I think that the most important part of the article is the following quote in the introduction:

> probabilities may not be well defined if the mechanism or method that produces the random variable is not clearly defined.

So comparing the answer of 1/2, where the chords are generated by moving along the radius in increments of dr, with the answer of 1/3, where the chords are generated based on the angle from the starting point.

The problem doesn't specify how to pick random points, though I suspect that if you were to choose two random points on the radius of a circle by picking a reference "going around the edge" of the circle clockwise by some amount in [0, 2pi), we get the angular situation that produces the 1/3 answer.

>> probabilities may not be well defined if the mechanism or method that produces the random variable is not clearly defined.

i don't agree. Because it is too vague and contradictory. If you defined probabilities it strongly implies that you defined Borel algebra of sets in the space. This is what meant by the "mechanism or method that produces the random variable". Without well defined Borel set (implicitly or explicitly) there is no probability at all, not just "well defined".

To illustrate - lets take our example of random chord. One would think - take the set of all possible chords and assign equal probability to each. As the number of chords uncountable, one can't really do such assignment and still obtain an additive measure (one of the condition of something to be a probability). We're forced to go more coarse grained. Find or choose a set of subsets such that we can assign numbers to them such that there would be additivity when the subsets are combined and other properties making that assignment of numbers a "probability", ie. we need to define a Borel algebra. The way one builds such set of subsets, an algebra, is the actual definition of the probability and it is "the mechanism or method that produces the random variable".

So, in the Wikipedia article, for example, taking random endpoints means that we're using, as the basis for the algebra, the open sets in the space x S1, while taking random midpoints means that we're building the algebra on the basis of the open sets in the space of the 2 dimensional sphere. It is pretty obvious that these 2 algebras are completely different and thus the probabilities they define are completely different, and one can't talk about probabilities without specifying the Borel algebra defining it.

The '"maximum ignorance" principle' section from the Wikipedia article addresses most of what you are saying best.

This is much better than the linked site.