Bertrand's Paradox (with 3blue1brown) - Numberphile

So points in a circle is pretty basic, but it does remind me of a nice 3Blue1Brown video ( [VIDEO] ) about random chords of a circle. Some of these 'generate a random sample' situations can be quite non-obvious. edit: The video is about 'Bertrand's Paradox' that another post references.

In a related vein, there's a Numberphile / 3B1B video about the pitfalls of plotting randomness in a circle: [VIDEO]

I was always surprised at how easily you get biased sampling when generating random points despite all input - something I often saw students do was essentially normalize({2 rand() - 1, 2 rand() - 1, 2 rand() - 1}) or variations of that (where rand() is "good" not the literal rand(3)), and there are numerous other ways that are more subtly wrong. IIRC the nominally correct way for a sphere specifically is something like a=rand() b=rand() and the random point is something* like { cos(a) sin(b), sin(b), cos(a) cos(b) }[1] I think the best illustration of "reasonable choices of what random values should be used leading to biased results" is Bertrand's paradox which I was introduced to via numberphile/3blue1brown: [VIDEO] and am just glad that nothing I have ever needed random sampling for has ever been important :D [1] please don't use this blindly, I'm really just going off very old recollection, if you need it google random sphere sampling :D