What is the probability that a randomly drawn chord will be longer than the radius of the circle?
When we talk about "a point of a circle is more likely to be at a distance less than √3r/2 from the center", we are saying that a region from the cartesian plane "has more points than" another region.
But the set of points from such a region is infinite and non-countable. When we take a small circle off a bigger one, the resulting "donut" "has less points" than the original circle? The Cantor sets have an infinite number of points, althought they are "scattered" around the convex sets that we use to construct them.
After I posted my previous comment, I thought about Cantor sets and now I think that all this calculation makes no sense...