The mathematics of roulette

We call a simple bet a bet that is made through a unique placement of chips on the roulette table. The table below notes the winning probabilities for each category of simple bet, for both American and European roulette:

Let us denote by R the set of all roulette numbers. Any placement for a bet is then a subset of R, or an element of P (R). Denote by A the set of the groups of numbers from R allowed for a bet made through a unique placement. A has 154 elements.

For example,

A (straight-up bet),

A (split bet),

A (corner bet),

A (odd bet),

A (the numbers 0 and 19 cannot be covered by an allowed unique placement).

We can define a simple bet as being a pair (A, S), where

A and S is a real number.

A is the placement (the set of numbers covered by the bet) and S is the basic stake (the money amount in chips). Because each simple bet has a payout defined by the rules of roulette, we can also look at a simple bet as at a triple

, where

is a natural number (the coefficient of multiplication of the stake in case of winning), which is determined solely by A. We have that

, according to the rules of roulette.

The probability of winning a simple bet becomes

, where

means the cardinality of the set A. Of course,

could be 38 or 37, depending on the roulette type (American or European, respectively).

, where R is the set of real numbers and

is the characteristic function of a set:

Function

is called the profit of bet B, applying the convention that profit can also be negative (a loss).

The variable e is the outcome of the spin. If

(the player wins bet B), then the player makes the positive profit

, and if

(the player loses the bet B), then the player makes a negative profit of – S (losing an amount equal to S as result of that bet).

Definition: We call a complex bet any finite family of pairs

with

A and

real numbers, for every

(I is a finite set of consecutive indexes starting from 1). Denote by B the set of all complex bets.

Definition: The complex bet B is said to be disjointed if the sets

are mutually exclusive.

Definition: Let

be a complex bet. The function

is called the profit of bet B.

Definition: A complex bet B is said to be contradictory if

for every

. This means such a bet will result in a loss, no matter the outcome of the spin.

Definition: The bets B and B' are said to be equivalent if functions

and

, as stair functions, take the same values respectively on sets of equal length. We write B ~ B'. This definition also applies to simple bets.

These are the basic definitions that stand for the base of the mathematical model of roulette betting. All about the complex bets, the profit function, the equivalence between bets and all their properties can be found in the book titled "ROULETTE ODDS AND PROFITS: The Mathematics of Complex Bets".

Statement 4: Two disjointed complex bets

and

for which

for every

are equivalent.

Statement 5: Let

be a simple bet and let

A such that they form a partition of

(

and

). Then:

if and only if S = T + R and

(

is the payout of

Statement 6: Let

be a complex bet. If

is a partition of

with

A and if

, then: B ~

Statement 10: The profits of two equivalent bets have the same mathematical expectation.

The proofs of these statements and other important results with direct application in the creation and management of the roulette betting systems are to be found in the book, along with examples and applications. This partition into equivalence classes of the set B of complex bets and the whole mathematical theory lead to the improved bets. A more precise definition for an improved bet is a bet obtained through a transformation upon an initial bet related to its stakes and/or placements, according to personal objective and/or subjective strategic criteria. A transformation is an act of choice over the equivalence classes of B or within a certain equivalence class. The mathematical theory of complex bets helps to restrain the area of choice and select the improved bets that fit a certain personal strategy.

This complex bet consists of a colour bet (payout 1 to 1) and several straight-up bets (payout 35 to 1) on numbers of the opposite colour. Let us denote by S the amount bet on each number, by cS the amount bet on the colour and by n the number of bets placed on single numbers (the number of straight-up bets). S is a positive real number (measurable in any currency), the coefficient c is also a positive real number and n is a non-negative natural number (between 1 and 18 because there are 18 numbers of one colour). The possible events after the spin are: A – winning the bet on colour, B – winning a bet on a number and C – not winning any bet. These events are mutually exclusive and exhaustive, so:

Now let us find the probability of each event and the profit or loss in each case:

A. The probability of a number of a certain colour winning is P(A) = 18/38 = 9/19 = 47.368%. In the case of winning the colour bet, the player wins cS – nS = (c – n)S, using the convention that if this amount is negative, that will be called a loss.

B. The probability of one of n specific numbers winning is P(B) = n/38. In the case of winning a straight-up bet, the player wins 35S – (n – 1)S – cS = (36 – n – c)S, using the same convention from event A.

C. The probability of not winning any bet is . In the case of not winning any bet, the player loses cS + nS = (c + n)S.

With this formula, increasing the probability of winning would be done by increasing n. But this increase should be done under the constraint of the bet being non-contradictory. Of course, this reverts to a constraint on the coefficients c. It is natural to put the condition of a positive profit in both cases A and B, which results in: n < c < 36 – n. This condition gives a relation between parameters n and c and restrains the number of subcases to be studied.

