While I think the connection with what GP wanted to convey is tenuous, Betrands paradox _is_ interesting, and even more so E.T. Jaynes' solution.
As with most paradoxes, it's about how a seemingly innocuous question turns out to either be underspecified to have an answer, or tricks even the mind of mathematicians to assume too much.
Here it's about what it means to have a "random chord of a circle"; several simple ways to generate random chords lead to very different results, but true randomness should not be easy distinguishable from other true randomness...
E.T.J's solution to invoke the maximum ignorance principle to require any source of random chords to be size- and translation-invariant (because those dimensions are not specified in the problem) seems elegant.
Now, comparing to orbits, or to balls on a train, the thing here is: all these different viewpoints, although very different, do lead to the same result. Even if you calculate stuff; and if I'm not mistaken, that's because of relativity (Galilean relativity should be sufficient for the train example to work, the more modern ones for the rest).
As with most paradoxes, it's about how a seemingly innocuous question turns out to either be underspecified to have an answer, or tricks even the mind of mathematicians to assume too much.
Here it's about what it means to have a "random chord of a circle"; several simple ways to generate random chords lead to very different results, but true randomness should not be easy distinguishable from other true randomness...
E.T.J's solution to invoke the maximum ignorance principle to require any source of random chords to be size- and translation-invariant (because those dimensions are not specified in the problem) seems elegant.
Now, comparing to orbits, or to balls on a train, the thing here is: all these different viewpoints, although very different, do lead to the same result. Even if you calculate stuff; and if I'm not mistaken, that's because of relativity (Galilean relativity should be sufficient for the train example to work, the more modern ones for the rest).