To rigorously study rifle mixtures, Diaconis used a powerful mathematical tool called the Markov chain.
“A Markov chain is a repeated action whose outcome depends only on the current state and not on how that state was reached,” says Sami Hayes Assaf, a mathematician at the University of Southern California. This means that Markov chains have no “memory” of what happened before. It’s a pretty good model for shuffling the cards, says Assaf. The outcome of the seventh shuffle depends solely on the order of the cards after the sixth shuffle, not how the deck was shuffled the previous five times.
Markov chains are widely used in statistics and computer science to deal with random sequences of events, whether card shuffling, vibrating atoms, or stock price fluctuations. In each case, the future “state” – the order of play, the energy of an atom, the value of an action – depends only on what happens now, not on what happened before. .
Despite their simplicity, Markov chains can be used to make predictions about the probability of certain events after many iterations. Google’s PageRank algorithm, which ranks websites in their search engine results, is based on a Markov chain that models the behavior of billions of Internet users randomly clicking on web links.
In collaboration with Dave Bayer, a mathematician from Columbia University in New York, Diaconis showed that the Markov chain describing riffle rearrangements has a sharp transition from ordinate to random after seven rearrangements. This behavior, known to mathematicians as a cutoff phenomenon, is a common feature of problems involving mixing. Remember to stir the cream in the coffee: when you stir, the cream forms fine white streaks in the black coffee before they suddenly and irreversibly mix together.
Knowing which side of the cut-off a deck of cards is on – whether it’s properly shuffled or if it still retains a memory of its original order – gives players a distinct advantage against the house.
In the 1990s, a group of Harvard and MIT students managed to beat the odds playing blackjack in casinos across the United States using card counting and other methods to detect whether the game was playing correctly. mix. Casinos responded by introducing more sophisticated card-shuffling machines and shuffling the game before it was fully played, as well as stepping up surveillance of players. But it is still rare to see a deck of cards shuffled seven times in a casino.
Casino executives may not have paid much attention to Diaconis and his research, but he continues to have a huge influence on mathematicians, statisticians, and computer scientists who study chance. At a conference held at Stanford in January 2020 to honor Diaconis’ 75th anniversary, colleagues from around the world lectured on the mathematics of genetic classification, how cereal settles in a shaking box and, well course, card shuffling.
Diaconis isn’t very fond of the game itself – he says there are better and more interesting ways to make a living. But he doesn’t blame players who try to gain an advantage by using their brains.
“Thinking is not cheating,” he says. “Thinking is thinking.”
* Shane Keating is a science writer and sLecturer in Mathematics and Oceanography at the University of New South Wales, Sydney
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