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of the theory in its present form. But it is possible to find a satisfactory
solution within the framework of the frequency theory, after this
axiom has been eliminated. It will be found by analysing the assumptions
which allow us to argue from the irregular succession of single
occurrences to the regularity or stability of their frequencies.
50 THE FREQUENCY THEORY OF VON MISES
probability 139
A frequency theory which provides a foundation for all the principal
theorems of the calculus of probability was first proposed by Richard
von Mises. 1 His fundamental ideas are as follows.
The calculus of probability is a theory of certain chance-like or
random sequences of events or occurrences, i.e. of repetitive events
such as a series of throws with a die. These sequences are defined as
‘chance-like’ or ‘random’ by means of two axiomatic conditions: the
axiom of convergence (or the limit-axiom) and the axiom of randomness. If a
sequence of events satisfies both of these conditions it is called by von
Mises a ‘collective’.
A collective is, roughly speaking, a sequence of events or occurrences
which is capable in principle of being continued indefinitely;
for example a sequence of throws made with a supposedly indestructible
die. Each of these events has a certain character or property; for
example, the throw may show a five and so have the property five. If we
take all those throws having the property five which have appeared up
to a certain element of the sequence, and divide their number by the
total number of throws up to that element (i.e. its ordinal number in
the sequence) then we obtain the relative frequency of fives up to that
element. If we determine the relative frequency of fives up to every
element of the sequence, then we obtain in this way a new sequence—
the sequence of the relative frequencies of fives. This sequence of frequencies is
distinct from the original sequence of events to which it corresponds,
1 R. von Mises, Fundamentalsätze der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 4, 1919,
p. 1; Grundlagen der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 5, 1919, p. 52; Wahrscheinlichkeit,
Statistik, und Wahrheit (1928), 2nd edition 1936, English translation by J. Neyman,
D. Sholl, and E. Rabinowitsch: Probability, Statistics and Truth, 1939; Wahrscheinlichkeitsrechnung
und ihre Anwendung in der Statistik und theoretischen Physik (Vorlesungen über angewandte Mathematik
1), 1931.