But my intuition suggests that constructing a proof for all chords should be relatively simple, and indeed 1/3 of them hold the property that they are longer than √3.
To solve the problem, you have to know from what distribution chords are drawn. It could be equal probability per unit angle, per unit distance along the circle, or something else altogether.
Similar difficulties emerge when someone gives you a problem that starts with a "random real number."
(This is almost just nitpicking, but in another sense it's important: viewing an apparently terminating decimal expansion as actually 0-padded lets one say correctly that "a real number is rational if and only if its decimal expansion eventually repeats", full stop, rather than having to add the qualification "… or is eventually 0.")
Another reason that it might be a good reason to have it count: by my definition, I can say "the decimal expansion of a number eventually repeats if and only if the binary expansion of that number eventually repeats." This becomes considerably more awkward if you don't consider a terminating expansion as consisting of eventually repeating 0's; for example, 1/5 has a terminating decimal expansion 0.2 but an immediately-repeating binary expansion 0.repeat(0011).
In a similar manner, a chord can by described by a point and a direction.
– The first (1/2) method selects a point from a flat distribution along x-axis, and defines that the direction is always vertical. (Edit: this is how I interpreted the animated gif, but Wikipedia says that actually from a flat distribution along a random radius, and then the direction is perpendicular to the radius.)
– The second (1/3) method selects a point from a flat distribution on the perimeter, and then actually selects another point also from the same distribution, to define the direction by connecting these points.
– The third (1/4) method selects a point from a flat distribution on the circle area, and the defines that the direction is always perpendicular to the radius.
The whole point of the paradox is to demonstrate that there is no single universally agreed upon way to define a flat distribution for the set of all chords. And when we start to try to define one, by breaking the definition of a cord into components, and sampling these components from a flat distribution (or just selecting one direction, because symmetry says they are all equal), we get different results depending on how we broke the concept of cord into components.
This is exactly the issue: when dealing with infinities and probabilities, you need to choose a measure. Depending on how that measure is chosen, you might get different answers.
The ninja story doesn't work here, since for any p1 I think there's exactly one p2 possible. The photographer story DOES work, since he can start at any point, and the mermaid can then be at any second point. I'm not sure about the dragon story, but I'm guessing it doesn't work either.
Of course a circle has no corners to complicate things like that, but it can still have biases. Just because we have found a way to uniquely identify cords does not mean it will lead to a uniform distribution.
This is the sentence that helped me understand the paradox.
Perhaps this analogy will help. Think about how technically picking a value at random from a normal distribution can result in any value at all, extending to infinity in both directions, but that doesn't mean all values are equally likely. Every real number can be uniquely expressed as a number of standard deviations from mean in a normal distribution, but that still has a bias.
So you are in favour of method 2.
But method 3 says: Take a random point inside the circle, and use this as a center point of the cord.
What do you find wrong with this method?
This method has different probability density from the random-center and random-endpoints methods. That's the whole point of the exercise: you get different probability density across the chord space depending on how you select from it.