Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

May 25, 2007

What's the Deal?

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.

Muse, May/June 2003, p. 23.

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