''But they didn't think they were doing anything wrong. The players ''admitted to it readily,'' Dr. These players had figured out that the cards were not being randomly shuffled, and that they could predict the distributions of cards by knowing what the deck looked like at the end of the previous hand. After watching play ''and doing a little thinking in between,'' Dr. He said a bridge club in New York State once consulted him, as a magician, to find out whether several players were cheating. Diaconis has found that many bridge players take advantage of the non-randomness of seemingly shuffled cards. Kaplan said, ''I probably will move up from four to five'' shuffles, a decision which, the research shows, will not appreciably improve the randomness of the shuffled cards.ĭr. Kaplan replied, ''There will be a few who will be affected and will doggedly shuffle seven times to the irritation of everyone else.'' As for himself, Mr. Asked whether he expected bridge players to change their shuffling habits, Mr. ''We've tested these shuffles and feel that they are random,'' he said, adding that ''no one has ever complained.''īridge players usually shuffle about four times, except in some tournaments when a computer randomly mixes the cards, said Edgar Kaplan, who is editor and publisher of Bridge World magazine. ''I know people who are out there doing that now.''Īt Trump Plaza in Atlantic City, blackjack dealers shuffle eight decks twice at the beginning of each game, said Howard Dreitzer, who is senior vice president of casino operations. ''There are people who go to casinos and make money on this,'' Dr. The realization that most shuffled decks are not actually random allows gamblers to improve their odds of winning. ''Most people shuffle cards three or four times. The usual shuffling produces a card order that ''is far from random,'' Dr. Diaconis, who is also a magician, has invented numerous card tricks and has been carefully watching casino dealers and casual card players shuffle for the past 20 years. Persi Diaconis, a mathematician and statistician at Harvard University who is the other author of the discovery, said the methods used are already helping mathematicians analyze problems in abstract mathematics that have nothing to do with shuffling or with any known real-world phenomena.ĭr. David Aldous, a statistician at the University of California at Berkeley. The new result ''definitely solves the problem,'' said Dr. Other problems in statistics, like analyzing speech patterns to identify speakers, might be amenable to similar approaches, he said. Dave Bayer, a mathematician and computer scientist at Columbia who is a co-author of the recent discovery. No one expected that the shuffling problem would have a simple answer, said Dr.

Riffle cards