Quote Originally Posted by D_Drmmr View Post
Actually, it's the other way round. It doesn't matter how many balls there are in the urn, as long as 1/4th of them are white and you draw balls uniform randomly (meaning each ball has the same chance of being draw each time) then the probability that you draw a white ball is independent of the number of balls in the urn. That's why it is reasonable to state that it also works with an infinite number of balls.

The problem would be very different if 1/4th of the top 100 balls are white and everything else black. If you then draw uniform randomly from the top 100 balls, the probability of drawing a white ball is 1/4. But that doesn't mean that 1/4th of all the balls are white.

That's the whole point. These two problems look the same, but they are very different. I believe your reasoning is flawed, because you are generalizing a consequence of one problem into a general rule.
Yes there are two ways of seeing it depending on whether you know the white ball fraction or whether you want to determine it by drawing balls.

You claim the triangles are not drawn with equal probability. How is that?

I may be mathematically naive but if you can define the infinite plane as a set of points then you can also pick points from this set at random. Why not? Each point has an equal probability of being picked and the probability of a specific point being picked is zero.

And if you pick three points independently at random in the infinite plane you've picked a triangle at random in the infinite plane. If you pick triangles like that doesn't each triangle have an equal chance of being picked? If not, exactly what triangles are favoured?