## Roulette System

### 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.

### How Casinos Struggle with a Roulette System

The result of applying the theory mentioned in the previous section would be encouraging, as it claims that the probability of winning a bet on even chances is over 99.9%. Not bad at all for game in a casino where one expects to take chances. But the problem is that it is impossible to put this brilliant method of enrichment into practice.

Gambling institutions have a simple way not to let a game transform into one where the solvent gambler would be practically “doomed” to win. Therefore, casinos limit the maximum bet alllowed.

In any casino in the world, every table – be it a roulette, blackjack or poker – specifies the rate of the minimum and maximum bet allowed. The range between them can be 10, 30 or even 100 times. Nonetheless, every table has a limit.

The restriction of the upper limit of bets is proof of the fact that a roulette system based on the principle of repeated increases in a bet are dangerous for the casino. Take, for example, a table where the minimum bet is €25, and the maximum is €1000. Why aren’t players allowed to bet more than €1000? Is it because they will not have enough money to pay? Or because the casino is afraid that players will win and then take the money home? What difference does it make, if in the neighboring VIP-hall €2,000 bets and even €10,000 bets are allowed! Real high-rollers can agree with the casino’s administration to make even higher bets. Clearly, there’s no shortage of cash at the casino. Another principle is at work: the ratio between the maximum and the minimum. If a maximum of €10,000 is established, the minimum bet will rarely be less than €250. Casinos don’t want to allow doubling to happen more than 5 times. Otherwise the players’ chances would become inadmissibly great.

Given this restriction at casinos, different types of a roulette system have been developed to use a small range bet variation. In order to insure that the bet stays for a long time between maximum and minimum, it is necessary to consider an arithmetic progression rather than a geometrical one. This increases the bet not by several times but in several units.

Let's consider one type of this roulette system, named after its author.

### The Thomas Donald Roulette System

The principal regulations of this roulette system are the following:

• For gaming players need to have 3000 times more capital than the initial bet.
• After each loss, the player should increase the following bet by one unit. After each win, the player should diminish the next bet by one unit.

This roulette system is based on the assumption by the author that during any certain interval of time – a day, week, month, or year – the number of losses and wins is approximately equal. The author promises winnings if the gambler will use his roulette system during such intervals of time and observe two more rules:

• Not to play if the player cannot freely dispose with his time within the target day or cannot afford betting 3000 times his initial bet.
• Not to play with someone else's money or with borrowed money.

#### Nekrasov’s Precepts

The last two items have no direct relevance to the Thomas Donald Roulette System and are more simply moral guidelines. However, we should note that these rules have a certain universal character. Many outstanding gamblers have commented on a mystical connection between their own attitude towards money and Luck. Even Russian poet N. A. Nekrasov, going to a big game, put the money he was prepared to loose in a separate pocket. «It is necessary to think of this money as if they are already lost,» he said. Luck loves a casual relationship towards money. If a player obsesses over every bet, if he takes money away from his family or any important affairs, he shouldn’t be playing.

Let’s now examine how the Thomas Donald Roulette System works in practice.

We will always bet on red, and our initial bet is €1. Assume that out of 37 spins, red is landed upon 18 times, black is landed upon the same number, and zero is landed upon once. Now assume red and black take turns in the following manner: 4 times red, 4 times black, 3 times red, 3 times black, 2 times red, 2 times black, down to one time each.

Bet № Winning Color Bet Win/Loss Balance
1 red 1 +1 +1
2 red 1 +1 +2
3 red 1 +1 +3
4 red 1 +1 +4
5 black 1 -1 +3
6 black 2 -2 +1
7 black 3 -3 -2
8 black 4 -4 -6
9 red 5 +5 -1
10 red 4 +4 +3
11 red 3 +3 +6
12 black 2 -2 +4
13 black 3 -3 +1
14 black 4 -4 -3
15 red 5 +5 +2
16 red 4 +4 +6
17 black 3 -3 +3
18 black 4 -4 -1
19 red 5 +5 +4
20 black 4 -4 0
21 red 5 +5 +5
22 black 4 -4 +1
23 red 5 +5 +6
24 black 4 -4 +2
25 red 5 +5 +7
26 black 4 -4 +3
27 red 5 +5 +8
28 black 4 -4 +4
29 red 5 +5 +9
30 black 4 -4 +5
31 red 5 +5 +10
32 black 4 -4 +6
33 red 5 +5 +11
34 black 4 -4 +7
35 red 5 +5 +12
36 black 4 -4 +8
37 zero 5 -5 +3

As a result we have won €3 even though we lost one more bet than we won. Note that the alternation of red and black was very unprofitable for us: for the first 4 times we won only €1 for every time. If the series had begun with 4 black, then the following 4 wins would bring not €4, but €14 (5+4+3+2). If we reversed the order that the red and black fell in the above series, we would have won €17.

