I'll try being more specific. This is what nuzzle did: he defined an angle preserving transformation f from the set of triangles T to the set of triangles; he parametrized the image of that transformation as a subset of the plane (what he describes as an "American football"); then he uniformly picked points from the football taking the probability that such points represent acute triangles. Recapitulating, he argues that : (*) being the transformation angle preserving and being triangle acuteness an angle dependent property then such probability is "the chance that a triangle is acute":

now, if A is the set of acute triangles, F is the "American football" set in R^2, you have

A -j-> T -f-> T -p-> F

( where j is the natural inclusion and p a function sending each transformed triangle to the corresponding point in the american football set )

the probability he computed is equal to L{ p(f(j(A))) } / L{ F } , where L{} is the Lebesgue measure in the plane ( aka the "area" ).

now, the statement (*) tantamounts saying that:

if f and f' preserve acuteness and F and F' are bounded then L{ p(f(j(A))) } / L{ F } = L{ p'(f'(j(A))) } / L{ F' }

this statement is false ( I provided a counter example in post #10, but it should be evident from the very definition of a measure ).

Quote Originally Posted by TheGreatCthulhu
Would it make any difference if the points could somehow... "pop-up" on their own? If there wasn't any sentient-being-introduced picking involved? Or this is just the same?
what do you mean ?