  The first detailed and sophisticated attempt at a bolder approach to defining randomness for a sequence with a single stochastic property was by von Mises (von Mises, 1957; von Mises, 1941). Suppose you were presented with any subsequence \(x_1 , \ldots ,x_{n-1}\) of (not necessarily consecutive members of) a sequence, and asked to predict the value of \(x_n\). If the sequence were really random, then this information—the values of any previous members of the sequence, and the place of the desired outcome in the sequence—should be of no use to you in this task. To suppose otherwise is to suppose that there is an exploitable regularity in the random sequence; a gambler could, for example, bet reliably on their preferred outcome and be assured of a positive expected gain if they were in possession of this information. A gambling system selects points in a sequence of outcomes to bet on; a successful gambling system would be one where the points selected have a higher frequency of ‘successes’ than in the sequence as a whole, so that by employing the system one can expect to do better than chance. But the failure of gambling systems to make headway in games of chance suggests that genuinely random sequences of outcomes aren’t so exploitable. Von Mises, observing the empirical non-existence of successful gambling systems, makes it a condition of randomness for infinite sequences that they could not be exploited by a gambling system (his ‘Prinzip vom ausgeschlossenen Spielsystem’). The idea is that without what effectively amounts to a crystal ball, there is no way of selecting a biased selection of members of a random sequence.