Bertrand's paradox (statistics)

Question

I just learned about Bertrand paradox in today's class, and am very shocked. I was wondering if there are indeed only 3 (known?) unique ways of picking a chord in a circle at random, with 3 different probability values for the lengths being shorter than a side of an inscribed equilateral triangle, or whether there are infinitely many unique ways of picking a random chord in a circle of this length (and consequently infinitely many unique probabilities corresponding to each random picking method [countably/uncountably infinite then?]). Thanks!

Answer 1

There are infinitely many different ways to randomly pick a chord from a circle. One way to construct infinitely many would be to create a probability distribution defined over the circle and pick two points accordingly, of which there are infinitely many. However, not all of these seem "natural" like the 3 in Bertrand's paradox, which is what makes it interesting.

Answer 2

Draw a circle "inside" the letter V, so that the top of the circle is level with the top tips of the V, and the circle does not intersect the V.

Draw a random chord by choosing a random point on the V (assume the points are uniformly distributed) and connecting this point to the top of the circle.

Let P = probability that the chord's length is longer than the side length of an equilateral triangle inscribed in the circle.

By changing the shape of the V (making it wider or taller), P can be any value between 0 and 1.