Today this work is generally thought of as the classic statement of the frequentist position on the foundations of probability as contrasted with a Bayesian approach. In its own day, and even more in the earlier years of von Mises research, the principal alternative to the frequentist position was not the Bayesian outlook of the past 50 years but rather the “equiprobable” outlook of many of the pioneers of probability theory.

These pioneers often tacitly assumed that one could list the possible outcomes and define the probability of an event as the number of “favorable” outcomes divided by the total number of outcomes. This works for one toss of one coin, but if you toss two, and list the outcomes as 0, 1 or 2 heads, then the results do not agree well with the real world behavior of coins. One head comes up about twice as often as none or two. What von Mises did to deal with this was to assert that probability pertained to limiting values of relative frequencies of repeated events. This made probability more empirical and less *a priori*. In addition, von Mises contributed ideas on determining what situations in the real world might reasonably be treated via probability. In particular, he had some interesting things to say on the nature of randomness.

Like many of the older books, this one is best understood in the context of its own time and still earlier times. Transposing it into issues of our own day may be risky. The author does mention subjective probability here and there, but barely mentions the ideas that today would be labelled “Bayesian.” Today that approach is most common in business. As an example of the difference in the two approaches, consider a manager who believes that a new product will appeal to 40% of its market. A survey might later indicate the appeal is closer to 20%. A frequentist would probably go with the 20% figure while the Bayesian would try to combine the two estimates into a new and intermediate one.

While the frequentist and Bayesian approaches are often treated as mutually exclusive, it might be more accurate to say that Bayesians allow frequentist probabilities but allow subjective probabilities as well. Mathematicians have a very different view of foundations, and some of us might simply say that anything that satisfies the Kolmogorov axioms is probability, while allowing that some situations are a better match than others, and that the proof of the pudding should be how well the results of applying probability theory match reality.

It may also be true that the two approaches suit different problems. A classic 1960 paper by John Tukey distinguishes between conclusions and decisions. Decisions are unique situations such as our example of marketing a new product. A decision has to be made once and for all. It that case the Bayesian approach has appeal. Conclusions are scientific beliefs based on an accumulation of evidence, and in science including the researcher’s prior opinions is not considered cricket. It is clear that von Mises’ interest is in the role of probability in science, so though he takes a dim view of subjective probabilities, he is evaluating them as a basis for scientific conclusions rather than practical decisions.

In light of all this, it might be better to read Tukey and more recent authors on frequentist versus Bayesian statistics. One might read von Mises to learn more about the history of probability and statistics and mathematics, and the connection between these disciplines and science. In its time, the frequentist outlook resolved some of the paradoxes and inconsistencies in earlier writings. As with any authentic source from an earlier age, the reader of this book will find some unfamiliar terminology, probably further complicated by the translation from German. However, for anyone interested in history of the issues it covers, this book is *the* book to read.

**Reference**

Tukey, J. W. (1960) “Conclusions versus Decisions” *Technometrics* **2** 424-433.

After a few years in industry, Robert W. Hayden ([email protected]) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He now teaches statistics online at statistics.com and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA’s Teaching Statistics.