Index

n!, 80π, estimation of, 43–46

absorbing Markov chain, 415absorbing state, 416AbsorbingChain (program), 421absorption probabilities, 420Ace, Mr., 241Ali, 178alleles, 348AllPermutations (program), 84ANDERSON, C. L., 157annuity, 246

life, 246terminal, 246

arc sine laws, 493area, estimation of, 42Areabargraph (program), 46asymptotically equal, 81

Baba, 178babies, 14, 250Banach’s Matchbox, 254BAR-HILLEL, M., 176BARNES, B., 175BARNHART, R., 11BAYER, D., 119Bayes (program), 147Bayes probability, 136Bayes’ formula, 146BAYES, T., 149beard, 153bell-shaped, 47Benford distribution, 195BENKOSKI, S., 40Bernoulli trials process, 95BERNOULLI, D., 227

BERNOULLI, J., 112, 149, 310–312Bertrand’s paradox, 47–50BERTRAND, J., 49, 181BertrandsParadox (program), 49beta density, 168BIENAYME, I., 310, 378BIGGS, N. L., 85binary expansion, 69binomial coefficient, 92binomial distribution, 98, 184

approximating a, 329Binomial Theorem, 102BinomialPlot (program), 98BinomialProbabilities (program), 98Birthday (program), 78birthday problem, 77blackjack, 247, 253blood test, 254Bose-Einstein statistics, 106Box paradox, 181BOX, G. E. P., 213boxcars, 27BRAMS, S., 179, 182Branch (program), 382branching process, 377

customer, 394BranchingSimulation (program), 388bridge, 181, 182, 199, 203, 287BROWN, B. H., 38BROWN, E., 425Buffon’s needle, 44–46, 51–53BUFFON, G. L., 9, 44, 50–51BuffonsNeedle (program), 45bus paradox, 164

calendar, 38cancer, 147

501

502 INDEX

canonical form of an absorbingMarkov chain, 417

car, 137CARDANO, G., 30–31, 111, 249cars on a highway, 66CASANOVA, G., 11Cauchy density, 218, 401cells, 347Central Limit Theorem, 325

for Bernoulli Trials, 330for Binomial Distributions, 328for continuous independent trials

process, 357for discrete independent random

variables, 345for discrete independent trials

process, 343for Markov Chains, 464proof of, 398

chain letter, 389characteristic function, 398Chebyshev Inequality, 305, 316CHEBYSHEV, P. L., 313chi-squared density, 216, 296Chicago World’s Fair, 52chord, random, 47, 54chromosomes, 348CHU, S.-C., 109CHUNG, K. L., 153Circle of Gold, 389Clinton, Bill, 196clover-leaf interchange, 39CLTBernoulliGlobal, 332CLTBernoulliLocal (program), 329CLTBernoulliPlot (program), 327CLTGeneral (program), 345CLTIndTrialsLocal (program), 342CLTIndTrialsPlot (program), 341COATES, R. M., 305CoinTosses (program), 3Collins, People v., 153, 202color-blindness, 424conditional density, 162conditional distribution, 134conditional expectation, 239

conditional probability, 133CONDORCET, Le Marquis de, 12confidence interval, 334, 360conjunction fallacy, 38continuum, 41convolution, 286, 291

of binomial distributions, 289of Cauchy densities, 294of exponential densities, 292, 300of geometric distributions, 289of normal densities, 294of standard normal densities, 299of uniform densities, 292, 299

CONWAY, J., 432CRAMER, G., 227craps, 235, 468Craps (program), 235CROSSEN, C., 161CROWELL, R., 468cumulative distribution function, 61

joint, 165customer branching process, 394cut, 119

Dartmouth, 27darts, 56, 57, 59, 60, 64, 71, 163, 164Darts (program), 58DAVID, F. N., 86, 337, 489DAVID, F. N., 32de MOIVRE, A., 37, 87, 148, 336, 489de MONTMORT, P. R., 85de MERE, CHEVALIER, 4, 31, 37degrees of freedom, 217DeMere1 (program), 4DeMere2 (program), 4density function, 56, 59

beta, 168Cauchy, 218, 401chi-squared, 216, 296conditional, 162exponential, 53, 66, 163, 205gamma, 207joint, 165log normal, 224Maxwell, 215, 295

