Wikipedia on Randomness’s Mathematics

“The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling, but later in connection with physics. Statistics is used to infer the underlying probability distribution of a collection of empirical observations. For the purposes of simulation, it is necessary to have a large supply of random numbers or means to generate them on demand.

Algorithmic information theory studies, among other topics, what constitutes a random sequence. The central idea is that a string of bits is random if and only if it is shorter than any computer program that can produce that string (Kolmogorov randomness)—this means that random strings are those that cannot be compressed. Pioneers of this field include Andrey Kolmogorov and his student Per Martin-Löf, Ray Solomonoff, and Gregory Chaitin. For the notion of infinite sequence, one normally uses Per Martin-Löf’s definition. That is, an infinite sequence is random if and only if it withstands all recursively enumerable null sets. The other notions of random sequences include (but not limited to): recursive randomness and Schnorr randomness which are based on recursively computable martingales. It was shown by Yongge Wangthat these randomness notions are generally different.[13]

Randomness occurs in numbers such as log (2) and pi. The decimal digits of pi constitute an infinite sequence and “never repeat in a cyclical fashion.” Numbers like pi are also considered likely to be normal, which means their digits are random in a certain statistical sense.”

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