A Tutorial-Based Approach to Teaching the Mathematics and Strategies
behind Gambling Payouts
M. SABRIGIRIRAJ
Department of Information Technology,
Hindusthan College of Engineering and Technology,
Coimbatore, Tamil Nadu,
INDIA
Abstract: – Roulette is a popular gambling game due to its simplicity and attraction of high payouts and it is
purely based on chance. It is essential to know about the various payout strategies adopted by gambling houses
before deciding to venture into any game of gambling. This tutorial paper mathematically analyzes the payout
strategies adopted by the gambling houses for the game Roulette based on simple probability concepts. From
the presented mathematical analysis, it can be easily verified that the payout strategy is fixed in favor of
gambling houses and there is rarely any chance for a gambler to win on a long-term average. Finally, to
maintain the neutrality of chance between the gambler and the gambling house, appropriate payouts are
proposed for one set of common bets and relaxations for another set of common bets in Roulette.
Key-Words: - Roulette, Probability, Gambling, Long term average, Payout, Casinos.
1 Introduction
Gambling games have a rich and diverse past that
spans across cultures and centuries, developing
from ancient rites to modern-day entertainment.
The roots of gambling can be traced back to ancient
civilizations. Dice, one of the initial known
gambling tools, dates to about 3000 BCE in
Mesopotamia. The Chinese were also initial
gamblers, developing rudimentary forms of lottery
games and games of chance as early as 200 BCE.
In ancient Rome and Greece, gambling was
both a widespread pastime and a root of concern,
leading to rules and prohibitions in certain times.
Card games appeared in China all through the 9th
century and progressively spread to Europe by the
14th century, developing into numerous forms
including modern poker and blackjack. The 17th
century saw the founding of the first casinos in
Italy, with the Ridotto in Venice opening in 1638 to
offer skillful gambling environments. The 19th and
early 20th centuries marked the upsurge of
gambling in the United States, where games like
roulette and slot machines added popularity in
saloons and later in Las Vegas, which became the
gambling capital of the world by the mid-20th
century. At present, gambling has extended into the
digital realm, with online casinos contributing a
wide range of games accessible globally.
Notwithstanding its development, the primary
appeal of gambling remains unaffected by the
excitement of chance and the confidence of
winning.
Roulette has been a widespread gambling game
since 1863 and it is pervasive everywhere in the
world, [1]. It is purely a chance-based game. The
game is very simple and cool to track, [2].
Gamblers throng to Casinos as the game seems to
be a charm and mystery with enthusiasm but not for
a person who can think rationally and precisely.
The rules for playing Roulette are almost alike
everywhere in the world. Since gambling strategy
is always designed in favor of the gambling house,
there certainly cannot be any method to reveal the
winning strategy. However, mathematicians have
widely used Roulette to teach probability theory [3]
and or propose strategies to increase the chances of
winning the games, [4]. A model with better
visualization of the betting game Roulette was
carried out in [5]. Roulette wheel game strategy
finds wide applications while solving various
optimization problems in engineering like test suite
minimization problems [6], declarative
programming [7], crowdsourcing [8], metabolic
pathway design [9], classification tasks [10], smart
building [11]. This paper is organized as follows:
Section 2 gives the playing methodology with a
Roulette wheel. In section 3, a mathematical
analysis is made which proves that playing with
Received: March 5, 2024. Revised: September 6, 2024. Accepted: October 7, 2024. Published: November 6, 2024.
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Roulette never pays off in the long run and
gambling strategies are fixed to favor gambling
houses. Section 4 discusses the results obtained and
section 5 concludes the paper and future research
avenues are indicated.
2 Roulette Wheel Game
Roulette is a wheel that houses 37 slots (numbered
from 0 to 36 randomly) as shown in Figure 1. Out
of 37 slots 18 are red, 18 are black and the
remaining 1 is green. In some places, the Roulette
wheel houses 38 randomly numbered slots with 18
red, 18 black, and 2 green. We consider only the
former and not the latter in the analysis. However,
the playing strategy is the same in both types. In
this game, a dealer spins the wheel in a particular
direction (clockwise or anticlockwise) and at the
same instant, a tiny metallic ball is made to roll in
the opposite direction, exactly around the wheel
rim. In this process, the ball moves fast initially but
after a time gap, comes to rest due to gravity. The
number and color at the resting place holds the key
to the game. A gambler is a winner if he is able to
predict exactly the number or color of the resting
place. Else, the gambler becomes a loser. That is,
the gambler can make a bet on the number or color.
Possible bets include predicting the color (either
red or black), odd or even number, low value (1-
18), or high-valued number (19-36), [3]. A gambler
is not permitted to bet for the number ‘0’ (or
‘green’) and if a ball lands here, the gambler loses.
This is the very first game strategy fixed to favor
the casino or the gambling house.
Fig. 1: A Roulette Wheel
3 Mathematical Analysis
Table 1 shows the payouts for various schemes of
this game. In this section, mathematical analysis is
done for the cases of betting on a number and
betting on a color. First, assume that a bet is made
on a number (the bet common name is straight up).
