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MatheMUSEments
What's the Deal?
By Ivars Peterson
Muse, May/June 2003, p. 23.
Have you ever been dealt a gin rummy hand and realized you
already had gin? Or an incredible run of hearts, so you were very
close even though you didn't quite have gin? Did you think you
were lucky? Or did you think that the dealer should have shuffled
more times?
Card players sometimes get lazy and fail to shuffle decks of
cards as fully as they should. That sloppiness leaves traces of
patterns in the order of the cards—patterns that experts
and gamblers can take advantage of to win more often than they
otherwise would.
When computer-shuffled decks were first used in bridge
tournaments, there was an outcry. The players thought there were
wild fluctuations in the distribution of cards of different suits.
Research showed the problem lay not in the computer but in the
players' expectations.
In bridge, cards tend to clump together in groups of four of
the same suit, and shuffling often didn't break up these groups.
In fact, the intuition of bridge players had been shaped by
generations of badly shuffled cards. Books on bridge recommended
strategies based on bad shuffles. When computer shuffling was
introduced, many of these strategies had to be changed.
How often should you shuffle a deck to be sure that the cards
are all mixed up? Many people think three shuffles are enough.
They're wrong. Statisticians David Aldous and Persi Diaconis have
studied shuffling, and they concluded that it takes about seven
riffle shuffles to put 52 cards in random order. In a riffle
shuffle, you cut the deck into two packets of cards, then holding
one packet in each hand, you run the cards past your thumbs to
raggedly interleave the cards.
Curiously, the transition from order to randomness occurs
quite abruptly. If you shuffle five times or fewer, the original
order disappears. You can see the same sort of sudden transition
in your kitchen when you stir together white flour and cinnamon.
At first you see thick streaks as the ingredients mingle. After a
few more strokes, the whole mixture suddenly smooths to a tan
color.
Not everyone agrees that you need as many as seven shuffles.
Other mathematicians, using different ways of measuring
randomness, say as few as five shuffles may work. Nonetheless,
it's pretty clear that three lackadaisical shuffles aren't enough
to truly mix up a deck of cards.
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