MagnetoHydroDynamics comments on Probability is in the Mind - Less Wrong
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Say you have a circle. On this circle you draw the inscribed equilateral triangle.
Simple, right?
Okay. For a random chord in this circle, what is the probability that the chord is longer than the side in the triangle?
So, to choose a random chord, there are three obvious methods:
WHAT NOW?!
The solution is to choose the distribution of chords that lets us be maximally indifferent/ignorant. I.e. the one that is both scale, translation and rotation invariant (i.e. invariant under Affine transformations). The second solution has those properties.
Wikipedia article