Alex's Adventures in Numberland - Alex Bellos [144]
When you are gambling in a casino, two factors come into play that determine how much money you will win or lose. Playing strategy is about how to win a game. Betting strategy is about money management – how much to bet and when. Is it worth, for example, betting your entire purse on one bet? Or is it worth dividing your money up into the smallest possible stakes? Different strategies can have a surprisingly large impact on how much money you can expect to make.
The best-known betting strategy is called the ‘martingale’, or doubling up, and was popular with French gamblers in the eighteenth century. The principle is to double your bet if you lose. Let’s say you are betting on the toss of a coin. Heads you win $1, tails you lose $1. Just say the first flip is tails. You lose $1. For the next bet you must stake $2. Winning on the second bet wins you $2, which recoups your $1 loss from the first bet and puts you $1 in profit. Say you lose the first five flips:
Lose $1 bet so next time bet $2
Lose $2 bet so next time bet $4
Lose $4 bet so next time bet £8
Lose $8 bet so next time bet $16
Lose $16 bet
You will be 1 + 2 + 4 + 8 + 16 = $31 down, so the sixth bet must be for $32. If you win you recoup your losses, and profit. But despite risking so much money, you are only ahead by $1, your original stake.
Martingale certainly has an appeal. In a game where the odds are almost 50:50 – like, say, betting on the red at roulette, which has a probability of 47 percent – you are very likely to win a fair percentage of plays and so have a good chance of staying ahead. But the martingale system is not fail-safe. For a start, you are only winning in small increments. And we know that in a run of 30 coin flips, a streak of five heads, or five tails, is more likely than not. If you start with a $40 bet and have a five-game losing streak, you will find yourself having to bet $1280. At the Peppermill Casino, though, you wouldn’t be able to – it has a maximum bet of $1000. One reason why casinos have maximum bets is to stop systems like martingale. The exponential growth of martingale bets on a losing streak often accelerates bankruptcy, rather than insuring against it. The system’s most famous champion, eighteenth-century Venetian playboy Giacomo Casanova, discovered this the hard way. ‘I still played on the martingale,’ he once said, ‘but with such bad luck that I was soon left without a sequin.’
Still, if you stood at the Peppermill roulette table playing martingale with a $10 starting stake on red, you would have to be very unlucky not to eventually win $10. The system would break down only if you lost six times in a row, and there is only a 1-in-47 chance of that happening. Once you have won, however, it would then be advisable to cash in your winnings and leave. By continuing to gamble, the chances of an unlucky streak will eventually become more likely than not.
Let’s consider a different system of betting. Imagine you are given $20,000 in a casino and told you must gamble it on red roulette table. What’s the best strategy for doubling your money? Is to be bold and bet the whole thing in one bet, or to be cautious and bet in the smallest possible amount, in stakes of $1? Even though it seems initially reckless, your chances of success are much better if you bet the whole amount in one go. In maths-speak, bold play is optimal. With a smidgen of reflection, this makes sense: the law of large numbers says that you will lose in the long run.