Bertrand's paradox

Problem

Randomly generate chords within a unit circle. What is the probability that such a randomly-generated chord has a length larger than \(\sqrt{3}\)?

Solution

It depends! How will you randomly generate the chords? Three methods to randomly generate chords are shown in the animation below:

Random endpoints

Randomly pick two points on the circumference of the circle. Create a chord by connecting them. The probability that this chord has a length exceeding \(\sqrt{3}\) is \(\frac{1}{3}\).

Random radial point

Create a temporary line that connects the origin to a random point on the circumference of the circle. Then, pick a random point along that line. Create a chord that passes through this point and is perpendicular to the temporary line. The probability that this chord has a length exceeding \(\sqrt{3}\) is \(\frac{1}{2}\).

Random midpoint

Randomly choose any point within the circle and use is as the midpoint of a chord. The probability that this chord has a length exceeding \(\sqrt{3}\) is \(\frac{1}{4}\).