This article is published in the April/May 2012 issue of MAA FOCUS.
Mathematical Reviews and its online version MathSciNet provide timely reviews and abstracts of articles and books that contribute to research in the mathematical sciences. Published monthly in print and continuously updated online, they are produced by the Mathematical Reviews Division of the American Mathematical Society.
Conger, Mark A.(1-MI); Howald, Jason(1-SUNYPD)
A better way to deal the cards. (English summary)
Amer. Math. Monthly 117 (2010), no. 8, 686–700.
Most models of the randomization of a deck of cards by shuffling assume that the cards in the deck are distinct, and measure the deviation from the uniform distribution on all orderings of the permutation. For example, the estimates from the standard model of riffle shuffling [D. Bayer and P. W. Diaconis, Ann. Appl. Probab. 2 (1992), no. 2, 294−313; MR1161056 (93d:60014)] use these assumptions. The authors consider decks with repeated cards, or decks in which the cards are dealt into hands as in most card games. They compute asymptotic formulas for both cases. In particular, the amount of shuffling needed for randomization is strongly dependent on how the cards are dealt; cutting a poker deck properly improves the randomization significantly, dealing bridge cards cyclically improves the randomization by a factor of 13 over cutting the deck into four piles (asymptotically; Monte Carlo simulations show that the asymptotic is close after five riffles), and dealing 1-2-3-4-4-3-2-1 improves the randomization by a further factor of 13 (asymptotically).
Reviewed by David J. Grabiner
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Adapted from an article by Michael A. Jones that appeared in the April 2010 Michigan Section−MAA Newsletter (pdf).
Michael A. Jones has been an associate editor at Mathematical Reviews since 2008, after 14 years as a faculty member at various institutions. He is a member of the editorial board of the College Mathematics Journal.
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