Bell's Theorem (1964) describes an inequality that should hold if quantum mechanics' randomness can be explained by certain types of hidden variables. In the time since, we've repeatedly observed that inequality violated in labs, leading most to presume that the normal types of hidden variables you would intuit don't exist. There are some esoteric loopholes that remain possibilities, but for now the position that matches our data the best is that there are not hidden variables and quantum mechanics is fundamentally probabilistic.
So to make sure I am understanding correctly, the normal distribution of the outcomes is itself evidence that other hidden factors aren't at play, because those factors would produces a less normal distribution?
I.e. if coin toss results skew towards heads, you can conclude some factor is biasing it that way, therefore if the results are (over the course of many tests) 'even', you can conclude the absence of biasing factors?
Or the Many Worlds Interpretation is correct. It is deterministic, we just don't know which branch we're in.