A point can by described by two numbers (the coordinates). If we are selecting a "random point in a square", everybody's intuition agrees that this means select

A point can by described by two numbers (the coordinates). If we are selecting a "random point in a square", everybody's intuition agrees that this means selecting the x coordinate and the y coordinate from a flat distribution. 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.