Yet as D. H. Lehmer stated in 1951: "A random sequence is a vague notion... in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians".
Using the concept of the impossibility of a gambling system, von Mises defined an infinite sequence of zeros and ones as random if it is not biased by having the frequency stability property i.e. the frequency of zeros goes to 1/2 and every sub-sequence we can select from it by a "proper" method of selection is also not biased.
During the 20th century various technical approaches to defining random sequences were developed and now three distinct paradigms can be identified.
But this method was considered too weak by Alexander Shen who showed that there is a Kolmogorov–Loveland stochastic sequence which does not conform to the general notion of randomness.
His original definition involved measure theory, but it was later shown that it can be expressed in terms of Kolmogorov complexity.