Random Chord Paradox

Bertrand's paradox asks for the length of a random chord in a unit circle. What are the odds that the length is greater than ? As shown in this Demonstration, the answer is , , or , depending on how random is defined.
In the first case, a value in (-1,1) is chosen, and a chord is drawn. Optionally, the chord can be rotated by a random angle. About half of the chords will have length greater than .
In the second case, a random point on the edge of the circle is chosen, and a chord is connected to the lowest point of the circle. Optionally, both points can be chosen randomly. About a third of the chords will have length greater than .
In the third case, a random point inside the circle is chosen, and this is used as the midpoint of a chord. About a quarter of the chords will have length greater than .


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