Probability That a Random Chord Exceeds the Side of an Inscribed Triangle

Bertrand's chord paradox has three valid answers depending on sampling method, one third, one half and one quarter, and interviewers usually mean one third.

Ask for the probability that a random chord of a circle is longer than the side of an inscribed equilateral triangle, and the honest answer is that the question is incomplete until the method of choosing the chord is specified. This is Bertrand's chord paradox, and the three standard methods of picking a random chord give three different, all mathematically valid, answers: one third, one half and one quarter. When an interviewer says choose two random points on the circle and draw the chord between them, the intended answer is one third.

An equilateral triangle is inscribed in a circle of radius r, so its side length equals r times the square root of three. A chord is a straight line segment connecting two points on the circle. The question is how likely a randomly drawn chord is longer than that fixed triangle side length, and the answer depends entirely on what probability distribution is placed on the space of chords.

Because a chord is fully determined by two real numbers under any of several natural parameterizations, endpoints, perpendicular distance and direction, or midpoint coordinates, there is no single canonical uniform distribution over the space of chords the way there is an obvious uniform distribution over a finite set of outcomes. That absence of a canonical choice is the entire source of the paradox.

Pick two points independently and uniformly on the circle's circumference and connect them. By symmetry, fix the first point anywhere, since the circle looks the same from every point on it. The triangle's three vertices divide the circle into three equal arcs of 120 degrees each. Placing one triangle vertex at the fixed first point, the chord to the second point exceeds the triangle's side length exactly when the second point lands on the 120 degree arc directly opposite the first point, the arc between the other two vertices that does not contain the first point. Since that arc is exactly one third of the circle, the probability is one third.

This method is the one most naturally implied by the phrase choose two random points on the circle, which is why it is usually the answer an interviewer expects unless a different sampling procedure is specified explicitly.