# Casino outsmarted at probability

There's a great story in The Atlantic about a man who broke the bank at an Atlantic City casino by winning \$6m at Blackjack. He won without counting cards: the practice of remembering previously-played cards that allows the player's expected return to exceed 100%. (Casinos consider this practice cheating, and will ban players whose play reflects card-counting strategy.) Don Johnson's carefully-observed plays were dependent only on the cards played that hand, and so he was allowed to continue his winning streak.

And this was no mere lucky streak: Johnson had also taken home multi-million dollar winnings in other nearby casinos in the last month. He did it by exploiting the odds ... and the desperation of hard-up casinos eaget to compete for the business of (hopefully) lucrative high-rollers. By convincing casino bosses to offer him loss discounts and lenient game rules, he had tuned the tables on the house edge: with enough time, money and skill, he was guaranteed to win.

For example, at the Trop, he was willing to play with a 20 percent discount after his losses hit \$500,000, but only if the casino structured the rules of the game to shave away some of the house advantage. Johnson could calculate exactly how much of an advantage he would gain with each small adjustment in the rules of play. He won’t say what all the adjustments were in the final e-mailed agreement with the Trop, but they included playing with a hand-shuffled six-deck shoe; the right to split and double down on up to four hands at once; and a “soft 17” (the player can draw another card on a hand totaling six plus an ace, counting the ace as either a one or an 11, while the dealer must stand, counting the ace as an 11). When Johnson and the Trop finally agreed, he had whittled the house edge down to one-fourth of 1 percent, by his figuring. In effect, he was playing a 50-50 game against the house, and with the discount, he was risking only 80 cents of every dollar he played. He had to pony up \$1 million of his own money to start, but, as he would say later: “You’d never lose the million. If you got to [\$500,000 in losses], you would stop and take your 20 percent discount. You’d owe them only \$400,000.”

Figuring the odds of a table game is fairly straighforward: in a standard Blackjack, for example, the house expects to rake in a profit of 1% of all bets laid over the long run. The key there is the long run: i.e. over millions of hands played by players of varying skill levels and strategies. When it comes to figuring the expected profit that comes from attracting a skilled high-roller, things get much trickier. Casinos have computer models for to estimate the expected wins (or risk of losses) from high rollers, incorporating not just long-run probability and expected returns, but also aspects of game theory (will he quit or continue if he gets \$1,000,000 up? What about \$2,000,000 down?) and even physiology (will he play differently when he gets sleepy or hungry?). In this case, with the aggressive incentives and the modified rules, it seems like Johnson had the better model: “I just think somebody missed the math when they did the numbers on it,” he said.