Includes bibliographical references and index.

CONTENTS
PREFACE page V
PREFACE TO THE THIRD GERMAN EDITION vii
FIRST LECTURE
The Definition of Probability
Amendment of Popular Terminology 1
Explanation of Words 2
Synthetic Definitions 3
Terminology 4
The Concept of Work m Mechanics 5
An Historical Interlude 6
The Purpose of Rational Concepts 6
The Inadequacy of Theories 7
Limitation of Scope 8
Unlimited Repetition 10
The Collective 1 1
The First Step towards a Definition 12
Two Different Pairs of Dice 13
Limiting Value of Relative Frequency 14
The Experimental Basis of the Theory of Games 16
The Probability of Death 16
First the Collective — ^then the Probability 18
Probability in the Gas Theory 20
An Historical Remark 21
Randomness 23
Definition of Randomness: Place Selection 24
The Principle of the Impossibility of a Gambling System 25
Example of Randomness 27
Summary of the Definition 28
SECOND LECTURE
The Elements of the Theory of Probability
The Theory of Probability is a Science Similar to Others 30
The Purpose of the Theory of Probability 31
The Beginning and the End of Each Problem must be Probabilities 32
Distnbution in a Collective 34
Probability of a Hit; Continuous Distribution 35
Probability Density 36
The Four Fundamental Operations 38
First Fundamental Operation: Selection 39
CONTENTS
Second Fundamental Operation: Mixing 39
Inexact Statement of the Addition Rule 40
Uniform Distribution 41
Summary of the Mixing Rule 43
Third Fundamental Operation: Partition 43
Probabihties after Partition 45
Initial and Final Probability of an Attribute 46
The So-called Probabihty of Causes 46
Formulation of the Rule of Partition 47
Fourth Fundamental Operation* Combination 48
A New Method of Forming Partial Sequences : Correlated Sampling 49 Mutually Independent Collectives 50
Derivation of the Multiplication Rule 51
Test of Independence 53
Combination of Dependent Collectives 55
Example of Noncombinable Collectives 56
Summary of the Four Fundamental Operations 57
A Problem of Chevalier de M6r6 58
Solution of the Problem of Chevalier de Mere 59
Discussion of the Solution 62
Some Final Conclusions 63
Short Review 64
THIRD LECTURE
Critical Discussion of the Foundations of Probability
The Classical Definition of Probability 66
Equally Likely Cases ... 68
... Do Not Always Exist 69
A Geometrical Analogy 70
How to Recognize Equally Likely Cases 71
Are Equally Likely Cases of Exceptional Significance? 73
The Subjective Conception of Probability 75
Bertrand’s Paradox 77
The Suggested Link between the Classical and the New Definitions of Probability 79
Summary of Objections to the Classical Defimtion 80
Objections to My Theory 81
Fmite Collectives 82
Testing Probability Statements 84
An Objection to the First Postulate 86
Objections to the Condition of Randomness 87
Restricted Randomness 89
Meaning of the Condition of Randomness 91
Consistency of the Randomness Axiom 92
A Problem of Terminology 93
Objections to the Frequency Concept 94
Theory of the Plausibility of Statements 95
The Nihilists 97
Restriction to One Single Initial Collective 98
Probability as Part of the Theory of Sets 99
Development of the Frequency Theory 101
Summary and Conclusion 102

