Art Of Probability

Richard W Hamming

• Probability * Introduction * Models in General * The Frequency Approach Rejected * The Single Event Model * Symmetry as the Measure of Probability * Independence * Subsets of a Sample Space * Conditional Probability * Randomness * Critique of the Model Some Mathematical Tools * Permutations * Combinations * The Binomial DistributionBernoulli Trials * Random Variables, Mean and the Expected Value * The Variance * The Generating Function * The Weak Law of Large Numbers * The Statistical Assignment of Probability * The Representation of Information Methods for Solving Problems * The Five Methods * The Total Sample Space and Fair Games * Enumeration * Historical Approach * Recursive Approach * Recursive Approach * The Method of Random Variables * Critique of the Notion of a Fair Game * Bernoulli Evaluation * Robustness * InclusionExclusion Principle Countably Infinite Sample Spaces * Introduction * Bernoulli Trials * On the Strategy to be Adopted * State Diagrams * Generating Functions of State Diagrams * Expanding a Rational Generating Function * Checking the Solution * Paradoxes Continuous Sample Spaces * A Philosophy of the Real Number System * Some First Examples * Some Paradoxes * The Normal Distribution * The Distribution of Numbers * Convergence to the Reciprocal Distribution * Random Times * Dead Times * Poisson Distribution in Time * Queing Theorem * Birth and Death Systems * Summary Uniform Probability Assignments Maximum Entropy * What is Entropy? * Shannons Entropy * Some Mathematical Properties of the Entropy Function * Some Simple Applications * The Maximum Entropy Principle Models of Probability * General Remarks * Maximum Likelihood in a Binary Choice * Von Mises Probability * The Mathematical Approach * The Statistical Approach * When The Mean Does Not Exist * Probability as an Extension of Logic * Di Finetti * Subjective Probability * Fuzzy Probability * Probability in Science * Complex Probability Some Limit Theorems * The Biomial Approximation for the case p=1/2 * Approximation by the Normal Distribution * Another Derivation of the Normal Distribution * Random Times * The Zipf Distribution * Summary An Essay on Simulation