There are things people know and believe, and there are things people know and don't believe. It's just a question of finding the right subject - religion, politics, your favourite football team - we all have some vulnerable area where the brain and heart diverge. Then we follow the heart.
Like everyone else, roulette players went to school and heard the maths teacher say, 'a coin has no memory'. If heads comes up ten times in a row, that doesn't affect the odds on the 11th try. It's still 50-50, as always. And they know the same applies to the roulette wheel.
They know, for instance, that there is such a thing as the 'house edge'. The European (not American) wheel has 37 numbers, which are 0-36. Looking only at the 1:1 bets, those numbers are divided into Red and Black, Odd and Even, Upper Eighteen and Lower Eighteen. Except for the '0', that is, which falls into none of these groups. Bet on any of those 1:1 chances, and you lose your money if '0' turns up. And that's how the casino makes a profit. Without the '0' they would break even in the long term. Every one of those 1:1 chances would turn up the same number of times as every other. They would lose as many bets as they won. However, there is a '0', and it's going to turn up on average once every 37 spins. Or to put it another way, the casino is going to win 1/37 of all the money staked over the course of a year.
The reverse truth applies to the player. Luck exists, but only in the short term. In the long term there is only statistics. You can win ten coin tosses in a row, or you can lose ten - a variation of 0 - 100%, often done, often seen. Over a hundred tosses, that wouldn't happen, though you might get 25% - 75%, though not often seen. Over a thousand the range would be even shorter, probably 45% - 55%. As the total number of tries increases the results tend more and more closely to 50%. At the roulette wheel, flat-betting on Red for a year, you would come out even-steven versus Black. But not versus the casino - because the '0' steals your money one time in 37. So you, the average statistical player, lose 1/37 of what you staked over a year.
However, people continue to play roulette, and a lot of them are people who call themselves 'professional gamblers', who have 'researched' the matter and can back up every assertion with extraordinarily detailed extracts, sometimes detailed to the point of pedantry, from the statistical bible, including all the sub-sections of the 'laws of chance'.
Bending the Arithmetic
There are ways of outwitting statistical certainty. These are called 'systems', and a lot of them are the inventions of legendary players of past times, mostly from the 19th Century, whose names oddly conjure up memories of your school chemistry book - D'Alembert, Labouchere, and so on. In the past it was rather difficult in practice to test the claims of these systems, simply because of the difficulty of first collecting enough data and, secondly, analysing it with paper and pencil. But nowadays we have technology. There is one casino in Hamburg which publishes its daily run on the Internet, all archived, all downloadable. If we download some, and if one of us can program computers, we can run those spins through the program and see what happens. And if one downloads one week's worth (2,440 spins), and one has programmed a 'Martingale' system (named after some English milord), what one ends up with, apart from a twee personal pronoun, is something that constitutes a cautionary tale.
The Martingale system is the least regarded of all the systems, even though it can win a lot of money in quite a short time. And that's because it can lose a lot more in a much shorter time. Here's how it works: you stake one unit on Red; if it wins you win one; if it loses, you double, staking two. If it loses again, you double again, and keep on doubling until you win. So you might have this staking sequence, or 'progression': 1, 2, 4, 8, 16. If, say, the fifth of those is a winning spin, it covers all previous losses leaving a profit of one unit. Now, sitting down at the table on an average evening, you'll probably win steadily for half an hour, one unit at a time. Then you hit a row of ten blacks (often done, often seen), leaving you with losses of 1+2+4+8+16+32+64+128+256+512, supposing that they let you bet that much. But there is always a house limit on stakes, and you've probably already hit it. That will limit your losses somewhat, but it'll still wipe out everything you won up to that time, even if you were winning all evening. What players do, or all except a few lunatics, is impose their own limit - a 'stop-loss' - abandoning a progression at, say, 64, and starting again at 1. So, in practice, a system like this is not open-ended. The stop-loss is part of the system, and a system's profitability obviously depends on winnings from progressions that succeed outweighing losses from progressions that fail.
It seems to be loading the dice somewhat to use the most primitive of systems as the subject for analysis. But the reason for avoiding other systems is not that they are cleverer than the Martingale - they aren't - but because their complexities simply blow an obscuring fog over simplicities that should be revealed. As might become apparent, it doesn't actually matter what system you use.
Setting a 'stop-loss' of 64, and running all six 1:1 chances through the computer program - Odd, Even, Red, Black, Upper, Lower - we find five of the six with a loss after 2,440 spins. The one that shows a profit is Even. So that's the one to look at. We get this table:
With progression systems it's a habit to think linearly. We look at the progression itself, work out the profit on winning ones and the loss on losing ones, and maybe we use statistical laws (if we think we know them) to calculate how many winning runs we can expect in x spins. But it would be instructive to think laterally, crossways. At each staking level we can calculate profit/loss by multiplying the two Won/Lost scores by the stake and then subtracting one from the other. The first line gives a deficit of 1x(657-611)=46, the second line a deficit of 2x(314-297)=34, and so on.
This is where the delightful simplicities emerge. The bets at the lowest stakes are losing ones overall, for reasons given earlier. Where the sample is large enough, statistics determines the result, not luck. At stakes=1 we have 1268 spins, at stakes=2 we have about half that number - and those totals go on roughly halving as the stakes rise. And as they do our fortunes improve. We start breaking free of statistical certainty. At the lower stakes the sheer number of bets ensures that we lose. But at higher stakes the bets start thinning out, and then luck becomes the determining factor. It is in the rarefied atmosphere of the 16s and 32s and 64s that our fortunes turn. The reason that 'Even' showed a profit against the Hamburg spins (unlike the other five chances) is almost entirely due to the 51 bets laid at the two highest stake levels, which we won against the casino in the ratio 3-2. But it was luck that won it for us, not the system.
Why Do Casinos still Run?
If the truth be told, there's no such thing as a 'system'. That's an illusion created by thinking linearly. Thinking crossways, we see that our table is just a collection of bets. It doesn't actually matter in what order they are laid - your 'progression' means nothing. D'Alembert and Labouchere are no better than Martingale, for all their complexities. In the end they can all be reduced to a table just like this one, revealing the same thing: you can only win in the short term. Eventually, the 32s and 64s will run into the same problem as the 1s, 2s and 4s. Too many bets at the same stake: you lose.
Remember, though, that the final result over a year is that you lose 1/37 of your money.
Do the casinos know this? Yes, but like the players, they don't believe it. We tend to think of the casino owners as the clever guys and the players as the mugs. But that is not so, or not entirely. Casino managers are subject to the same superstitions as players. They cannot rid themselves of the notion that there might be such a thing as an unbeatable system. And that's why they are always on the alert, sidling up to people writing things in notebooks, watching for hand-signals across the floor, trying to spot the 'teams', those drilled squads with their pooled resources and pooled optimism who are going to 'take' Vegas in a single night. The casino bosses have seen all the movies too.