INDEX 503

normal, 212t-, 360uniform, 60, 205

derangement, 85DIACONIS, P., 119, 251Die (program), 225DieTest (program), 297distribution function, 1, 19

properties of, 22Benford, 195binomial, 184geometric, 184hypergeometric, 193joint, 142marginal, 143negative binomial, 186Poisson, 187uniform, 183

DNA, 348DOEBLIN, W., 449DOYLE, P. G., 87, 470Drunkard’s Walk example, 416, 419–

421, 423, 427, 443Dry Gulch, 279

EDWARDS, A. W. F., 107Egypt, 30Ehrenfest model, 410, 433, 441, 460,

461EHRENFEST, P., 410EHRENFEST, T, 410EhrenfestUrn (program), 462EISENBERG, B., 160elevator, 88, 115Emile’s restaurant, 75ENGLE, A., 445envelopes, 179, 180EPSTEIN, R., 287equalization, 472equalizations

expected number of, 479ergodic Markov chain, 433ESP, 250, 251EUCLID, 85Euler’s formula, 202

Eulerian number, 126event, 18events

attraction of, 160independent, 139, 164repulsion of, 160

existence of God, 245expected value, 226, 268exponential density, 53, 66, 163, 205extinction, problem of, 379

factorial, 80fair game, 241FALK, R., 161, 176fall, 130fallacy, 38FELLER, W., 11, 106, 191, 201, 218,

254, 344FERMAT, P., 4, 32–35, 111–112, 156Fermi-Dirac statistics, 106figurate numbers, 107financial records

suspicious, 196finite additivity property, 23FINN, J., 178First Fundamental Mystery of Proba-

bility, 232first maximum of a random walk, 496first return to the origin, 473Fisher’s Exact Test, 193FISHER, R. A., 252fixed column vector, 435fixed points, 82fixed row vector, 435FixedPoints (program), 82FixedVector (program), 437flying bombs, 191, 201Fourier transform, 398FRECHET, M., 466frequency concept of probability, 70frustration solitaire, 86Fundamental Limit Theorem for Reg-

ular Markov Chains, 448fundamental matrix, 419

for a regular Markov chain, 457

504 INDEX

for an ergodic Markov chain, 458

GALAMBOS, J., 303GALILEO, G., 12Gallup Poll, 14, 336Galton board, 100, 351GALTON, F., 281, 345, 350, 377GaltonBoard (program), 100Gambler’s Ruin, 426, 486, 487gambling systems, 241gamma density, 207GARDNER, M., 181gas diffusion

Ehrenfest model of, 410, 433, 441,460, 461

GELLER, S., 176GeneralSimulation (program), 9generating function

for continuous density, 394moment, 366, 395ordinary, 370

genes, 348, 411genetics, 345genotypes, 348geometric distribution, 184geometric series, 29GHOSH, B. K., 160goat, 137GONSHOR, H., 425GOSSET, W. S., 360grade point average, 343GRAHAM, R., 251GRANBERG, D., 161GRAUNT, J., 246Greece, 30GRIDGEMAN, N. T., 51, 181GRINSTEAD, C. M., 87GUDDER, S., 160

HACKING, I., 30, 148HAMMING, R. W., 283, 284HANES data, 345Hangtown, 279Hanover Inn, 65hard drive, Warp 9, 66

Hardy-Weinberg Law, 349harmonic function, 428Harvard, 27hat check problem, 82, 85, 104heights

distribution of, 345helium, 106HEYDE, C., 378HILL, T., 196Holmes, Sherlock, 91HorseRace (program), 6hospital, 14, 250HOWARD, R. A., 406HTSimulation (program), 6HUDDE, J., 148HUIZINGA, F., 389HUYGENS, C., 147, 243–245hypergeometric distribution, 193hypotheses, 145hypothesis testing, 100

