Question

Probability that length of randomly chosen cord of a circle lies between

Answer 1

The history of probability theory has been influenced strongly by paradoxes, results that seem to defy intuition. Many of these have been reviewed in a recent book by Prakash Gorroochurn [2012]. We will have a look at Bertrand’s Paradox (1889), a simple result in geometric probability.

Let’s start with an equilateral triangle and add an inscribed circle and a circumscribed circle. It is a simple geometric result that the radius of the outer circle is twice that of the inner one.Bertrand’s problem may be stated thus:

Problem: Given a circle, a chord is drawn at random. What is the probability that the chord length is greater than the side of an equilateral triangle inscribed in the circle?

We will consider three ways of drawing a chord in the outer circle:

Fix the end-points of the chord.

Choose the chord centre on a fixed diameter.

Fix the mid-point of the chord