Consider a fixed number $k > 3$ of random points in the plane, each independently distributed according to a 2D standard normal distribution. What is the probability that the convex hull of these $k$ points is a triangle?
Presumably there is no closed form formula (or is there?), so I would also be interested in some sort of bounds on this probability.
I have coded the following (lazy) way to generate random quadrilaterals in the plane:
Pick $k=10$ points from a 2D standard normal distribution
Extract 4 points from the boundary of the convex hull of those 10 points
This is convenient since there is a simple and fast 1-liner in my programming language to do step 2 (convhull command in Matlab). It would be nice to know how likely it is that this algorithm will fail, and/or how large I need to make $k$ so that I can be reasonably certain it will "never" fail.