Randomness is Random - Numberphile

There is a very accessible "numberphile" video which I think is closely related to this, about playing the odds on predicting elements of a nominally-random coin-flip sequence using the expected run length. [VIDEO]

I would argue that the problem with frequentist statistics is that it aligns with humans' flawed intuition of how randomness works. People are inherently obsessed with finding patterns to support their hypotheses. The problem is that what we perceive as random and extremely unlikely events are in fact much more probable than what we estimate from using Gaussian methods. And the frequentist approach helps to create this distortion by ignoring black swans. Here's a great video demonstrating how people tend to misunderstand randomness: [VIDEO]

Oh, I remember good old fallout 1-2 when you have 42% hit chance but still miss 7 times in a row. There is also a good Numberphile video on what we think random really is (it is not) [VIDEO]

This may indirectly answer your question: [VIDEO] TLDW: We don't pick numbers at random. At all. We're so bad at it that you can make money off this fact.