Bibliography: The Jungles of Randomness

The Jungles of Randomness

From slot machines and amusement park rides to dice games and shuffled cards, chance and chaos pervade everyday life. Sorting through the various meanings of randomness and distinguishing between what we can and cannot know with certainty proves to be no simple matter. Inside information on how slot machines work, the perils of believing random number generators, and the questionable fairness of dice, tossed coins, and shuffled cards illustrate how tricky randomness can be.

Online Biliography

A list of Web articles, books, and other resources that deal with various aspects of randomness.

General References

Beasley, J.D. 1989. The Mathematics of Games. Oxford, England: Oxford University Press.

Bennett, D.J. 1998. Randomness. Cambridge, Mass.: Harvard University Press.

Ekeland, I. 1993. The Broken Dice and Other Mathematical Tales of Chance. Chicago: University of Chicago Press.

Huff, D., and U. Geis. 1959. How to Take a Chance. New York: W.W. Norton.

Peterson, I. 1995. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.

Weaver, W. 1963. Lady Luck: The Theory of Probability. New York: Wiley.

Ivars Peterson's MathTrek

A Catalog of Random Bits

Randomness, Risk, and Financial Markets

Heads or Tails?

The Bias of Random Number Generators

Card Shuffling Shenanigans

Rainbow Randomness

Lava Lamp Randomness

Disorder in the Deck

Tilt-A-Whirl Chaos I

Tilt-A-Whirl Chaos II

Unfair Dice

A Penny Surprise

Deep in the Jungles


Catlin, D. 2002. Non-random randomness. Part 1. Casino City Times (June 1).

Heidel, R. 1996. Slot machines. Scientific American 275(August):112.

Koeppel, D. 2002. Money wired. Popular Science 261(November):76-86.

Mezrich, B. 2002. Hacking Las Vegas. Wired 9(September):130-139.

Orkin, M. 1991. Can You Win? The Real Odds for Casino Gambling, Sports Betting, and Lotteries.. New York: W.H. Freeman.

Scarne, J. 1986. Scarne's New Complete Guide to Gambling. New York: Simon & Schuster.

Wykes, A. 1964. The Complete Illustrated Guide to Gambling. Garden City, N.Y.: Doubleday.

Ace's Guide to Gambling: Odds on Slot Machines

History of Slot Machines


Ditto, W.L., and L. Pecora. 1993. Mastering chaos. Scientific American 269(August):78-84.

Kautz, R.L., and B.M. Huggard. 1994. Chaos at the amusement park: Dynamics of the Tilt-A-Whirl. American Journal of Physics 62(January):59-66.

Lorenz, E.N. 1993. The Essence of Chaos. Seattle: University of Washington Press.

Shinbrot, T., et al. 1992. Chaos in a double pendulum. American Journal of Physics 60(June):491-499.

Kautz, R.L., and B.M. Huggard. 1994. Chaos at the amusement park: Dynamics of the Tilt-A-Whirl. American Journal of Physics 62(January):59-66.

The Sellner Manufacturing Company, which makes the Tilt-A-Whirl, has a Web site at

To learn more about the mathematics underlying chaos you can try The Chaos Hypertextbook at or visit the University of Maryland's Chaos Group Web site at

Pinball History

Random Number Generators

Bauke, H., and S. Mertens. 2004. Pseudo random coins show more heads than tails. Journal of Statistical Physics 114(February):1149-1169.

Ferrenberg, A.M., D.P. Landau, and Y.J. Wong. 1992. Monte Carlo simulations: Hidden errors from "good" random number generators. Physical Review Letters 69(Dec. 7):3382-3384.

Hayes, B. 2001. Randomness as a resource. American Scientist 89(July/August):300-304.

______. 1993. The wheel of fortune. American Scientist 81(March/April):114-118.

McNichol, T. 2003. Totally random. Wired 11(August).

Mertens, S., and H. Bauke. 2004. Entropy of pseudo-random-number generators. Physical Review E 69(May):055702.

Warnock, T. 1987. Random-number generators. Los Alamos Science 15:137-141.

Information about the lavarand random number generator is available at


Diaconis, P., and J.B. Keller 1989. Fair dice. American Mathematical Monthly 96(April):333-348.

Coin Tossing

Ford, J. 1983. How random is a coin toss? Physics Today 36(April):40-47.

Keller, J.B. 1986. The probability of heads. American Mathematical Monthly 93(March):191-197.

Klarreich, E. 2004. Toss out the toss-up: Bias in heads-or-tails. Science News 165(Feb. 28):131-132.

Peterson, I. 2003. Flipping a coin. Muse 7(April):19.

______. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.

Vulovic, V.Z., and R.E. Prange. 1986. Randomness of a true coin toss. Physical Review A 33(January):576-582.

For a mathematical introduction to coin tossing, see

2004. Magician-turned-mathematician uncovers bias in coin flipping. Stanford University news release. June 4.

Card Shuffling

Aldous, D., and P. Diaconis. 1986. Shuffling cards and stopping times. American Mathematical Monthly 93(May):333-348.

Bayer, D., and P. Diaconis. 1992. Trailing the dovetail shuffle to its lair. Annals of Applied Probability 2:294-313.

Berger, P. 1973. On the distribution of hand patterns in bridge: Man-dealt versus computer-dealt. Canadian Journal of Statistics 1:261-266.

Diaconis, P. 1996. The cutoff phenomenon in finite Markov chains. Proceedings of the National Academy of Sciences 93(Feb. 20):1659-1664.

Folger, T. 1991. Shuffling into hyperspace. Discover 12(January):66-67.

Keller, J.B. 1995. How many shuffles to mix a deck? SIAM Review 37(March):88-89.

Klarreich, E. 2002. Coming up trumps. New Scientist 175(July 20):42-44.

Kolata, G. 1990. In shuffling cards, 7 is winning number. New York Times (Jan. 9).

______. 1985. Prestidigitator of digits. Science 85 6(April):67-72.

Mackenzie, D. 2002. The mathematics of . . . shuffling the Stanford flip. Discover 23(October):22-23.

Trefethen, L.N., and L.M. Trefethen. 2000. How many shuffles to randomize a deck of cards? Proceedings of the Royal Society, London A 456(Oct. 8):2561-2568.

A brief outline of the mathematics of card shuffling can be found at


Bassein, S. 1996. A sampler of randomness. American Mathematical Monthly 103(June-July):483-490.

Compagner, A. 1991. Definitions of randomness. American Journal of Physics 59(August):700-705.

Eckhardt, R. 1987. Stan Ulam, John von Neumann, and the Monte Carlo method. Los Alamos Science 15:131-137.

Kac, M. 1983. What is random? American Scientist (July/August):405-406.