Randomization

There are exactly 52 factorial (expressed in shorthand as 52!) possible orderings of the cards in a 52 card deck. In other words there are 52 × 51 × 50 × 49 × ··· × 4 × 3 × 2 × 1 possible combinations of card sequence. This is approximately 8×10⁶⁷ possible orderings or specifically 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. The magnitude of this number means that it is exceedingly improbable that two randomly selected, truly randomized decks will be the same. However, while the exact sequence of all cards in a randomized deck is unpredictable, it may be possible to make some probabilistic predictions about a deck that is not sufficiently randomized.

Sufficient number of shuffles

The number of shuffles which are sufficient for a “good” level of randomness is a fundamental question, and the answer depends on the type of shuffle and the measure of “good enough randomness”, which in turn depends on the game in question. Broadly, for most games, four to seven good riffle shuffles (GRS) are both necessary and sufficient: for unsuited games such as blackjack, four GRSs are sufficient, while for suited games with strict conditions on randomness, seven GRSs are necessary. There are some games, however, for which even seven GRSs are insufficient.

In practice the number of shuffles required depends both on the quality of the shuffle and how significant non-randomness is, particularly how good the people playing are at noticing and using non-randomness. Two to four shuffles is good enough for casual play. But in club play, good bridge players take advantage of non-randomness after four shuffles, and top blackjack players supposedly track aces through the deck; this is known as “ace tracking”, or more generally, as “shuffle tracking”.

Research

Following early research at Bell Labs, which was abandoned in 1955, the question of how many shuffles was required remained open until 1990, when it was convincingly solved as seven shuffles, as elaborated below. Some results preceded this, and refinements have continued since.

A leading figure in the mathematics of shuffling is mathematician and magician Persi Diaconis, who began studying the question around 1970, and has authored many papers in the 1980s, 1990s, and 2000s on the subject with numerous co-authors. Most famous is (Bayer & Diaconis 1992), co-authored with mathematician Dave Bayer, which analyzed the Gilbert-Shannon-Reeds model of random riffle shuffling and concluded that the deck did not start to become random until five good riffle shuffles, and was truly random after seven, in the precise sense of variation distance described in Markov chain mixing timae; of course, you would need more shuffles if your shuffling technique is poor.Recently, the work of Trefethen et al. has questioned some of Diaconis’ results, concluding that six shuffles are enough. The difference hinges on how each measured the randomness of the deck. Diaconis used a very sensitive test of randomness, and therefore needed to shuffle more. Even more sensitive measures exist, and the question of what measure is best for specific card games is still open. Diaconis released a response indicating that you only need four shuffles for un-suited games such as blackjack.

On the other hand variation distance may be too forgiving a measure and seven riffle shuffles may be many too few. For example, seven shuffles of a new deck leaves an 81% probability of winning New Age Solitaire where the probability is 50% with a uniform random deck. One sensitive test for randomness uses a standard deck without the jokers divided into suits with two suits in ascending order from ace to king, and the other two suits in reverse. (Many decks already come ordered this way when new.) After shuffling, the measure of randomness is the number of rising sequences that are left in each suit.