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Probability, Statistics, and Truth

Richard von Mises
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Robert W. Hayden
, on

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.


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

After a few years in industry, Robert W. Hayden ( 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 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.



FIRST LECTURE The Definition of Probability
Amendment of Popular Terminology
Explanation of Words
Synthetic Definitions
The Concept of Work in Mechanics
An Historical Interlude
The Purpose of Rational Concepts
The Inadequacy of Theories
Limitation of Scope
Unlimited Repetition
The Collective
The First Step towards a Definition
Two Different Pairs of Dice
Limiting Value of Relative Frequency
The Experimental Basis of the Theory of Games
The Probability of Death
First the Collective-then the Probability
Probability in the Gas Theory
An Historical Remark
Definition of Randomness: Place Selection
The Principle of the Impossibility of a Gambling System
Example of Randomness
Summary of the Definition

SECOND LECTURE The Elements of the Theory of Probability
The Theory of Probability is a Science Similar to Others
The Purpose of the Theory of Probability
The Beginning and the End of Each Problem must be Probabilities
Distribution in a Collective
Probability of a Hit; Continuous Distribution
Probability Density
The Four Fundamental Operations
First Fundamental Operation: Selection
Second Fundamental Operation: Mixing
Inexact Statement of the Addition Rule
Uniform Distribution
Summary of the Mixing Rule
Third Fundamental Operation: Partition
Probabilities after Partition
Initial and Final Probability of an Attribute
The So-called Probability of Causes
Formulation of the rule of Partition
Fourth Fundamental Operation: Combination
A New Method of Forming Partial Sequences: Correlated Sampling
Mutually Independent Collectives
Derivation of the Multiplication Rule
Test of Independence
Combination of Dependent Collectives
Example of Noncombinable Collectives
Summary of the Four Fundamental Operations
A Problem of Chevalier de M‚r‚
Solution of the Problem of Chevalier de M‚r‚
Discussion of the Solution
Some Final Conclusions
Short Review

THIRD LECTURE Critical Discussion of the Foundations of Probability
The Classical Definition of Probability
Equally Likely Cases ...
... Do Not Always Exist
A Geometrical Analogy
How to Recognize Equally Likely Cases
Are Equally Likely Cases of Exceptional Significance?
The Subjective Conception of Probability
Bertrand's Paradox
The Suggested Link between the Classical and the New Definitions of Probability
Summary of Objections to the Classical Definition
Objections to My Theory
Finite Collectives
Testing Probability Statements
An Objection to the First Postulate
Objections to the Condition of Randomness
Restricted Randomness
Meaning of the Condition of Randomness
Consistency of the Randomness Axiom
A Problem of Terminology
Objections to the Frequency Concept
Theory of the Plausibility of Statements
The Nihilists
Restriction to One Single Initial Collective
Probability as Part of the Theory of Sets
Development of the Frequency Theory
Summary and Conclusion

FOURTH LECTURE The Laws of Large Numbers
Poisson's Two Different Propositions
Equally Likely Events
Arithmetical Explanation
Subsequent Frequency Definition
The Content of Poisson's Theorem
Example of a Sequence to which Poisson's Theorem does not Apply
Bernoulli and non-Bernoulli Sequences
Derivation of the Bernoulli-Poison Theorem
Bayes's Problem
Initial and Inferred Probability
Longer Sequences of Trials
Independence of the Initial Distribution
The Relation of Bayes's Theorem to Poisson's Theorem
The Three Propositions
Generalization of the Laws of Large Numbers
The Strong Law of Large Numbers
The Statistical Functions
The First Law of Large Numbers for Statistical Functions
The Second Law of Large Numbers for Statistical Functions
Closing Remarks

FIFTH LECTURE Application Statistics and the Theory of Errors
What is Statistics?
Games of Chance and Games of Skill
Marbe's Uniformity in the World'
Answer to Marbe's Problem
Theory of Accumulation and the Law of Series
Linked Events
The General Purpose of Statistics
Lexis' Theory of Dispersion
The Mean and the Dispersion
Comparison between the Observed and the Expected Variance
Lexis' Theory and the Laws of Large Numbers
Normal and Nonnormal Dispersion
Sex Distribution of Infants
Statistics of Deaths with Supernormal Dispersion
Solidarity of Cases
Testing Hypotheses
R. A. Fisher's Likelihood'
Small Sample Theory
Social and Biological Statistics
Mendel's Theory of Heredity
Industrial and Technological Statistics
An Example of Faulty Statistics
Some Results Summarized
Descriptive Statistics
Foundations of the Theory of Errors
Galton's Board
Normal Curve
Laplace's Law
The Application of the Theory of Errors

SIXTH LECTURE Statistical Problems in Physics
The Second Law of Thermodynamics
Determinism and Probability
Chance Mechanisms
Random Fluctuations
Small Causes and Large Effects
Kinetic Theory of Gases
Order of Magnitude of 'Improbability'
Criticism ofthe Gas Theory
Brownian Motion
Evolution of Phenomena in Time
Probability 'After Effects'
Residence Time and Its Prediction
Entropy Theorem and Markoff Chains
Svedberg's Experiments
Prediction of Time Intervals
Marsden's and Barratt's Experiments
Recent Development in the Theory of Gases
Degeneration of Gases: Electron Theory of Metals
Quantum Theory
Statistics and Causality
Causal Explanation Newton's Sense
Limitations of Newtonian Mechanics
Simplicity as a Criterion of Causality
Giving up the Concept of Causality
The Law of Causality
New Quantum Statistics
Are Exact Measurements Possible?
Position and Velocity of a Material Particle
Heisenberg's Uncertainty Principle
Consequences for our Physical Concept of the World
Final Considerations