The Bertrand paradox asks the probability of a random chord being longer than the side of an inscribed equilateral triangle. At least three (see link) answers are possible depending on how "random" is employed in drawing the chord. Jaynes argues that one of the answers is "best." Here's a cute question, not of my making, that might also lead to a preferred answer from among the three. Or maybe not.
The question first notes that the sum of the lengths of all sides and diagonals emanating from a vertex of a regular n-gon inscribed in the unit circle is 2 cot (pi/2n) and then asks us to use this fact to find the average length of a chord of the unit circle.
Does the answer to this question give credence to one of the three answers (again see link) over the other two? Is it Jaynes' choice?
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bonanova
The Bertrand paradox asks the probability of a random chord being longer than the side of an inscribed equilateral triangle. At least three (see link) answers are possible depending on how "random" is employed in drawing the chord. Jaynes argues that one of the answers is "best." Here's a cute question, not of my making, that might also lead to a preferred answer from among the three. Or maybe not.
The question first notes that the sum of the lengths of all sides and diagonals emanating from a vertex of a regular n-gon inscribed in the unit circle is 2 cot (pi/2n) and then asks us to use this fact to find the average length of a chord of the unit circle.
Does the answer to this question give credence to one of the three answers (again see link) over the other two? Is it Jaynes' choice?
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