Solution
Bertrand's Chord
Show the problem again
A chord of a circle is drawn by choosing its two endpoints independently and uniformly on the circumference. What is the probability the chord is longer than a side of the inscribed equilateral triangle? Express as a fraction.
Worked solution
The answer is 1/3. Fix the first endpoint and inscribe an equilateral triangle with a vertex there. The chord exceeds the triangle's side length iff the second endpoint falls on the arc between the other two vertices, which is 1/3 of the circumference. (Bertrand's paradox: other "random chord" procedures give 1/2 or 1/4, which is why the sampling method must be specified.)
Source: Bertrand's paradox, from Joseph Bertrand, 'Calcul des probabilites' (1889). Statement written for AxiomIQ.