Editing Shuffling (section)

==Randomization==
{{Redirect|52!|information on the mathematical function|factorial}}
There are 52 [[factorial]] (expressed in shorthand as 52[[factorial|!]]) 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 {{val|8.0658|e=67}} (80,658{{nbsp}}[[Names of large numbers|vigintillion]]) 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 [[Orders of magnitude (numbers)#1042 to 10100|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.

===Sufficiency===
The number of shuffles that are sufficient for a "good" level of randomness depends on the type of shuffle and the measure of "good enough randomness", which in turn depends on the game in question. For most games, four to seven riffle shuffles are sufficient: for [[suit (cards)|unsuited]] games such as [[blackjack]], four riffle shuffles are sufficient, while for suited games, seven riffle shuffles are necessary. There are some games, however, for which even seven riffle shuffles are insufficient.<ref name="vzs2004">{{Harv|Van Zuylen|Schalekamp|2004}}</ref>

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 [[contract bridge|bridge]] players take advantage of non-randomness after four shuffles,<ref name="kolata" /> and top blackjack players supposedly track aces through the deck; this is known as "ace tracking", or more generally, as "[[shuffle track]]ing".{{Citation needed|date=February 2007}}

===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.<ref name="kolata" /> Some results preceded this, and refinements have continued since.

A leading figure in the mathematics of shuffling is [[mathematician]] and [[stage magic|magician]] [[Persi Diaconis]], who began studying the question around 1970,<ref name="kolata" /><!-- 20 years before 1990 is 1970 --> and has authored many papers in the 1980s, 1990s, and 2000s on the subject with numerous co-authors. Most famous is {{Harv|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 [[Total variation distance of probability measures|variation distance]] described in [[Markov chain mixing time]]; of course, you would need more shuffles if your shuffling technique is poor.<ref name="kolata">{{Cite web
| title = In Shuffling Cards, 7 Is Winning Number
| first = Gina
| last = Kolata
| date = January 9, 1990
| work = [[The New York Times]]
| access-date = 2012-11-14
| url = https://www.nytimes.com/1990/01/09/science/in-shuffling-cards-7-is-winning-number.html
}}</ref> Recently, the work of Trefethen et al. has questioned some of Diaconis' results, concluding that six shuffles are enough.<ref>{{Harv|Trefethen|Trefethen|2000}}</ref> 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.{{Citation needed|date=February 2007}} Diaconis released a response indicating that you only need four shuffles for un-suited games such as [[blackjack]].<ref name=science_news>{{cite magazine
 | url = http://www.sciencenews.org/view/generic/id/38434/title/Shuffling_the_cards_Math_does_the_trick
 | title = Shuffling the cards: Math does the trick
 | date = November 7, 2008
 | access-date = 14 November 2008
 | magazine = Science News
 | archive-url = https://web.archive.org/web/20090111090252/http://www.sciencenews.org/view/generic/id/38434/title/Shuffling_the_cards_Math_does_the_trick
 | archive-date = 2009-01-11
 | quote = Diaconis and his colleagues are issuing an update. When dealing many gambling games, like blackjack, about four shuffles are enough.
 }}</ref><ref name=persi_at_stanford>{{cite web
 | url = http://www-stat.stanford.edu/~cgates/PERSI/papers/redblack.pdf
 | title = A Rule of Thumb for Riffle Shuffling
 | access-date = 14 November 2008
 | last = Assaf
 | first = Sami
 |author2=Persi Diaconis |author3=K. Soundararajan
  | publisher = t.b.a.
 }}</ref>

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.<ref name="vzs2004" /><ref>{{Harv|Mann|1994|loc=section 10}}</ref> One sensitive test for randomness uses a standard deck without the [[joker (playing card)|joker]]s 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.<ref name="vzs2004" />