Impossibility of a gambling system

It states that in a random sequence, the methodical selection of subsequences does not change the probability of specific elements.

[1][2] The principle states that no method for forming a subsequence of a random sequence (the gambling system) improves the odds for a specific event.

For instance, a sequence of fair coin tosses produces equal and independent 50/50 chances for heads and tails.

As a mathematical consequence of computability theory, more complicated betting strategies (such as a martingale) also cannot alter the odds in the long run.

With this property, the frequency of zeroes in the sequence stabilizes at 1/2, and every possible subsequence selected by any systematic method is likewise not biased.