Introduction to Probability

Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continued to influence such early researchers as Huygens, Bernoulli, and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a wellestablished branch of mathematics that finds applications in every area of scholarly activity from music to physics, and in daily experience from weather prediction to predicting the risks of new medical treatments. | Introduction to Probability Charles M. Grinstead Swarthmore College J. Laurie Snell Dartmouth College To our wives and in memory of Reese T. Prosser Contents 1 Discrete Probability Distributions 1 Simulation of Discrete Probabilities . . . . . . . . . . . . . . . . . . . 1 Discrete Probability Distributions . . . . . . . . . . . . . . . . . . . . 18 2 Continuous Probability Densities 41 Simulation of Continuous Probabilities . . . . . . . . . . . . . . . . . 41 Continuous Density Functions . . . . . . . . . . . . . . . . . . . . . . 55 3 Combinatorics 75 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Card Shuffling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4 Conditional Probability 133 Discrete Conditional Probability . . . . . . . . . . . . . . . . . . . . 133 Continuous Conditional Probability . . . . . . . . . . . . . . . . . . . 162 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5 Distributions and Densities 183 Important Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 183 Important Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6 Expected Value and Variance 225 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Variance of Discrete Random Variables . . . . . . . . . . . . . . . . . 257 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . 268 7 Sums of Random Variables 285 Sums of Discrete Random Variables . . . . . . . . . . . . . . . . . . 285 Sums of Continuous Random Variables . . . . . . . . . . . . . . . . . 291 8 Law of Large Numbers 305 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . 305 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . 316 v vi CONTENTS 9 Central Limit Theorem 325 .