There's a famous puzzle that goes like this: Suppose we pick 3 points at random from a given circle. What's the probability that the triangle they form is obtuse? The answer happens to be 3/4 and it's relatively easy to arrive to.
Naturally, one tries to "generalize" this by asking - what if we pick the 3 points at random from within the circle? What is the probability of the same event (obtuseness)?
Then the question of what picking a random point from a disk is arises, or maybe from other regions like an equilateral triangle. A quick google search gives you:
Yet another way of thinking about random triangles is given by this story (http://www.intmath.com/blog/random-triangles/758) about a smart student by the name of Kurt, who arrived at the conclusion that "in general" random triangles are obtuse at a slightly above 80% rate, utilizing the following sampling procedure on the angles alone:
A = a random number between 0 and 180; B = a random number between 0 and 180-A; C = 180 – A – B
Is there a sub-field of math that finds questions like listed below worthy of research: How do we construct algorithms for random point sampling from sets of certain geometry? How are said geometries affecting things like probabilities, distributions, etc. (almost as some sort of an action - a well known distribution distorted by the change of sampling enacted by said geometry).
I'm not sure if my questions make much sense. But to relate to the above "random triangle" setting we have 3 geometries:
1) A circle (answer is 75%) 2) A disk (not sure what the answer is, but certainly not 75%) 3) Kurt's sampling (it won't be correct to say that the geometry here is the plane, but it comes as close to the most general well-defined idea of a "random triangle" so in that sense the set of sample points is unbounded just like the plane - answer is a little over 80%)