Finch (2010) gives an overview of known results for random Gaussian triangles.
Let the vertices of a triangle in dimensions be normal (normal) variates. The probability that a Gaussian triangle in dimensions is obtuse is
(1)
| |||
(2)
| |||
(3)
| |||
(4)
| |||
(5)
|
where is the gamma function, is the hypergeometric function, and is an incomplete beta function.
For even ,
(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).