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!
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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.