Bertrand's paradox

Paradox described in the Calcul des probabilités (1889) of the French probabilist J. Bertrand (1822–1900). What is the probability that a chord drawn at random in a circle is longer than the side of an equilateral triangle whose three corners touch the circle? It is longer if its midpoint falls nearer the circumference than the centre of a radius bisecting it, so the probability is ½. Keeping one end of the chord fixed, it is longer if the angle at which it is drawn is within the 60° arc of the triangle, so the probability is 1/3. Or, it is longer if its midpoint lies in the area of the inner circle with radius one half of the original; this circle occupies one quarter of the area of the original, so the chance is 1/4. Bertrand used the paradox to show that there is no unique best way of applying the principle of indifference to such a case.