### Variation on Thomas Donald – The Donald-Natanson System

Several years ago one of the authors of this article, a professional mathematician, made a critical change to the Thomas Donald System. He argued the following:

Assume one always bet on red and the initial bet is €1. After black turns up he increases his bet by a unit, and after red turns up he reduces his bet by a unit. But what should he do, if he have bet €1 on red and have won? According to T. Donald, the bet should remain invariable since there is no such thing as a zero-sum bet or a negative bet. “But why?” asked the mathematician. And when he analyzed the problem, he came to an interesting conclusion.

The literal application of his roulette system, of course, is impossible. If a player bets €1, the next bet should be equal to zero. According to Natanson, the zero bet is simple: the player passes during the next spin of the wheel. Nonetheless, the player plays as if he had bet on red. And the player must be alert to the results of the round, so as to know how to bet next time. Assume that the ball falls on red again. The player has won and now should reduce the bet again. The following bet (according to the roulette system) should be equal to -1.

And what is the negative bet on red? It is the bet on black! Therefore, whatever happens, there is only one rule:

• when black is fallen on, the bet increases, and when red is, the bet decreases.

Imagine, for example, that at the first three starts of roulette red turns up every time. After the first round we have won €1, we pass on the second round, and bet -€1 the third turn by placing a euro on black.

Before the fourth spin, we decrease the bet to -€2. We put €2 on black.

Therefore, it can be proven that from 2N starts of roulette red and black turn up N times the winnings will equal N initial units. Irrespective of number of times red is landed upon (and accordingly, black), the “property of invariance” holds true: the sequence in which red and black alternate does not influence the size of the win.

Let us assume that the roulette is started 36 times. Your income (positive or negative) is shown in the table.

Numbers of Times Red Wins Income
14 -22
15 -6
16 +6
17 +14
18 +18
19 +18
20 +14
21 +6
22 -6
23 -22

For example, if red is landed upon 20 times then with an initial bet of €1, player wins €14. If red is landed upon 17 times, the player also wins €14. Interestingly, the income distribution is symmetrical in the middle of the table.

The above table demonstrates that what happens when the frequencies of red and black differ is insignificant (with other types of roulette system, players would lose). Donald counted on the fact that over time the frequencies would be roughly similar. Natanson followed in his footsteps, but intensified the system.

Finally, we can’t forget zero.

According to Donald, the next bet should be risen when zero is landed on. In Natanson’s modification, it should be raised modularly. In other words, if the bet is positive, it should be raised by one and if it is negative it should be lowered by one. Unfortunately, the appearance of zero breaks the beautiful “property of invariance,” and makes it impossible to determine one’s income. However, consider what happens when zero turns up only once in 36 spins.

First, assume zero is fallen upon when the bet was positive. In this case, the zero is the equivalent to black, and therefore the income is defined according to the same table as above. For example, when red is fallen upon 20 times, black 15 times, and zero once, the winnings are €14. However, this doesn’t mean that zero does not have any influence: it reduces the expected number red wins.

Now, assume zero is fallen upon when the bet is negative. Now it is equivalent to red. If red is fallen upon 20 times, then because of zero the number of its occurrences equals 21. Instead of €14 (according to the table) we only win €6. But if red was fallen upon less than 18 times, our income would increase.

Finally, assume zero is fallen upon when there’s no bet. We can do whatever we like: with rise of the bet zero will be equivalent to black, with a reduction, it’ll be equivalent to red. However don’t forget the background: if red is fallen upon more often than black, it is necessary to raise the bet, and visa versa. The more frequently both colors turn up, the better. Mr. Donald would be pleased.

### Biarritz or the Makarov Roulette System

Another simple roulette system named after the French resort was proposed by Alexander Makarov, a well-known computer programmer who wrote the program «Marriage» and who used a mathematical method called Monte Carlo Modelling in his work. This roulette system is very aggressive.

The bet is always done on one and the same number, with a payout of 35:1. The bet is repeated with every loss. The bet’s size remains constant, for example, €1. The gambler finishes playing either after the first occurrence of his number, or after 36 unsuccessful starts. The following variations are possible:

• The chosen number turns up on the 36th spin. The player neither wins nor loses, since the €35 payout compensates for the previous 35 failures.
• The chosen number turns up earlier. The earlier it happens, the more the player wins.
• The chosen number does not turn up at all. The player loses €36.

The likelihood of the last outcome (36/37)36, or approximately 0.37. The chances that at some point the player’s number will fall is essentially above 50%. This is another system that counts on consistency “out of the gate”.

