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# Do Casino Dealers Shuffle Enough to Randomize the Cards?

16 September 2003

Computer-automated shufflers were tested at bridge tournaments in the early 1970s. Players reportedly were quick to notice, and gripe about, the highly variable distributions of cards in the hands they received. These folks weren't just imagining things. But the problem wasn't with computerization. It was with the manual procedures to which everyone was accustomed.

Experience with manual shuffling conditioned bridge players to expect lots of hands with three or four cards of each suit. The machines were dealing startlingly less-evenly distributed combinations. Scrutiny showed that during play, cards get bunched and collected by suit in sets of three or four. Manual shuffling didn't break up the clumps and randomize the cards especially well, so new deals retained vestiges of this regularity. In contrast, the computers delivered hands having wider variances.

The same basic dilemma arises in casino card games. Standard edge calculations presume random drawing from a deck or shoe. But, cards tend to get clustered and set aside in characteristic ways as the action unfolds. The question therefore arises: do the common manual shuffles adequately mix the cards, or do remnants of the ordering remain from previous play? And, if the latter, do any or all of the three following possibilities pertain? 1) Solid citizens may unconsciously develop instincts to optimize their play other than by rules that presuppose randomness. 2) Bettors may become more successful at anticipating cards to be drawn than expected by chance. 3) Card games may exhibit hot or cold phases more extreme or frequent than the laws of probability suggest.

Mathematicians have examined how thoroughly a stack of cards must be shuffled to be randomized in an operative sense. In what are generally considered the two classic analyses, the researchers reached somewhat different conclusions depending on how they defined randomness. In both cases, though, only "riffling" (splitting the stack roughly into halves, and "fanning" the piles together to interleave the cards) was employed for the shuffle.

The first of these studies measured randomness in terms of the probability that card sequences could be predicted after a shuffle from a knowledge of their prior order. Predictability remained high for small numbers of riffles, then dropped off sharply. The conclusion was that cards became randomized abruptly, for most practical purposes, after seven riffles for one deck, 11 for six decks, and 12 for eight decks.

The second of the investigations modeled randomness more abstractly, in terms of information theory. The "entropy" or uncertainty of a totally random deck was measured by the number of ways it could be arranged. Repeated riffling arithmetically increased uncertainty from zero to the maximum value. This approach yielded no sharp cut-off. Moreover, the point where the cards could be deemed effectively randomized was lower -- after five riffles for one deck and eight for six and eight decks.

Not as many riffles are needed for blackjack or minibaccarat than predicted by shoe size alone, because there are only 10 different elements -- nine occurring four times per deck and one 16 times. Possible arrangements are accordingly fewer than when each card is unique. The information model gives the entropy for six decks in these games as 931 bits; that of 312 distinct cards is 2,140 bits. At eight decks, the respective figures are 1,250 and 3,025 bits. Actual numbers of possible arrangements are astronomical: 16 followed by 279 zeros for six-deck blackjack or minibaccarat shoes and 21 followed by 643 zeros for 312 distinct cards.

How many riffles do they do at your favorite casino? It's usually no more than four. And, the coarseness and consistency of the interleaving has an impact. However, statisticians say casinos rarely randomize the cards. Can this phenomenon be exploited? I'll leave this, for now, lest you think we're into superstition. Which, strictly speaking, we are. Still, it pays to contemplate the counsel of the shufflers' Shakespeare, Sumner A Ingmark:

A theorist is oft a preacher,
Experience, an awesome teacher,
Wise bettors won't spurn either feature.