Solution
Crossing Chords
Show the problem again
Four points A, B, C, D are chosen independently and uniformly at random on a circle. Chord AB and chord CD are drawn. What is the probability that the two chords intersect? Give a reduced fraction.
Worked solution
The answer is 1/3. Whatever the four positions, they occur in some cyclic order around the circle. There are exactly three ways to split four points into two pairs, and given the cyclic order exactly one of those three pairings (the one joining opposite points) gives crossing chords. The labels A, B, C, D are assigned to the positions completely at random, so the pairing {A, B}, {C, D} is equally likely to be any of the three pairings. Hence the chords cross with probability 1/3.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.