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”.


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.

Glossary of blackjack terms


Blackjack: Name of game, also known as 21.

Anchorman Third Baseman: The player who sits on the last   chair and plays card last.One initial card with two cards (one Ace and one of 10 points -10 or picture-), which gives 3:2 of the bet placed by player.

Chips: Roundlike coins given by the   casino in exchange for money, to be used in the game.

Dealer: An employee of the casino who is responsible for BLACKJACK game at table. He deals cards, accepts and pays out on bets.

Doubling down: Double of player’s initial bet, who is to receive only one more card.

Ten points card: Ten points or picture card, value of ten points.

Twenty one: Another name of the game BLACKJACK.

Dangerous card: Any card which may bust (or burn). Like a  hard 12-16 (taking over 21).

Hole card: The face down card of the dealer.

Insurance: A bet made by the player when the dealer shows his first card   as Ace. This bet wins if the dealer makes BLACKJACK (and rest of the players bets lose).

Bust (or burn): When card is taken and more than 21 is calculated on hand.   Player or dealer lose.

Burning of card: The removal of the first card from the dealer in first round, before dealing begins.

Soft total/card: Cards that contain an Ace that is counted as eleven.

Card count: Observation of cards that have been already played. This is very difficult when a game is played with 6 sets of cards, but does happen in casinos.

Natural: Term meaning BLACKJACK. Many sets:
The use of more than one set of cards for BLACKJACK in casino.

Push: Tie between dealer and player, so money does not change hands.

Round: A full round in game so that all players and dealer plays their cards.

Shoe: A box containing 4-6 sets of cards which dealer uses for BLACKJACK etc.

Hard card: Total of players cards which don?t include an Ace, or include other countable as unit (not as 11).

Splitting pairs: Separation of two cards even value (e.g two eights) so they can be played as separate cards.

Standing, Standing Pat: Not taking other cards.

Upcard: The face up card of dealer which players see before playing their own cards.

Card/Hand: The total of cards which one player holds and will play with.

First Baseman: The player who is dealt first and plays first.Usually seated in the first seat at the table

Hit and draw: The taking of more cards for the initial card.

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