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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:

http://mathworld.wolfram.com/DiskPointPicking.html http://mathworld.wolfram.com/TrianglePointPicking.html

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%)

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I imagine Kurt's "intended" distribution on the angles of a triangle should be the uniform distribution on the set $A+B+C=180^\circ$, $0^\circ<A,B,C<180^\circ$. This is not the same as the distribution that arises from the described sampling procedure, even after permutation. To say nothing of whether the angles of a triangle formed by picking three points uniformly from a planar region follows anything like this distribution... –  Rahul Feb 14 '13 at 5:50

1 Answer 1

The problem with Kurt's sampling is that it breaks a symmetry we would like to expect from a "correct" sampling method: The probability that $A\ge90$ is $\frac12$, the probability that $B\ge90$ is much smaller ...

As long as we want to sample points within a nice subset of the plane, "good" sampling is quite well-defined by the Lebesgue measure (the 2-dimensional one if the set has positive measure, the 1-dimensional one if the set has zero measure but is a one-dimensional manifold, say, as is the case for the circle). In absence of a standard measure, the concept of random choice is not defined - or more precisely: whenever one speaks about it, one must specify the underlying measure/random distribution, it is only a lesser sin to forget this if there already is a canonical standard measure and distribution. (What is the probability that a ranom natural number is even? A prime? A perfect square? Has last digit $1$? Has leading digit $1$?)

We have worse problems when sampling something else but points (or finite sets of points). For example, how to sample a random chord of a circle? Select 2 random points on the circle and join them? Select a random point inside the circle and draw the chord having it as center? Toss a coin on a parquetry floor with lumber width equal to coin diameter and check where a lumber edge intersects the coin? It turns out that in order to talk about random choices in such context one must be specific and describe the distribution one has in mind (which may depend on what "physical" phenomenon one wants to model oin the first place).

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I'm not sure I see the problem with symmetry breaking in Kurt's sampling - can't you fix that by claiming that the above sampling procedure "covers" triangles under transformations that permute the vertices? In other words, his sampling describes a family of triangles (6 of them, for all different vertex labelings A,B,C of 3 points) but for a property that does not depend on orientation like obtuseness, the answer is still "valid". –  Meina222 Feb 11 '13 at 15:45
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Let me make my other questions a bit more concrete. In the context of the triangle problem: 1) If you're given a more general region - say any polygon - what is the "algorithm" for a picking a point at random? 2) Looking at probability of obtuseness - how does the region affect that number? Intuitively, a circle should have a lower probability than a very elongated ellipse (with the extreme case of a line where every "deformed" triangle is obtuse). Can this measure be expressed as a formula of the ellipse radii ratio? Even for a circle - what is the answer (not obvious)? –  Meina222 Feb 11 '13 at 15:46

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