CONTENTS
fourth lecture
The Laws of Large Numbers
Poisson’s Two Different Propositions 104
Equally Likely Events 106
Arithmetical Explanation 107
Subsequent Frequency Definition 109
The Content of Poisson’s Theorem 1 lo
Example of a Sequence to which Poisson’s Theorem does not Apply 111 Bernoulli and non-Bernoulli Sequences 112
Derivation of the Bernoulli-Poisson Theorem 113
Summary 115
Inference 1 1 6
Bayes’s Problem 117
Initial and Inferred Probability 118
Longer Sequences of Trials 120
Independence of the Initial Distribution 122
The Relation of Bayes’s Theorem to Poisson’s Theorem 124
The Three Propositions 125
Generalization of the Laws of Large Numbers 125
The Strong Law of Large Numbers 127
The Statistical Functions 129
The First Law of Large Numbers for Statistical Functions 131
The Second Law of Large Numbers for Statistical Functions 1 32
Closing Remarks 133
FIFTH LECTURE
Application in Statistics and the Theory of Errors
What is Statistics ? 135
Games of Chance and Games of Skill 136
Marbe’s ‘Uniformity in the World’ 138
Answer to Marbe’s Problem 139
Theory of Accumulation and the Law of Series 141
Linked Events 142
The General Purpose of Statistics 144
Lexis’ Theory of Dispersion 145
The Mean and the Dispersion 146
Comparison between the Observed and the Expected Variance 148
Lexis’ Theory and the Laws of Large Numbers 149
Normal and Nonnormal Dispersion 151
Sex Distribution of Infants 1 52
Statistics of Deaths with Supernormal Dispersion 153
Solidarity of Cases 154
Testing Hypotheses 1 55
R. A. Fisher’s ‘Likelihood’ 157
Small Sample Theory 158
Social and Biological Statistics 160
Mendel’s Theory of Heredity 160
Industrial and Technological Statistics 161
An Example of Faulty Statistics 162
Correction 163
Some Results Summarized 165
CONTENTS
Descriptive Statistics 166
Foundations of the Theory of Errors 167
Gallon’s Board 169
Normal Curve 1 70
Laplace’s Law 171
The Application of the Theory of Errors 172
SIXTH LECTURE
Statistical Problems in Physics
The Second Law of Thermodynamics 174
Determinism and Probability 175
Chance Mechanisms 177
Random Fluctuations 178
Small Causes and Large Eflfects 179
Kinetic Theory of Gases 181
Order of Magnitude of ‘Improbability’ 183
Criticism of the Gas Theory 184
Brownian Motion 186
Evolution of Phenomena in Time 187
Probability ‘After-Etfects’ 188
Residence Time and Its Prediction 189
Entropy Theorem and Markoff Chains 192
Svedberg’s Experiments 194
Radioactivity 195
Prediction of Time Intervals 197
Marsden’s and Barratt’s Experiments 198
Recent Development in the Theory of Gases 199
Degeneration of Gases: Electron Theory of Metals 200
Quantum Theory 202
Statistics and Causality 203
Causal Explanation in Newton’s Sense 204
Limitations of Newtonian Mechanics 206
Simplicity as a Cnterion of Causality 208
Giving up the Concept of Causality 209
The Law of Causality 210
New Quantum Statistics 211
Are Exact Measurements Possible? 213
Position and Velocity of a Material Particle 214
Heisenberg’s Uncertainty Principle 215
Consequences for our Physical Concept of the World 217
Final Considerations 218
SUMMARY OF THE SIX LECTURES IN SIXTEEN PROPOSITIONS 221
NOTES AND ADDENDA 224
SUBJECT INDEX 237
NAME INDEX 243

PREFACE
THE present second English edition is not merely a reprinting of the first edition, which has been out of print for several years, but rather a translation of the third German edition, revised by the author in 1951; this last differs in many ways from the second German edition, on which the original Enghsh translation was based.
The change consists essentially in the author’s omitting some of the discussions of the early controversies regarding his theory, and making instead various additions: The concept of randomness, which plays a central role in the author’s theory, is reconsidered — in particular, with respect to the problem of mathematical consistency — and carefully reformulated. The question of substituting for it some ‘limited randomness’ is taken up, and the author concludes that, as far as the basic axioms are concerned, no such restriction is advisable. Systematic consideration is given to recent work concerned with the basic definitions of probability theory: in the ideas of E. Tomier and of J. L. Doob, the author sees a remarkable development of his theory of frequencies in collectives. The analysis of the two Laws of Large Numbers has often been considered an outstanding section of the book; in the 1951 edition, on the basis of new mathematical results, the discussion of the Second Law is deepened and enlarged, with the aim of clarifying this highly controversial subject. Comments are added on the testing of hypotheses, (as based on the inference theory originated by T. Bayes), on R. A. Fisher’s ‘likelihood’, and on a few related subjects. These and all other additions are selected as well as discussed in relation to the basic ideas advanced in the book.
The deviations in content of the present Enghsh version from the third German edition are insignificant. Some passages which seemed of more local (Austrian) interest have been omitted. In a few instances the text has been changed (with explanation in the Notes when necessary). The Notes, historical, bibliographical, etc., have been somewhat modified. References have been brought up to date; indications of German translations have been replaced by corresponding English works; several notes are new. A subject index has been added.
The present text is based on the excellent translation (1939) of Messrs. J. Neyman, D. Scholl, and E. Rabinowitsch; it has been supplemented by all the new material in the 1951 edition, and amended in the light of notes made by the author in anticipation of a new English edition. In addition, the entire text was given a careful editorial revision. The sixth chapter, on ‘Statistics in Physics’ was essentially retranslated.
In these various aspects, and in particular with regard to the new translations, I enjoyed the valuable assistance of Mrs. R. Buka. My sincere thanks go to Professor J, Neyman for his understanding encouragement, to Professor E. Tornier for significant advice regarding a few difficult passages, to Dr. A. O’Neill who prepared the index, to Mr. F. J. Zucker, who kindly read the translation of the sixth chapter, and to Mr. J. D. Elder, who was good enough to check the text with respect to uniformity and general consistency of style. I am very grateful to the Department of Mathematics, particularly to Professor G. Birkhoff, and to the Division of Engineering and Applied Physics of Harvard University, who together with the Office of Naval Research sponsored this work. Finally, I thank the publishers, Allen & Unwin, Ltd., London, and the Macmillan Company, New York, for the cooperation they gave freely whenever needed, and the Springer-Verlag, Vienna, for granting permission to bring out this edition.