Inclusion-Exclusion Principle, 103independence of events, 139, 164

mutual, 141independence of random variables

mutual, 143, 165independence of random

variables, 143, 165independent trials process, 144, 168interarrival time, average, 208interleaving, 119irreducible Markov chain, 433Isle Royale, 202

JAYNES, E. T., 49JOHNSONBOUGH, R., 153joint cumulative distribution

function, 165joint density function, 165joint distribution function, 142joint random variable, 142

KAHNEMAN, D., 38Kemeny’s constant, 469, 470KEMENY, J. G., 200, 406, 466KENDALL, D. G., 378

INDEX 505

KEYFITZ, N., 383KILGOUR, D. M., 179, 182KINGSTON, J. G., 157KONOLD, C., 161KOZELKA, R. M., 344

Labouchere betting system, 12, 13LABOUCHERE, H. du P., 12LAMPERTI, J., 267, 324LAPLACE, P. S., 51, 53, 350last return to the origin, 482Law (program), 310Law of Averages, 70Law of Large Numbers, 307, 316

for Ergodic Markov Chains, 439Strong, 70

LawContinuous (program), 318lead change, 482LEONARD, B., 256LEONTIEF, W. W., 426LEVASSEUR, K., 485library problem, 82life table, 39light bulb, 66, 72, 172LINDEBERG, J. W., 344LIPSON, A., 161Little’s law for queues, 276Lockhorn, Mr. and Mrs., 65log normal density, 224lottery

Powerball, 204LUCAS, E., 118

MAISTROV, L., 150, 310MANN, B., 119margin of error, 335marginal distribution function, 143Markov chain, 405

absorbing, 415ergodic, 433irreducible, 433regular, 433

Markov ChainsCentral Limit Theorem for, 464Fundamental Limit Theorem for

Regular, 448

MARKOV, A. A., 464martingale, 241, 242, 428

origin of word, 11martingale betting system, 11, 14, 248matrix

fundamental, 419MatrixPowers (program), 407maximum likelihood

estimate, 198, 202Maximum Likelihood

Principle, 91, 116Maxwell density, 215, 295maze, 440, 453McCRACKEN, D., 10mean, 226mean first passage matrix, 455mean first passage time, 452mean recurrence matrix, 455mean recurrence time, 454memoryless property, 68, 164, 206milk, 252modular arithmetic, 10moment generating function, 366, 395moment problem, 368, 398moments, 365, 394Monopoly, 469MonteCarlo (program), 42Monty Hall problem, 136, 161moose, 202mortality table, 246mule kicks, 201MULLER, M. E., 213multiple-gene hypothesis, 348mustache, 153mutually independent events, 141mutually independent random

variables, 143

negative binomial distribution, 186NEGRINI, M., 196New York Times, 340New York Yankees, 117, 253New-Age Solitaire, 129NEWCOMB, S., 196NFoldConvolution (program), 287

506 INDEX

normal density, 47, 212NormalArea (program), 322nursery rhyme, 84

odds, 27ordering, random, 126ordinary generating function, 370ORE, O., 30, 31outcome, 18Oz, Land of, 406, 439

Pascal’s triangle, 93, 103, 107PASCAL, B., 4, 32–35, 107, 111–112,

156, 242, 245paternity suit, 222PEARSON, K., 9, 351PENNEY, W., 432People v. Collins, 153, 202PERLMAN, M. D., 45permutation, 79

fixed points of, 82Philadelphia 76ers, 15photons, 106Pickwick, Mr., 153Pilsdorff Beer Company, 279PITTEL, B., 255point count, 287Poisson approximation to the

binomial distribution, 189Poisson distribution, 187

variance of, 262poker, 95polls, 333Polya urn model, 152, 174ponytail, 153posterior probabilities, 145Powerball lottery, 204PowerCurve (program), 102Presidential election, 336PRICE, C., 86prior probabilities, 145probability

Bayes, 136conditional, 133frequency concept of, 2

of an event, 19transition, 406vector, 407

problem of points, 32, 111, 147, 156process, random, 127PROPP, J., 256PROSSER, R., 200protons, 106POLYA, G., 15, 17, 475

quadratic equation, roots of, 73quantum mechanics, 106QUETELET, A., 350Queue (program), 208queues, 186, 208, 275quincunx, 351

