One could just as easily make an argument for a ratio of sqrt(3)/2. The chord is selected by randomly picking a length out of the range [0, 2], placing it arbi

One could just as easily make an argument for a ratio of sqrt(3)/2. The chord is selected by randomly picking a length out of the range [0, 2], placing it arbitrarily on the circle, and then rotating it randomly between zero and one revolution[1]. Before we can select a random chord, we must first map the set of chords onto the set of real numbers (or, equivalently, tuples of real numbers). This is required in order to meaningfully apply a distribution to them. The problem is that there are multiple ways to do this. Which mapping we select has as much effect on the outcome as the probability distribution. Likewise, if we want to "count" all random chords, we need to do the same thing (map them onto the set of real numbers). We will get different "counts" depending on which mapping we select. [1]- Edit: This is not a very satisfying approach, but it does demonstrate the problem well. Being satisfying isn't a requirement for a mathematical construct.