## Martingales And Other Illusions

A friend of mine who is passionate about casino games used to claim that it is possible to win at roulette . I told him he was wrong because it is mathematically proven that you cannot win. What was my surprise when he replied: “Whoever plays until he has nothing left will lose everything. So there is a way (at least) to play badly at the casino.

But then there is certainly also a way to play well. ” I was taken aback and I did not know how to answer him immediately. I have the spirit of the staircase, so I’m answering him here.

**RULES AND PROBABILITIES**

In any game, each time you bet, there is a probability p of winning and a probability q of losing your stake. It is certain that either we win or we lose! So the sum p + q is equal to 1, and q is equal to 1 – p. Suppose that, as with a coin toss, the one who wins gets his stake plus an amount equal to his stake; when you lose, you leave your bet to the other player, whom we will call the bank. In casinos, games of this type, such as red and black in roulette, give bettors a chance of winning less than 1/2. Gambling is then not fair, and it is suspected that this is why casinos are such profitable businesses. We will see in what follows that the profitability of casinos in the long term is guaranteed by mathematics.

**What is the probability p of the games offered to you? **If you find a friend who agrees to bank and you flip a coin with an untested coin, your probability p of winning mario88 casino is equal to 1/2. This is the same as if you were playing a flush at fair roulette. In simple roulette, if you play on black, you win when one of the 18 black numbers comes up; you lose when one of the 18 red numbers comes out and when the zero (which is green) comes out.

Thus the probability of winning p is equal to 18/37, or 0.4864…, because there are 37 numbers from 0 to 36. In French roulette , the rule is a little more complicated, because when the zero comes out your bet remains prisoner until the next throwing of the ball causes you to lose or recover it (without gain), and your probability of winning p is equal to 36/73, or 0.4931… (see the explanation of this 36/73 in Box 1). In American roulette there is a green zero and double zero, both bank-friendly, with no trap system. Your probability of winning p is 18/38, or 0.4736… There is also a Mexican roulette, with a triple zero, corresponding to a p = 18/39 = 0.4615, but it is for the gringos.

To calculate the bank’s payout, let’s imagine that $ 100 is wagered on black and count what, on average, the bank loses (i.e. pays) and wins. In a proportion of p cases, she loses 100 euros; in a proportion of 1 – p case, she earns 100 euros. Also the bank wins on average: 100 (1 – p) – 100 p euros, or 100 (1 – 2p) euros, which is sometimes called the mathematical expectation of winning of the bank for 100 euros wagered. As soon as p is less than 1/2, then 1 – 2p is positive, and therefore the bank wins on average.

This average bank win for 100 euros wagered is 0 francs toss (this game is not offered in casinos!); from 1.36 francs to French roulette (with the system of prisoner bets); 2.70 euros for single roulette, 5.26 euros for American roulette; of 7.69 euros for the Mexican.

Is it possible, by cunning behavior, to bypass this average theoretical gain of the bank and return it in its favor? The question is historical: the idea of studying the best ways of playing by the theory of probability is not recent. Pascal, d’Alembert, Moivre, Lagrange, Laplace and many other mathematicians, among the most eminent, were interested in it. If many results were obtained very early, even in this century, new principles allow a better understanding of roulette game strategies. To compare the different playing methods, called martingales, we put ourselves in the shoes of a player who initially has a capital of A euros and who wants to leave the casino as soon as he has B euros. How should he go about playing with black and red?

**THE STRATEGY OF CONSTANT BETS** : The strategy of constant bets consists in playing K euros at a time until either having lost your capital of A euros, or having reached the objective of B euros. This martingale is basic, yet many people use it.

**D’ALEMBERT’S MARTINGALE** :d’Alembert’s Martingale (or d’Alembert’s rising) consists each time you win to reduce the stake by K euros (because you think that luck has just been favorable to you, and that it is less likely to be the next move), and each time you lose to increase it by K euros (luck should now serve you better, do you think).

**THE GEOMETRIC MARTINGALE** : The geometric martingale is the most popular of all, as its principle is simple and, at first glance, infallible. This is La Martingale. We start by betting K euros; if you lose, you double the bet, and you double it that way until you win.

**THE THEOREM OF THE HARDI GAME** : The precept is moral, it is necessary to spend as little time as possible in front of the green carpet.