An older version of the Biarritz roulette system requires preliminary statistical research, and advises players to observe the course of the game during 111 starts (3 times about 37) and to bet on a number that was fallen on less than three times. From a mathematical perspective, this recommendation does not stand up to criticism since the ball does not have memory. At any moment, regardless of what happened earlier, all numbers are equally likely. On the other hand, statistical research sometimes reveals a poorly regulated roulette wheel upon which some numbers turn up less often than others or do not turn up at all. But in this case, it wouldn’t make sense to bet on those numbers which do not turn up because of some internal defects of the wheel.

### Other Versions of Martingale

We have already considered the probability of winning by results of several games by the continuous doubling of bets after each loss. The table below illustrates the financial implications nine consecutive losses preceding a win.

Bet Result Balance
1 1 - -1
2 2 - -3
3 4 - -7
4 8 - -15
5 16 - -31
6 32 - -63
7 64 - -127
8 128 - -255
9 256 - -511
10 512 + +1

While using this roulette system, it is necessary to remember the table’s limits. Let us modify the schedule for the table which has a minimum bet of €25.

Bet Result Balance
1 25 - -25
2 50 - -75
3 100 - -175
4 200 - -375
5 400 - -775
6 800 - -1575
7 1600 - -3175
8 3200 - -6375
9 6400 - -12775
10 12800 + +25

In general, any table where the minimum €25 is placed on the bet on even chances, the top limit on the same table is, as a rule, €1000. It is apparent from the above table that our experiment will finish after the 6th loss, or after losing €1,575. If we continue to play and, with no opportunity to double, simply make the maximum bet of €1,000 (based on the assumption that “our day will come”) on 7th, 8th and 9th throw, we lose another €3,000. Finally, on the 10th, we win back €1,000. Then we’d be in the hole €3,575.

If only one condition of the roulette system is impossible to fulfill, then the roulette system no longer works.

In systems like martingale rarely can be used the reverse of the principle. The player adds the sum of the initial bet to the next bet irrespective of the result of each spin. He plays on even chances. The player decides when to stop the game whenever he wants. Here is an example of such strategy:

Bet Result Balance
1 1 - -1
2 2 - -3
3 3 - -6
4 4 - -10
5 5 - -15
6 6 - -21
7 7 + -14
8 8 + -6
9 9 + +3

This type of roulette system has very little bearing on probability calculation, or on mathematics in general. What would the player do and which strategy will he select if he lost on 8th and 9th spin?

### Easy Does It

Here is an example of another roulette system based on the martingale principle. After each win the player remakes the initial bet. After each loss, the bet is doubled and increased by one unit. The player always bets on even chances.

Bet Result Balance
1 1 - -1
2 3 - -4
3 7 + +3
4 1 + +4
5 1 - +3
6 3 - 0
7 7 - -7
8 15 - -22
9 31 + +9
10 1 + +10

The author of this roulette system proceeds from the assumption that there are alternating series in the game. He tries to compensate the negative result of each losing series (black) with a win on the next one. It is easy to believe that with each winning spin, the player’s total winnings are equal to an initial sum at the beginning of the game.

But this roulette system is so aggressive that anyone using it risks hitting the table’s maximum limit too early. Assume that the 9th spin is a losing spin:

Bet Result Balance
... ... ... ...
8 15 - -22
9 31 - -63
10 63

The following bet should be 63 times higher than the initial one. If the minimum bet is €25, the new bet should be €1,575, which is much more than the table’s maximum. This roulette system can’t handle five consecutive loses.

### Cancellation Roulette System

This roulette system appears to be a little complex and intricate. In fact, the same principle of increasing bets after loses forms its basis. The player writes out a column of numbers from 1 to 10 and makes a bet equal to the sum of the top and the bottom line (11).

When the player wins, used numbers are scratched out, as sums up the following pair of numbers.

 1 2 3 4 5 6 7 8 9 10

However, the bet after each win remains 11. After a loss nothing is cancelled but the lowermost remaining number in a column increases by 11. The new bet is made with the same principle – the top number plus the bottom. If, for example, you win twice, and then lose, the column looks like that:

 1 2 3 4 5 6 7 19 9 10

The game continues until the all the numbers on the initial list are scratched out.

Bet Result Balance
1 1+10=11 + +11
2 2+9=11 + +22
3 3+8=11 - +11
4 3+19=22 + +33
5 4+7=11 - +22
6 4+18=22 + +44
7 5+6=11 - +33
8 5+17=22 + +55

It’s easy to see that after losses the increase in the initial bet in an arithmetic progression takes place. But if in Thomas Donald roulette system after a series of loses the decrease of a player’s stack of chips occurs smoothly and after a win the bet decreases by one unit, then in Cancellation roulette system the player continues to play from the initial large sum even after having won. Like in all roulette systems of this type, the prize is practically provided... if the player’s money doesn’t run out, and the size of the bet doesn’t exceed the limit established at the casino.