These formulas return the next tables of values, in which n increases from 1 to 17 and c increases by increments of 0.5. S is left as a variable for players to replace with any basic stake according to their own betting behaviors and strategies.

		Winning the bet on colour		Winning a bet on a number		Not winning any bet
n	c	Odds	Profit	Odds	Profit	Odds	Loss
1	1.5	47.36%	0.5 S	2.63%	33.5 S	50%	2.5 S
1	2	47.36%	S	2.63%	33 S	50%	3 S
1	2.5	47.36%	1.5 S	2.63%	32.5 S	50%	3.5 S
1	3	47.36%	2 S	2.63%	32 S	50%	4 S
1	3.5	47.36%	2.5 S	2.63%	31.5 S	50%	4.5 S
1	4	47.36%	3 S	2.63%	31 S	50%	5 S
1	4.5	47.36%	3.5 S	2.63%	30.5 S	50%	5.5 S

1	34	47.36%	33 S	2.63%	1 S	50%	35 S
1	34.5	47.36%	33.5 S	2.63%	0.5 S	50%	35.5 S
2	2.5	47.36%	0.5 S	5.26%	31.5 S	47.36%	4.5 S
2	3	47.36%	1 S	5.26%	31 S	47.36%	5 S
2	3.5	47.36%	1.5 S	5.26%	30.5 S	47.36%	5.5 S
2	4	47.36%	2 S	5.26%	30 S	47.36%	6 S
2	4.5	47.36%	2.5 S	5.26%	29.5 S	47.36%	6.5 S
2	5	47.36%	3 S	5.26%	29 S	47.36%	7 S
2	5.5	47.36%	3.5 S	5.26%	28.5 S	47.36%	7.5 S
2	6	47.36%	4 S	5.26%	28 S	47.36%	8 S
2	6.5	47.36%	4.5 S	5.26%	27.5 S	47.36%	8.5 S
2	7	47.36%	5 S	5.26%	27 S	47.36%	9 S
2	7.5	47.36%	5.5 S	5.26%	26.5 S	47.36%	9.5 S
2	8	47.36%	6 S	5.26%	26 S	47.36%	10 S
2	8.5	47.36%	6.5 S	5.26%	25.5 S	47.36%	10.5 S
2	9	47.36%	7 S	5.26%	25 S	47.36%	11 S

15	20	47.36%	5 S	39.47%	1 S	13.15%	35 S
15	20.5	47.36%	5.5 S	39.47%	0.5 S	13.15%	35.5 S
16	16.5	47.36%	0.5 S	42.10%	3.5 S	10.52%	32.5 S
16	17	47.36%	1 S	42.10%	3 S	10.52%	33 S
16	17.5	47.36%	1.5 S	42.10%	2.5 S	10.52%	33.5 S
16	18	47.36%	2 S	42.10%	2 S	10.52%	34 S
16	18.5	47.36%	2.5 S	42.10%	1.5 S	10.52%	34.5 S
16	19	47.36%	3 S	42.10%	1 S	10.52%	35 S
16	19.5	47.36%	3.5 S	42.10%	0.5 S	10.52%	35.5 S
17	17.5	47.36%	0.5 S	44.73%	1.5 S	7.89%	34.5 S
17	18	47.36%	1 S	44.73%	1 S	7.89%	35 S
17	18.5	47.36%	1.5 S	44.73%	0.5 S	7.89%	35.5 S

You will find the complete table in the book Roulette Odds and Profits: The Mathematics of Complex Bets, which also holds other important categories of improved bets, along with all their parameters:

Assume we place a colour bet. We denote by A the event a number of the chosen colour occurs. After each spin, event A may occur with probability p = 18/38 = 9/19 and does not occur with probability q = 1 – p = 10/19. The probability for event A to occur exactly m times in n spins (

) is given by the formula

, according to Bernoulli's formula.

The next table notes the numerical returns of this formula for n increasing from 10 to 100 spins in increments of 10. The numerical values are written in scientific notations. To convert them to decimal notations, we must move the decimal point to the left with the number of decimal places indicated by the number written after “E-“. For example, 505.77E-6 converts to 0.00050577, which means a 0.050577% probability. To use the table, choose the number of spins (n) and the number of occurrences (m) of the expected event. At the intersection of column n and row m we find the probability for that event to occur exactly m times after n spins. For example, if we want to find the probability of a red number occurring 15 times after 50 spins, we search at the intersection of column n = 50 and row m = 15 and find 5.3493E-3, which is 0.0053493 = 0.53493%.

It is helpful to find the probability of the expected event to occur at least a certain number of times after n spins.

Because events

are mutually exclusive, we can add their probabilities to find the probability of event A to occur at least a certain number of times.

