images/equations/GaussianTrianglePicking/Inline11.svg) |
(3)
|
 |  | ![(3B(1/4;1/2n,1/2n)Gamma(n))/([Gamma(1/2n)]^2)](/images/equations/GaussianTrianglePicking/Inline14.svg) |
(4)
|
 |  | ![(3B(3/4;1/2n,1/2)Gamma(n))/(2^n[Gamma(1/2n)]^2),](/images/equations/GaussianTrianglePicking/Inline17.svg) |
(5)
|
where 
is the gamma function, 
is the hypergeometric
function, and 
is an incomplete
beta function.
For even 
with positive integer 
,
 |
(6)
|
(Eisenberg and Sullivan 1996).
The first few cases are explicitly
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
(OEIS A102519 and A102520). The even cases are therefore 3/4, 15/32, 159/512,
867/4096, ... (OEIS A102556 and A102557)
and the odd cases are 
, where 
, 9/8, 27/20, 837/560, ... (OEIS A102558
and A102559).
Dimensions." Amer. Math. Monthly 103, 308-318, 1996.Finch,
S. "Random Triangles." Jan. 21, 2010.