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What about lines across a square? They must also all hit two points on its perimeter, but randomly choosing p1, p2 in that case will show a clear bias towards the diagonals.

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.

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> 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.

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