Impossibility of a gambling system
The principle of the impossibility of a gambling system is a concept in probability. It states that in a random sequence, the selection of sub-sequences does not change the probability of specific elements. Although the concept had been vaguely discussed in various forms for some time, it is generally attributed to Richard von Mises, who used the term collective rather than sequence.
Intuitively speaking, the principle states that it is not possible to select a sub-sequence of a random sequence in a way to improve the odds for a specific event. For instance, if a coin toss sequence is random with equal and independent 50/50 chances for heads and tails, then betting on heads every 3rd, 7th, or 21st toss, etc. does not change the odds of winning in the long run. Richard von Mises likened the principle of the impossibility of a gambling system to the principle of the conservation of energy, a law that can not be proven, but has held true in repeated experiments.
Elsewhere von Mises had also discussed impossibility of other issues in science and human understanding, e.g. in his book on Positivism he discussed the impossibility of exact descriptions due to linguistic constraints. And von Mises was also supportive of the notion of the impossibility of strict determinism in physics.
As a framework for the impossibility of a gambling system, Richard von Mises defined an infinite sequence of zeros and ones as a random sequence 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.
The sub-sequence selection criterion imposed by von Mises is important, because although 0101010101... is not biased, by selecting the odd positions, we get 000000... which is not random. Von Mises never totally formalized his definition of a proper selection rule for sub-sequences, but in 1940 Alonzo Church defined it as any recursive function which having read the first N elements of the sequence decides if it wants to select element number N+1. Church was a pioneer in the field of computable functions, and the definition he made relied on the Church Turing Thesis for computability.
In the mid 1960s, A. N. Kolmogorov and D. W. Loveland independently proposed a more permissive selection rule. In their view Church's recursive function definition was too restrictive in that it read the elements in order. Instead they proposed a rule based on a partially computable process which having read any N elements of the sequence, decides if it wants to select another element which has not been read yet.
The principle influenced modern concepts in randomness, e.g. the work by A. N. Kolmogorov in considering a finite sequence random (with respect to a class of computing systems) if any program that can generate the sequence is at least as long as the sequence itself.
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