How Many Shuffles to Mix a Deck?

A simple probabilistic model is devised to determine the number of shuffles required for the bottom card of a deck to become uniformly distributed with a specified tolerance. This number is a lower bound on the number of shuffles needed for the entire deck to be randomly mixed. We wish to find a lower bound on the number of riffle shuffles required to mix a deck of k cards. In such a shuffle, the deck is cut at random and then the two parts are riffled together. For simplicity we shall assume that the location of the cut is uniformly distributed, so that the probability is 1 /(k 1) that the cut is just below any one of the first k 1 cards. Furthermore, we assume that the bottom cards of the two parts are equally likely to be the bottom card after the riffle. Therefore each has probability 1/2 of being the new bottom card. To obtain a lower bound on the number of shuffles, we consider the original bottom card. It will be the bottom card of one of the two parts after the first cut, so it has probability 1/2 of still being on the bottom after one shuffle. Therefore, the probability that it remains on the bottom throughout n shuffles is 1/2 n. When a 52-card deck is well mixed, the probability that any given card is on the bottom must lie between (1 e) and (1 + e) for some e > 0, which indicates how close the deck is to random. Therefore, for the deck to be mixed, it is necessary that 1/2," < (1 + e) so n > log2[ 52 20 .625, n must be > 6. ]. This shows that for e < For a deck of k cards this argument yields n > log 2 k log 2 (1 + e) , log 2 k e / In 2. The last expression holds for e << 1. We shall now calculate exactly the probability Pn that the original bottom card is on the bottom after n shuffles of a k card deck. To do so we observe that p,'+l is given by The first term in brackets is the probability that the card was on the bottom after n shuffles. The second term is the probability Pn that it was not on the bottom, times the probability that the deck …