If the game is played, there are two possible
outcomes: either the ball lands on the bet number
(success/win) or the ball does not land on the bet
number (failure/loss). The payout for this type is
35:1 as dictated by the gaming house. In this
succeeding mathematical analysis, the initial
registration fee paid to become a member and enter
a gambling house or casino is ignored. In the case
of winning the bet, the gambler gets $35 and in
case of losing the bet, the loss is $1 for the gambler.
Let the sample space associated with this
experiment be {0,1,...,36}. This sample space can
be mapped to a Bernoulli trial with two possible
outcomes, one for success and another for failure. It
is to be noted that all Bernoulli trials are
independent trials. Thus, the outcome of this
experiment can be mapped onto a random variable
which can take a value ‘-1’ for a loss and ‘+35’ for
a win. Hence, Probability of win = 1/37 and
Probability of loss = 36/37. Also, the Probability of
win = (Number of wins) / (Number of games)
which needs to be equal to 1/37. For an easy
mathematical analysis, consider a long-term
average with 3700 games, then a number of wins =
100 and a number of losses = 3600, based on the
above probability values. The total amount likely
to be gained after playing 3700 games is (100 x
35)-(3600x1) = 3500-3600 = -100, which will
eventually be a loss for gamblers. It is to be noted
that for any number of games, there will always be
a loss for a gambler with this payout. Therefore, the
conclusion is to stop gambling as the gambling
houses can only end up making a profit in the long
term!
Table 1. Wages and Payouts for Roulette, [3]
What if the payout is changed to 36:1, which
may logically look truthful as the sample space has
37 elements out of which one element corresponds
to win and the rest corresponds to loss? In such a
scenario, the total amount likely to be gained after
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playing 3700 games would be (100 x 36)-(3600x1)
= 0, which leads to neither a win nor a loss for both
the gambler and the gambling house. However, a
gambler has a certainty to win on a long-term
average, only if the payout is 37:1, as (100 x 37)-
(3600x1) = 100.
Now, assume that a bet is made on the color
(let the bet's common name be Red, without loss of
generality). Payout for this type is 1:1 as dictated
by the gaming house and shown in Table 2. If the
game is played, there are two possible outcomes:
either the ball lands on the Red color (success/win)
or the ball does not land on the Red color
(failure/loss). It is to be noted that the gambling
house has fixed the payout safely in its favor. That
is, for a win, the ball needs to land on a Red color
but if the ball lands on either Green or Black color,
it is interpreted as a loss. After the gambler bets on
color and if the gambler wins the game, he gets $1.
If the gambler loses, his loss is $1. The sample
space is {Red, Not a Red}. However, this sample
space can be mapped to a Bernoulli trial with two
possible outcomes, one for success and another for
failure. Thus, the outcome of this experiment can
be mapped onto a random variable which can take a
value ‘-1’ for a loss and ‘1’ for a win. Hence,
Probability of win = 18/37
Probability of loss = 19/37
Probability of win = (Number of wins) / (Number
of games) = 18/37.
Consider a long-term average with 3700
games, then a number of wins = 1800 and a number
of losses = 1900, based on the same probability
values. So, the total amount likely to be earned
after playing 3700 games would likely be -100
(1800 x 1 – 1900 x1) It is to be noted that for any
number of games, there will always be a loss for a
gambler with this payout. However, if the gambling
houses ignore the scenario of the ball landing on
Green color as neither success nor failure, then
such a scenario puts both the gambler and the
gaming house at par with each other as the
probability of both success and failure are equally
likely. But, gambling houses won’t resort to such a
payout as it may not favor the gambling house.
4 Results and Discussion
A mathematical analysis carried out in the previous
section clearly reveals that payouts are fixed by the
gambling houses in favor of them. It also shows
that a gambler is more likely to lose on a long-term
average. In a game of chance, playouts should
favor neither gamblers nor the gambling house.
Fair payouts are proposed for one set of the bets
and relaxations required to maintain neutrality are
proposed for another set of the bets in Table 2.
Hence, it is to be clearly understood that stopping
gambling is the only option for a gambler to avoid
a loss as payouts favor only gambling houses.
