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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another way:to select a chord of random length varying from 0 to 2r. favorable selection length varies from 1.7r(length of side) to 2r. so probability~2r-1.7r/2r=3/20. –  user36519 Jul 25 '12 at 16:27