TITLE: True random number set
QUESTION [7 upvotes]: Consider the double-slit experiment. If we measure position of electron on the screen multiple times, we get different results. But if we repeat the experiment many times we obtain a nice histogram which converges (smoothens) when more experiments are done. So the numbers obtained by this random process are not truly random. By the above statement I mean that the set of all numbers is giving a smooth histogram. If the laws of the universe are not assumed to be that nice, we can imagine a scenario where the histogram doesn't actually converge but keeps changing. How can I make this idea mathematically clear? The probability courses I've studied always assume a distribution for numbers before doing anything.
To elucidate what I'm thinking, consider this example. I could be flipping coins and find after 10 flips I had gotten 85% heads, then after 100 flips I had gotten 65% heads, after 1000 flips I had gotten 95% heads, after 10,000 flips I had gotten 15% heads etc. I want to talk about the situation where the fraction doesn't converge.
I apologize if the question is vague. Even I'm not clear with what exactly I'm looking for, hence I'm asking if there's some mathematical treatment of probability where the histogram doesn't converge. If the question is not suitable for this site, can someone please suggest the correct stack site.

REPLY [0 votes]: Some of the previous answers were more philosophical (e.g. David Hume), others provided great examples (e.g. the Cauchy Distribution).  I want to focus on this question of OP's:

...we can imagine a scenario where the histogram doesn't actually
converge but keeps changing. How can I make this idea mathematically
clear?

Ergodicity
A random process which reaches a steady state is called ergodic; In OP's example, if we observe an ergodic process at any intervals, the frequency of different observations will eventually converge to a nice histogram.  If we observe a non-ergodic process, the observations will not converge.
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