Table 2. Existing payouts and proposed fair
payouts
Bet
Common
Name
Winning Spaces
Existing
Payout
Straight
Up
Any Single Number
Including ‘0’
35 to 1
Split
Any two adjoining
numbers vertical or
horizontal
17 to 1
Basket
(0, 1, 2) OR (0, 2, 3)
11 to 1
Street
Any 3 numbers
horizontal
11 to 1
Corner
Any four adjoining
numbers in a block
8 to 1
Six Line
Any 6 numbers from
2 rows
5 to 1
1st
Column
1,4,7,10,13,16,19,22,
25,28,31,34
2 to 1
2nd
Column
2,5,8,11,14,17,20,23,
26,29,32,35
3rd
Column
3,6,9,12,15,18,21,24,
27,30,33,36
1st
Dozen
1 through 12
2nd
Dozen
13 through 24
3rd
Dozen
25 through 36
Odd
1, 3, 5,…, 35
1 to 1
Even
2, 4, 6, …, 36
1 to 18
1, 2, 3,…, 18
19 to 36
19, 20, 21,…, 36
Red
Red numbers
Black
Black numbers
The mathematical analysis made in the
preceding section is applicable only to straight and
common bets in Roulette. Other common more
risky betting strategies include D'Alembert
systems, Martingale, and the Fibonacci. In the
D'Alembert system type bet, bet value is increased
or decreased by one unit based on the win or loss of
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the previous bet respectively. Another worse risky
bet is the Fibonacci strategy, which follows the
Fibonacci sequence to determine the bet size for the
current bet based on the sum of the immediate past
two bets. Finally, the worst risky bet is Martingale's
strategy, which involves doubling the bet after each
loss, aiming to recover all previous losses with a
single win. The chance of the above risky versions
of gambling strategies beneficial to gamblers may
happen only in case of equally likely payouts for
both gamblers and gambling houses.
5 Conclusion
In this paper, a mathematical study is carried out to
show that playing Roulette never pays off for a
gambler with the existing payouts. Further, it is
shown that gambling payouts are fixed in favour of
gambling houses and there is little chance for a
gambler to earn on a long-term average. An
interesting future work is analyzing other popular
gambling games and revealing how the payouts are
designed to favor gambling houses. Another
interesting and much-needed future work is to
review the application of Roulette and other
gambling games in optimization techniques across
multiple domains of science, engineering, and
technology. A much-awaited work is the
deployment of artificial intelligence and machine
learning principles to bring out profits for the
gambler with the existing payouts.
References:
[1] Stan Lipovetsky, “Mathematics of The Big
Four Casino Table Games: Blackjack,
Baccarat, Craps & Roulete, Technometrics,
Vol.65, No.4, 2023, pp.613-614.
https://doi.org/10.1080/00401706.2023.22628
98.
[2] Stan Lipovetsky, “Luck, Logic, and White
Lies: The Mathematics of Games”,
Technometrics, Vol.65, No.4, 2023, pp.604-
606.
https://doi.org/10.1080/00401706.2023.22628
89.
[3] Marshall, Jennings B., "Probability with
roulette", Teaching Statistics, Vol.29, No.3,
2007, pp.74-79.
https://doi.org/10.1111/j.1467-
9639.2007.00274.x.
[4] Croucher, John S, "A comparison of strategies
for playing even money bets in
roulette", Teaching Statistics, Vol. 27, No.1,
2005, pp.20-23.
DOI: 10.1111/j.1467-9639.2005.00193.x
[5] N. Kunanets, V. Morokhovych, V. Kut and D.
Popchenko, "Project for the development of
an online betting game “Live Roulette””, 2023
IEEE 18th International Conference on
Computer Science and Information
Technologies (CSIT), Lviv, Ukraine, 2023,
pp. 1-5.
DOI: 10.1109/CSIT61576.2023.10324019
[6] Hager Hussein, Ahmed Younis, Walid
Abdelmoez, "Quantum-inspired genetic
algorithm for solving the test suite
minimization problem", WSEAS Transactions
on Computers, Vol.19, 2020, pp.143-155.
DOI: 10.37394/23205.2020.19.20
[7] Golemanova, Emilia, and Tzanko Golemanov,
"Genetic Algorithms in a Visual Declarative
Programming”, WSEAS Transactions on
Information Science and Applications, Vol.19,
2022, pp.138-152.
DOI: 10.37394/23209.2022.19.14
[8] Gomez, Luis, Andres Rey, and Angel Lozada.
"Aggregation of results in Crowdsourcing by
means of an Evolutionary Algorithm that
calculates the Approximate Median
String", WSEAS Transactions on Computers,
Vol.18, 2019, pp.1-10.
[9] Cao, Yahui, Tao Zhang, Xin Zhao, Xue Jia,
and Bingzhi Li. "MooSeeker: A Metabolic
Pathway Design Tool Based on Multi-
Objective Optimization
Algorithm", IEEE/ACM Transactions on
Computational Biology and Bioinformatics,
vol. 20, no. 6, 2023, pp. 3609-3622.
DOI: 10.1109/TCBB.2023.3307363
[10] Li, Jingyi, Guocheng Liao, Lin Chen, and Xu
Chen. "Roulette: A Semantic Privacy-
Preserving Device-Edge Collaborative
Inference Framework for Deep Learning
Classification Tasks", IEEE Transactions on
Mobile Computing, vol. 23, no. 5, 2023, pp.
5494-5510.
DOI: 10.1109/TMC.2023.3312304
[11] El-Afifi, Magda I., Bishoy E. Sedhom,
Abdelfattah A. Eladl, Mohamed Elgamal, and
Pierluigi Siano., "Demand Side Management
Strategy for Smart Building Using Multi-
Objective Hybrid Optimization Technique",
Results in Engineering, Vol.22, Issue: June
2024, 2024, pp.1-16.
https://doi.org/10.1016/j.rineng.2024.102265.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author solely contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
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