## How does a roulette system operate?

Let’s be critical: we will consider several well-known types of a roulette system and analyze them from the mathematical point of view. First of all, we should ask a question: whether mathematics can help in principle?

Suppose that playing with me in chuck-farthing you want to win. It doesn’t matter how much, assume €1. Can you win for sure? The answer is: in the real life – yes, you can, only if you observe two conditions:

- if I take your rules of the game;

- if you have a significant capital, allowing to play according to a certain system.

You suggest me to toss up a coin and bet €1 that the heads will fall out. If you win, you reach your goal, and the game can be stopped at once. If the tails fell out, you bet again, but this time it will be €2 – that the heads will fall out. If after the second throw you hit the heads, then according to the results of two throws you win €1. If the tails will fall out once more, you bet €4... You proceed this way until the heads will fall out even one time. It can be easily assured that if you double your bets after each loss, the very first win will make your balance positive. It will be +€1.

What is the probability that the heads will never fall out? Let’s calculate. The probability that the heads won’t fall out after the first throw is 1/2. The probability that the heads won’t fall out neither after the first nor after the second throw is (1/2)^{2} or 1/4. Then the probability decreases in geometric progression. After three throws – 1/8, after four – 1/16... after ten – 1/1024.

In such a way, the probability that the heads will fall out even one time after 10 throws is more than 99.9%.

Can we claim that playing with me you will win €1?

Of course, we can: the probability of 0.999 is close to 100%. First of all, I must agree to play on these conditions, and secondly, you must have enough money: because if the heads won’t fall out before the tenth throw, you will pay me out €511 (1+2+4+8+16+32+64+128+256), and the bet size in the tenth throw will be €512.

The same is applied to roulette if you bet on so-called even chances: red or black, even or odd, 1-18 or 19-36. The only difference is that the probability that one of these chances will fall out is slightly less than a half – not 1/2, but 18/37 (we consider applying this system to roulette with one zero).

This system is calculated with the same strategy for several successive bets. Assume that you bet on the red. The probability that the red won’t fall out after the first spin is 19/37 or 0.513513. The probability that the red won’t fall out neither after the first nor after the second spin is (19/37)^{2} or 0.263696. Probability values for the majority of spins are given in the table:

The amount of spins | The probability that the red will never fall out |

1 | 0.513513 |

2 | 0.263696 |

3 | 0.135411 |

4 | 0.069535 |

5 | 0.035707 |

6 | 0.018336 |

7 | 0.009416 |

8 | 0.004835 |

9 | 0.002483 |

10 | 0.001275 |

As the table shows, the probability that the red will fall out even one time out of ten spins is almost thousand times as much as the probability that the black will fall out ten times in a row. To be more precise, the probability that the red will fall out even one time is 99.8725%.

The majority of types of a roulette system are based on this principle of repeated increases in the bet after a loss. Martingale is the most well known of this roulette system. In fact, Martingale is less a roulette system than a principle itself, and on this principle innumerable systems have been constructed, including types of a roulette system. Some suggest increasing the bet after each loss, others suggest on the inverse, or increasing the bet after a win, while a third type of roulette system apply more nuanced schemes. Below, we analyze some of the most interesting types of a roulette system and we shall test them in a simulated game.

It is worth noting that the word «martingale» has four different meanings. Its original meaning is a part of harness that would prevent a frightened horse from throwing its head back. This word was also used to mean the half-belt on a coat or overcoat. Game systems of the same name have “constraining” functions: they are intended to keep the puzzled player from panicking. Finally, the famous mathematician Paul Levy, who was studying the paradoxes of gambling at the beginning of the XX century, introduced the precise and complex term «martingale» into the theory of probability.

It is also interesting that all the systems (including types of a roulette system) based on the Martingale principle fall under the label of the D’Alembert system. This label is intended derisively, and takes its name from the great French mathematician and encyclopaedist Jean le Rond d’Alembert. Ironically, d’Alembert considered the use of his “law of balance” in game systems erroneous, since the law is true only for a continuous and infinite number of events, while any game consists of finite number of events and is limited by time and human perception.

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