[olliej]

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({2rand() - 1, 2rand() - 1, 2rand() - 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: https://www.youtube.com/watch?v=mZBwsm6B280 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

[hinkley]

I used to think “normal distributions are everywhere” but the more math and science I watch on YouTube the more the central limit theorem pops up. It’s the CLT that’s everywhere, it just brings normal distribution as it’s +1.

[richrichie]

Indeed. The fundamental insight behind CLT i.e. sample average is normally distributed even when population distribution is not normal is intuitive, yet the the theorem is magical.

[hinkley]

When you have more than one variable, the outcomes get clumpy.