i`m wondering why the law of large numbers holds for pseudorandom numbers. Is this a empirical observation such as that the law of large numbers holds for coin tosses, or is this somehow programmed into the way the pseudorandom number generator works?
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This is one of the criteria of quality of pseudorandom generators. They are designed for this (among other things).
The law of great number could be expressed this way: any sequences of bits of size $n$ will be observed with empirical probability $2^{-n}$ for a long observational sequence.
Note that the larger $n$, the more it becomes unmanageable to make such observations. Hence this property is only tested for "small" $n$ say < 32. For any generator, there is always an $n$ large enough where this property will be strongly violated, but it requires an $n$ most often beyond usual observational limits.
