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# Probability, Statistics, and Truth

Richard von Mises

PREFACE
PREFACE TO THE THIRD GERMAN EDITION

FIRST LECTURE The Definition of Probability
Amendment of Popular Terminology
Explanation of Words
Synthetic Definitions
Terminology
The Concept of Work in Mechanics
An Historical Interlude
The Purpose of Rational Concepts
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
Randomness
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 Mâ€šrâ€š
Solution of the Problem of Chevalier de Mâ€šrâ€š
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
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
Summary
Inference
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'
Theory of Accumulation and the Law of Series
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
Correction
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

SUMMARY OF THE SIX LECTURES IN SIXTEEN PROPOSITIONS