RABELAIS, F., 12racquetball, 157radioactive isotope, 66, 71RAND Corporation, 10random integer, 39random number generator, 2random ordering, 126random process, 127random variable, 1, 18

continuous, 58discrete, 18functions of a, 210joint, 142

random variablesindependence of, 143mutual independence of, 143

random walk, 471in n dimensions, 17

RandomNumbers (program), 3RandomPermutation (program), 82rank event, 160raquetball, 13rat, 440, 453records, 83, 234Records (program), 84regression on the mean, 282regression to the mean, 345, 352regular Markov chain, 433

INDEX 507

reliability of a system, 154restricted choice, principle of, 182return to the origin, 472

first, 473last, 482probability of eventual, 475

reversibility, 463reversion, 352riffle shuffle, 119RIORDAN, J., 86rising sequence, 120rnd, 42ROBERTS, F., 426Rome, 30ROSS, S., 270, 276roulette, 13, 237, 432run, 229RENYI, A., 167

SAGAN, H., 237sample, 333sample mean, 265sample space, 18

continuous, 58countably infinite, 28infinite, 28

sample standard deviation, 265sample variance, 265SAWYER, S., 412SCHULTZ, H., 254SENETA, E., 378, 444service time, average, 208SHANNON, C. E., 465SHOLANDER, M., 39shuffling, 119SHULTZ, H., 256SimulateChain (program), 439simulating a random variable, 211snakeeyes, 27SNELL, J. L., 87, 175, 406, 466snowfall in Hanover, 83spike graph, 6Spikegraph (program), 6spinner, 41, 55, 59, 162spread, 266

St. Ives, 84St. Petersburg Paradox, 227standard deviation, 257standard normal random

variable, 213standardized random variable, 264standardized sum, 326state

absorbing, 416of a Markov chain, 405transient, 416

statisticsapplications of the Central Limit

Theorem to, 333stepping stones, 412SteppingStone (program), 412stick of unit length, 73STIFEL, M., 111STIGLER, S., 350Stirling’s formula, 81STIRLING, J., 87StirlingApproximations

(program), 81stock prices, 241StockSystem (program), 241Strong Law of Large

Numbers, 70, 314suit event, 160SUTHERLAND, E., 182

t-density, 360TARTAGLIA, N., 109tax returns, 196tea, 252telephone books, 255tennis, 157, 424tetrahedral numbers, 107THACKERAY, W. M., 14THOMPSON, G. L., 406THORP, E., 247, 253time to absorption, 419TIPPETT, L. H. C., 10traits, independence of, 216transient state, 416transition matrix, 406

508 INDEX

transition probability, 406tree diagram, 24, 76

infinite binary, 69Treize, 85triangle

acute, 73triangular numbers, 107trout, 198true-false exam, 267Tunbridge, 154TVERSKY, A., 14, 38Two aces problem, 181two-armed bandit, 170TwoArm (program), 171type 1 error, 101type 2 error, 101typesetter, 189

ULAM, S., 11unbiased estimator, 266uniform density, 205uniform density function, 60uniform distribution, 25, 183uniform random variables

sum of two continuous, 63unshuffle, 122USPENSKY, J. B., 299utility function, 227

VANDERBEI, R., 175Vandermonde determinant, 369variance, 257, 271

calculation of, 258variation distance, 127VariationList (program), 127volleyball, 158von BORTKIEWICZ, L., 201von MISES, R., 87von NEUMANN, J., 10, 11vos SAVANT, M., 40, 86, 136, 176,

181

Wall Street Journal, 161watches, counterfeit, 91WATSON, H. W., 378WEAVER, W., 465

Weierstrass Approximation Theorem,315

WELDON, W. F. R., 9Wheaties, 117, 253WHITAKER, C., 136WHITEHEAD, J. H. C., 181WICHURA, M. J., 45WILF, H. S., 90, 474WOLF, R., 9WOLFORD, G., 159Woodstock, 154

Yang, 129Yin, 129

ZAGIER, D., 485Zorg, planet of, 90