Probability, Statistics and Truth (Dover Books on Mathematics)

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 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 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 Méré Solution of the Problem of Chevalier de Méré 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 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' 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 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 of the 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 Radioactivity 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 NOTES AND ADDENDA SUBJECT INDEX NAME INDEX