Journal of Theoretical Probability

, Volume 32, Issue 1, pp 417–446 | Cite as

Shuffling Large Decks of Cards and the Bernoulli–Laplace Urn Model

  • Evita Nestoridi
  • Graham WhiteEmail author


In card games, in casino games with multiple decks of cards and in cryptography, one is sometimes faced with the following problem: How can a human (as opposed to a computer) shuffle a large deck of cards? The procedure we study is to break the deck into several reasonably sized piles, shuffle each thoroughly, recombine the piles, perform a simple deterministic operation, for instance a cut, and repeat. This process can also be seen as a generalised Bernoulli–Laplace urn model. We use coupling arguments and spherical function theory to derive upper and lower bounds on the mixing times of these Markov chains.


Bernoulli–Laplace urn model Shuffling large decks Mixing times Path coupling Spherical functions Dual Hahn polynomials Gelfand pairs 

Mathematics Subject Classification (2010)

60C05 60J10 



We would like to thank Persi Diaconis for suggesting this problem, as well as for many helpful discussions and comments.


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Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA

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