The University of Michigan Combinatorics Seminar
If a standard deck of 52 cards is repeatedly riffle shuffled, the probability distribution over the 52! possible orderings becomes closer and closer to the uniform distribution. In 1992, Bayer and Diaconis showed that 7 riffle shuffles mix a deck of 52 cards. The model of riffle shuffles and other assumptions behind this result will be described in the talk. For many card games, it is not necessary for all the 52! orderings to be equally likely. For example, in the game of bridge, if cards are dealt in some order to each of the four players, it is necessary that each player gets a random set of 13 cards for the game to be fair. But the order in which the 13 cards are dealt to a player is of no significance. Another example where all 52! orderings need not be equally likely is the game of black-jack, where the distinction between suits is ignored. We will consider these situations and show that, in fact, fewer than 7 riffle shuffles mix a deck in these situations. All new results presented in this talk were obtained in joint work with Mark Conger.