Chapter 5 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with the relative frequency theory of probability.
The relative frequency theory was originally developed in the 19th century by John Venn (1834–1923) and Robert Leslie Ellis (1817–1859), and then by the early 20th century empiricists Hans Reichenbach and Richard von Mises (1919 and 1961), who were influenced by the Vienna circle (Gillies 2000: 88).
Richard von Mises saw probability theory as an empirical science dealing with repeated events of the same type, such as in games of chance, biological statistics, and natural phenomena (Gillies 2000: 89). Such repeatable events or mass phenomena were called “collectives” by von Mises (which could be “empirical collectives” or “mathematical collectives”), and a “sample space” is a set of possible outcomes in such a collective.
To establish objective probabilities by means of the relative frequency approach the phenomenon in question must yield stable relative frequencies for the outcomes it exhibits in the long run, and von Mises called this the “law of stability of statistical frequencies” (Gillies 2000: 92).
Richard von Mises thus believed that real numeric probabilities are confined to processes where stable relative frequencies can be obtained, and indeed “probability” in the scientific sense is limited to this (Gillies 2000: 97–98).
Critics of von Mises countered that his frequency theory approach does not explain the qualitative probabilities obtained in inductive arguments (Gillies 2000: 99), and thus seems to be a highly restricted definition of probability.
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.
Mises, Richard von. 1919. “Grundlagen der Wahrscheinlichkeitsrechnung,” Mathematische Zeitschrift 5: 52–99.
Mises, Richard von. 1961. Probability, Statistics and Truth (2nd edn.). Allen and Unwin, London.