  Shorn of the inessential presentational device of selecting an outcome to bet on based on past outcomes, von Mises contends that it is a property of stochasticity that a random sequence should not be such that information about any initial subsequence \(x_1 x_2 \ldots x_{k-1}\) provides information about the contents of outcome \(x_k\). He formally implements this idea by defining a place selection as ‘the selection of a partial sequence in such a way that we decide whether an element should or should not be included without making use of the [value] of the element’ (von Mises 1957: 25). He then defines a random sequence as one such that every infinite subsequence selected by an admissible place selection retains the same relative digit frequencies as in the original sequence (so one cannot select a biased subsequence, indicating that this is a genuine property of stochasticity). In our case this will mean that every admissibly selected subsequence will meet the property of large numbers with equal frequency of 1s and  0s.[7]  One way to characterise the resulting set of von Mises-random (vM-random) sequences is that it is the largest set which contains only infinite sequences with the right limit frequency and is closed under all admissible place selections. If the limit frequency of a digit is 1, say in the sequence \(111\ldots\), it is true that every admissible place selection determines a subsequence with the same limit frequency. Von Mises intends this result, for this is what a random sequence of outcomes of trials with probability 1 of obtaining the outcome 1 looks like. This sequence does not meet the property of large numbers, however. So we modify von Mises’ own condition, defining the vM-random sequences as the largest set of infinite sequences which have the limit frequency \(\frac{1}{2}\), and which is closed under all admissible place selections.