Casino Auto-Shuffler Fails Test Conducted By California Mathematicians

Written By Dave Champagne on October 25, 2022 - Last Updated on October 31, 2022
Mathematicians find flaw in auto-shuffler

Eager to maximize the number of blackjack hands that can be dealt per hour while minimizing payroll costs by hiring less-skilled people to deal their games, casino managers have enthusiastically embraced automatic card shufflers.

“I believe the deck is sufficiently randomized,” to quote Lt. Commander Data on “Star Trek: The Next Generation.”

Perhaps not.

Machines not so random after all

The original auto-shufflers were exposed several years ago when hackers used a hidden video camera to record the workings of the card shuffler through a glass window. The images, transmitted to an accomplice outside in the casino parking lot, were played back in slow motion to figure out the sequence of cards in the deck, which was then communicated back to the gamblers inside.

This proved costly. As noted in a recent BBC story, the damages ran into millions of dollars.

Manufacturers of the machines were not about to let that happen again. Enter Persi Diaconis, a professor at Stanford University. Diaconis, a mathematician with a background in magic (how appropriate!), is regarded as the world’s foremost expert on the mathematics of card shuffling.

Contacted by a manufacturer to evaluate its updated auto-shuffler, Diaconis eagerly jumped at the opportunity.

Review found a flaw in the system

Along with Stanford statistician Susan Holmes, Diaconis traveled to the company’s Las Vegas showroom to examine a prototype of their new machine. The pair soon discovered a flaw.

Although the mechanical shuffling action appeared random, the mathematicians noticed that the resulting deck still had rising and falling sequences, which meant they could make predictions about the card order.

Diaconis and Holmes devised a simple technique for guessing which card would be turned over next. If the first card flipped was the five of hearts, say, they guessed that the next card was the six of hearts, on the assumption that the sequence was rising. If the next card was actually lower – a four of hearts, for instance – this meant they were in a falling sequence, and their next guess was the three of hearts.

With this simple strategy, the mathematicians were able to correctly guess nine or 10 cards per deck – one-fifth of the total. That’s enough to double or triple the advantage of a competent card-counter.

Card-counting is a practice in which a player keeps track of which cards have been dealt in order to have a slight advantage predicting the probability that the next card is a winner or loser.

While legal, casinos tend to crack down heavily on card-counting at table games like blackjack. They think it diminishes the house’s advantage. The use of technology to assist a card-counter is outlawed.

Casinos use different types of shufflers

Tribal casinos are the only legal places to play Vegas-style blackjack in California. A different version where players play against each other can be found at numerous California cardrooms.

Some casinos in California use machines that shuffle one shoe’s worth of cards as a second shoe is being dealt. That way, dealers switch the cards only once a shoe is expended. A shoe generally holds one to six decks, but some hold up to eight.

Other casinos opt for different machines, like the auto-shuffler. With it, expended cards are simply fed back into the machine, so the shoe is endless. Players often blame these for long strings of “bad” cards, as cards are constantly recycled back into the game.

As simple – or complex – as you want to make it

A deck of playing cards contains a multitude of possibilities. A standard 52-card deck can be shuffled into an astronomical number of possible arrangements, or permutations.

There are 52 possible values for the first card, 51 for the second, 50 for the third and so on down through the deck. This is called the 52-factorial, or just 52.

Back to the drawing board

The manufacturer of the machines reacted predictably, with executives horrified.

“We are not pleased with your conclusions,” they wrote to Diaconis, “but we believe them and that’s what we hired you for.” The company quietly shelved the prototype and switched to a different machine.

Diaconis doesn’t care for gambling much himself. He says there are better and more interesting ways to make a living. But he doesn’t begrudge players who try to get an edge by using their brains.

“Thinking isn’t cheating,” he says. “Thinking is thinking.”

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Dave Champagne

Dave Champagne is a freelance writer with a wealth of varied experience with local newspapers, websites, and podcasts, gained in tandem with an ongoing I/T career spanning the better part of five decades. Based in upstate New York, Dave entertains himself with sports and horse race betting as well as the occasional attempt at tournament golf.

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