Bertrand's Paradox

Simple question: Given a circle, what's the average length of a chord of the circle? OK, maybe that's too messy; try something simpler: What's the probability that a chord will be longer than the side of an inscribed equilateral triangle (radius * √ 3)?

Depending on what method you use to define the chord, you get 3 different answers, all of which look perfectly reasonable. The first is the most obvious--fix one point at one corner of that inscribed triangle, and then just pick other points on the circle to draw the chords to.

If you draw the picture, it seems instantly obvious that you should get 1/3 of the chords being longer than the specified value. But if you look closely, you might notice that for the same small angular range, the density of chords for points close to the original point is higher than for those chords reaching to the opposite side of the circle--is this the kind of uniformity of distribution we want? It would seem to give too many short chords, for a lower probability of long ones.

Other approaches give 1/4 and 1/2 for the probability. Therein lies the paradox--which of the 3 values is it? The video (3blue1brown) doesn't need much more than high school geometry; give it a whirl.

Bottom line--sometimes you have to be very very careful to define what you are trying to measure.