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Einstein is reported to have said:

No one can win at roulette unless he steals money from the table while the croupier isn’t looking

In contrast, the internet is full of playing systems that either guarantee you will win, or at least maximise your chances of doing so.

So who is right? Einstein or the internet?

Roulette – like most casino games – is a game with negative expected gain, whatever you choose to bet on. In European casinos a roulette wheel contains the numbers 1 to 36, as well as the number 0. That’s a total of 37 numbers. There are different type of bets available – betting on single numbers, betting on the colours of the numbers, of which – excluding zero – half are black and half are red, or betting on whether the result is odd or even etc. We’ll focus on bets on a single number, but the same argument applies to any type of bet.

The casino states odds for a single number bet of 35/1, which means that if you bet (say) $1 and your number comes up, you win $35 plus the return of your $1 stake. Otherwise you lose your stake. But since there are 37 numbers on the wheel, each of which is equally likely, your chance of winning is 1/37. So, the amount you expect to win with any such bet is:

In other words, for every $100 you gamble at roulette in this way, you will lose an average of $2.70. Had the stated odds been 36/1 you can easily check that the expected winnings would be zero, in which case the game is said to be fair. But casinos aren’t there to be fair, they’re there to make money. And by reducing the payout odds from 36/1 to 35/1 they are guaranteed to do so.

Incidentally, in the US and some other countries, the standard roulette wheel includes a ’00’ as well as a ‘0’, while the payout odds are kept the same. This means that your chances of winning are reduced to 1/38 and the expected loss per spin is almost doubled.

Now, you might get lucky, and win anyway. But the fact that you will lose in the long run, at a rate of $2.70 per $100 bet on a European table, is a mathematical certainty.

Except…. this assumes that successive spins of a roulette wheel are entirely unpredictable, even given the history of previous spins. And there are two reasons why, in theory at least, this assumption could be incorrect:

- There is a bias in the wheel – perhaps due to misalignment or wheel manufacture – which means that some outcomes are more likely than others;
- There is a serial dependence in the numbers, perhaps due to the pattern of ball spinning by the croupier, which means that given the sequence of previous spins, it’s possible to predict the outcome of the next spin with a better probability than that provided by completely random spins.

In practice, the precision of equipment used in casinos eliminates the first of these possibilities, while the chaotic physics of ball and wheel spin as well as the complexity of the roulette design – which includes deflectors that interrupt the ball’s trajectory as it slows – eliminates the second.

Nonetheless, there’s something of an industry on the internet – here and here, for example – of people trying to sell methods for predicting roulette outcomes based on the sequence of previous spins. Many of these even claim to be based on statistical analyses. In some cases the results they publish are impressive. But of course, it’s not clear how trustworthy these results are, nor indeed how they should be offset against the unknown results which are unpublished. Moreover, it seems reasonable to assume that if the developers of these methods had a foolproof system for winning at roulette, they might not need to be peddling the methods for a few dollars on the internet.

One method which has been shown to work, if conducted carefully, involves the use of cameras to monitor the ball and wheel speed, and a computer to calculate the updated probabilities of the outcome based on the visual information provided by the cameras. I’ll discuss this approach in a future post.

A different approach to serious gambling on roulette is the use of structured betting systems. The most famous one, the so-called martingale system, is to bet $1 on either red or black numbers (or odd or even numbers), which have odds of 1/1. So, if you win, you win $1 plus return of stake. Again, this type of bet has negative expected gain because the probability of winning is just 18/37, since 0 is neither red nor black. In fact, you can easily check that the expected loss for a $1 bet is $0.027, just like for a single number bet.

But the idea of the martingale system is this: if you win, you stop and keep your $1 profit. Otherwise, play again but raising your stake to $2. If you then win, you keep the $2 winnings, offset against the $1 lost in the previous round. So, you again win $1 overall. If you lose, you play again, raising your stake to $4. Again, if you win, you keep the $4, offset against $3 lost in the previous rounds, for an overall $1 win. And you keep playing this way, doubling your stake every time you lose, until eventually you win and stop, with guaranteed overall winnings of $1. Of course, you don’t have to start with $1. Start with $1000, doubling each time you lose, and your method would guarantee that you win $1000.

At least, that’s the theory. Maybe you can see the flaw in the approach. If not, drop me a line and I’ll explain.

But this raises a question: even though roulette has a negative expected gain on each spin, is there any betting strategy which could lead to an expected gain? For example, the gambler can choose when to play and when to stop, whereas the casino is obliged to accept all legally placed bets. So, one might ask whether, for example, a strategy where you decide to stop if you hit winnings above a certain level, and stop if you lose below a lower level, could possibly generate positive average wins.

Unfortunately, the answer to this also turns out to be no. There is a mathematical result called the Optional Stopping Theorem which basically says, if you have no method for predicting future results, there is no strategy which gives the gambler an expected gain. In other words, no matter how you play roulette, you will lose in the long run.

Damn Einstein!