In practice, in the table we must add the results of the column of the chosen n, starting from the row of the chosen m down to the last non-empty cell.

n m	10	20	30	40	50	60	70	80	90	100
1	14.679E-3	47.885E-6	117.15E-9	254.78E-12	519.44E-15	1.0167E-15	1.9346E-18	3.6062E-21	6.617E-24	11.992E-27
2	59.451E-3	409.42E-6	1.5289E-6	4.4713E-9	11.454E-12	26.992E-15	60.069E-18	128.2E-21	265.01E-24	534.23E-27
3	142.68E-3	2.2109E-3	12.842E-6	50.973E-9	164.93E-12	469.67E-15	1.2254E-15	2.9999E-18	6.9963E-21	15.706E-24
4	224.73E-3	8.4565E-3	78.017E-6	424.35E-9	1.7442E-9	6.0235E-12	18.473E-15	51.972E-18	136.95E-21	342.79E-24
5	242.7E-3	24.355E-3	365.12E-6	2.7498E-6	14.442E-9	60.717E-12	219.46E-15	710.98E-18	2.12E-18	5.9235E-21
6	182.03E-3	54.798E-3	1.3692E-3	14.436E-6	97.481E-9	500.91E-12	2.1397E-12	7.9986E-15	27.03E-18	84.409E-21
7	93.614E-3	98.637E-3	4.225E-3	63.108E-6	551.46E-9	3.4778E-9	17.607E-12	76.101E-15	291.93E-18	1.0201E-18
8	31.595E-3	144.26E-3	10.932E-3	234.29E-6	2.6677E-6	20.736E-9	124.79E-12	624.98E-15	2.7259E-15	10.673E-18
9	6.319E-3	173.11E-3	24.051E-3	749.72E-6	11.204E-6	107.83E-9	773.69E-12	4.4998E-12	22.352E-15	98.194E-18
10	568.71E-6	171.38E-3	45.456E-3	2.0917E-3	41.344E-6	494.93E-9	4.2476E-9	28.754E-12	162.95E-15	804.21E-18
11		140.22E-3	74.382E-3	5.1342E-3	135.31E-6	2.0247E-6	20.852E-9	164.68E-12	1.0666E-12	5.9219E-15
12		94.647E-3	105.99E-3	11.167E-3	395.77E-6	7.4409E-6	92.269E-9	852.23E-12	6.3194E-12	39.529E-15
13		52.42E-3	132.09E-3	21.647E-3	1.0412E-3	24.727E-6	370.49E-9	4.012E-9	34.125E-12	240.82E-15
14		23.589E-3	144.35E-3	37.572E-3	2.4765E-3	74.71E-6	1.3576E-6	17.28E-9	168.92E-12	1.3469E-12
15		8.492E-3	138.58E-3	58.613E-3	5.3493E-3	206.2E-6	4.5615E-6	68.43E-9	770.26E-12	6.9499E-12
16		2.3884E-3	116.92E-3	82.424E-3	10.532E-3	521.94E-6	14.112E-6	250.2E-9	3.2495E-9	33.229E-12
17		505.77E-6	86.661E-3	104.73E-3	18.957E-3	1.2158E-3	40.344E-6	847.73E-9	12.731E-9	147.77E-12
18		75.866E-6	56.33E-3	120.44E-3	31.279E-3	2.614E-3	106.91E-6	2.6703E-6	46.467E-9	613.26E-12
19		7.1873E-6	32.019E-3	125.51E-3	47.412E-3	5.2005E-3	263.34E-6	7.8424E-6	158.48E-9	2.382E-9
20		323.43E-9	15.849E-3	118.6E-3	66.139E-3	9.5949E-3	604.37E-6	21.527E-6	506.33E-9	8.6824E-9
21			6.7926E-3	101.66E-3	85.036E-3	16.448E-3	1.2951E-3	55.356E-6	1.519E-6	29.768E-9
22			2.5009E-3	79.018E-3	100.88E-3	26.243E-3	2.596E-3	133.61E-6	4.2877E-6	96.206E-9
23			782.9E-6	55.656E-3	110.53E-3	39.022E-3	4.8761E-3	303.24E-6	11.409E-6	293.64E-9
24			205.51E-6	35.481E-3	111.92E-3	54.142E-3	8.594E-3	648.16E-6	28.665E-6	847.88E-9
25			44.39E-6	20.437E-3	104.75E-3	70.169E-3	14.232E-3	1.3067E-3	68.108E-6	2.3198E-6

You will find the complete table in the book Roulette Odds and Profits: The Mathematics of Complex Bets, which holds all the categories of repeated bets, along with their calculations for both American and European roulette.

Author

The author of this page is Catalin Barboianu (PhD). Catalin is a games mathematician and problem gambling researcher, science writer and consultant for the mathematical aspects of gambling for the gaming industry and problem-gambling institutions.

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