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 European and American roulette:
Simple
bet

Probability
(odds)
European
roulette

Probability
(odds)
American
roulette

Payout

Straight
Up

1/37
= 2.70% (36 : 1)

1/38
= 2.63% (37 : 1)

35
to 1

Split
Bet

2/37
= 5.40% (17.5 : 1)

2/38
= 5.26% (18 : 1)

17
to 1

Street
Bet

3/37
= 8.10% (11.3 : 1)

3/38
= 7.89% (11.6 : 1)

11
to 1

Corner
Bet

4/37
= 10.81% (8.2 : 1)

4/38
= 10.52% (8.5 : 1)

8
to 1

Line
Bet

6/37
= 16.21% (5.1 : 1)

6/38
= 15.78% (5.3 : 1)

5
to 1

Column
Bet

12/37
= 32.43% (2 : 1)

12/38
= 31.57% (2.1 : 1)

2
to 1

Dozen
Bet

12/37
= 32.43% (2 : 1)

12/38
= 31.57% (2.1 : 1)

2
to 1

Colour
Bet

18/37
= 48.64% (1.0 : 1)

18/38
= 47.36% (1.1 : 1)

1
to 1

Even/Odd
Bet

18/37
= 48.64% (1.0 : 1)

18/38
= 47.36% (1.1 : 1)

1
to 1

Low/High
Bet

18/37
= 48.64% (1.0 : 1)

18/38
= 47.36% (1.1 : 1)

1
to 1

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
(straightup 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).
For
a given simple bet B, we can define the following function:
R,
, where R is the set of real numbers and
is the characteristic
function of a set:
can be also written as:
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
R,
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".
Here
are a few of the properties of the equivalence between 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 ~.
This
complex bet consists of a colour bet (payout 1 to 1) and several
straightup 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 straightup bets). S
is a positive real number (measurable in any currency), the coefficient c
is also a positive real number and n is a nonnegative 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 straightup 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.
The
overall winning probability is
.
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
noncontradictory. 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

.....................missing
part.....................
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

....................missing
part.....................
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:
Betting
on a colour and on numbers of the opposite colour

Betting
on a column and on outside numbers

Betting
on the third column and on the colour black

Betting
on streets and on the opposite of the predominant colour

Betting
on corners and on the opposite of the predominant colour

Betting
on lines and on the opposite of the predominant colour

Betting
on a colour and on splits of the opposite colour

Betting
on High/Low and on splits of low/high numbers


Repeated
bets
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, 505.77E6
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
5.3493E3, 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.
Therefore,
the probability of A occurring at least m times after n
spins is
.
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
nonempty cell.
....................missing
part.....................
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.
Sources 
The
entire mathematics of roulette, along with the main categories and
subcategories of improved betting systems whose data fill dozens of
tables, can be found in the book ROULETTE ODDS AND PROFITS:
The Mathematics of Complex Bets. The book presents a rigorous
mathematical model for the roulette bets, which can be generalized to
several types of betting. See the Books
section for details. 

