To study rifle dribbling accurately, Diaconis used a powerful mathematical tool called a Markov chain.
“A Markov chain is any repeated action where the result depends only on the current state and not on how that state was arrived at,” explains Sami Hayes Assaf, a mathematician at the University of Southern California. This means that Markov chains have no “memory” of what happened before. This is a very good example of shuffling, Assaf says. The result of the seventh shuffle depends only on the order of the cards after the sixth shuffle, and not on how the five playing cards were shuffled prior to that.
Markov chains are widely used in statistics and computer science to handle random sequences of events, whether they are shuffles of cards, vibrating atoms, or fluctuations in stock prices. In each case, the future “state” – surface arrangement, atom energy, stock value – depends only on what is happening now, not on what happened before.
Despite their simplicity, Markov chains can be used to make predictions about the probability of certain events occurring after many iterations. Google’s PageRank algorithm, which ranks websites in their search engine results, is based on the Markov chain that models the behavior of billions of Internet users who randomly click on web links.
Working with Dave Baer, a mathematician at Columbia University in New York, Diaconis showed that a Markov chain describing shuffles with guns has a sharp transition from ordered to random after seven shuffles. This behavior, known to mathematicians as a separating phenomenon, is a common feature of problems involving mixing. Think of stirring cream in coffee: While stirring, the cream forms thin white streaks in black coffee before mixing abruptly and irreversibly.
Knowing which side of the deck is on—whether it’s shuffled correctly or if it still retains some memory from its original arrangement—gives gamblers a clear advantage against the house.
In the 1990s, a group of students at Harvard and MIT were able to beat the odds of playing blackjack in casinos across the United States using card counting and other methods of detecting whether the deck had been mixed correctly. Casinos responded by introducing more sophisticated shuffle machines, shuffling before they were fully operational, as well as stepping up player monitoring. But it is still rare to see a deck of cards shuffled to the machine as required seven times in the casino.
Casino executives may not have paid much attention to Diaconis and his research, but he still has a tremendous influence on mathematicians, statisticians, and computer scientists who study randomness. At a conference at Stanford in January 2020 to honor Diakonis’ 75th birthday, colleagues from around the world gave talks about the mathematics of genetic classification, how grains settle in a shaker box, and of course shuffling cards.
Diaconis doesn’t care much about gambling himself – he says there are better and more interesting ways to make a living. But he does not envy players who try to gain an advantage using their brains.
“Thinking is not deception,” he says. “Thinking is thinking.”
*Shane Keating He is a science writerSenior Lecturer in Mathematics and Oceanography at the University of New South Wales, Sydney
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