Beneath the varnish of flashing lights and free cocktails, casinos stand on a bedrock of mathematics, engineered to slowly bleed their patrons of cash. For years mathematically inclined minds have tried to turn the tables by harnessing their knowledge of probability and game theory to exploit weaknesses in a rigged system.
An amusing example played out when the American Physical Society held a conference in Las Vegas in 1986, and a local newspaper reportedly ran the headline “Physicists in Town, Lowest Casino Take Ever.” The story goes that the physicists knew the optimal strategy to outwit any casino game: don’t play.
Despite the warranted pessimism about beating casinos at their own games, a simple betting system based in probability will, in theory, make you money in the long run—with a huge caveat.
Consider betting on red or black at the roulette table. The payout is even. (That means if you bet $1 and win, you win $1. But if you lose, you lose your $1.) And, for simplicity, assume that you really have a 50–50 shot of calling the correct color. (Real roulette tables have some additional green pockets on which you lose, giving the house a slight edge.) We’ll also suppose that the table has no maximum bet.
Here’s the strategy: Bet $1 on either color, and if you lose, double your bet and play again. Continue doubling ($1, $2, $4, $8, $16, and so on) until you win. For example, if you lose the first two bets of $1 and $2 but win your third bet of $4, that means you lose a total of $3 but recoup it on your win—plus an additional $1 profit. And if you first win on your fourth bet, then you lose a total of $7 ($1 + $2 + $4) but make out with a $1 profit by winning $8. This pattern continues and always nets you a dollar when you win. If $1 seems like a measly haul, you can magnify it by either repeating the strategy afresh multiple times or beginning with a higher initial stake. If you start with $1,000, double to $2,000, and so on, then you will win $1,000.
You might object that this strategy makes money only if you eventually call the right color in roulette, whereas I promised guaranteed profit. The chance that your color will hit at some point in the long run, however, is, well, 100 percent. That is to say, the probability that you’ll lose every bet goes to zero as the number of rounds increases. This holds even in the more realistic setting where the house enjoys a consistent edge. If there is at least some chance that you’ll win, then you will win eventually because the ball can’t land in the wrong color forever.
So should we all empty our piggy banks and road-trip to Reno, Nev.? Unfortunately, no. This strategy, called the martingale betting system, was particularly popular in 18th-century Europe, and it still draws in bettors with its simplicity and promise of riches—but it is flawed. Gambling ranked among the many vices of notorious lothario Jacques Casanova de Seingalt, and in his memoirs he wrote, “I still played on the martingale, but with such bad luck that I was soon left without a sequin.”
Do you spot a flaw in the profit-promising reasoning above? Say you have $7 in your pocket, and you’d like to turn it into $8. You can afford to lose the first three bets in a row of $1, $2 and $4. It’s not very likely that you will lose three in a row, though, because the probability is only one in eight. So one eighth (or 12.5 percent) of the time you’ll lose all $7, and the remaining seven eighths of the time you’ll gain $1. These outcomes cancel each other out: −1/8 × $7 + 7/8 × $1 = $0.
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