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# A Comparison of Strategies for Playing Even Money Bets in Roulette

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This article discusses some strategies for playing roulette, making use of the binomial distribution and Normal approximation.
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Teaching Statistics. Volume 26, Number 3, Autumn 2004
20
TEST_155 Pages:
4
Yang wei
Blackwell Publishing, Ltd.Oxford, UKTESTTeaching Statistics0141-982X© Blackwell Publishing 2004263Original Article
A Comparison of Strategies for Playing Even Money Bets
in Roulette
KEYWORDS:
Teaching;
Gambling;
Strategy;
Binomial approximation.
John S. Croucher
Macquarie University, Sydney, Australia.
e-mail: jcrouche@gsm.mq.edu.au
Summary
roulette, making use of the binomial distribution
and Normal approximation.
INTRODUCTION
T
he popularity of roulette was established in
1863 with the opening of the Monte Carlo
Casino and today it is one of the most widely
played casino games around the world, although
the rules are slightly different between countries.
There is no shortage of specialist books on the
market that provide speciﬁc advice for playing
roulette, with statistical analysis being provided in
Croucher (2003), Epstein (1977) and Malmuth
(1994). None of these has been able to uncover
the elusive ‘winning strategy’, since unfortunately
one does not exist. However, they do explain why
it is difﬁcult to win in the long run. Roulette, as with
most games of chance, also provides an excellent
teaching opportunity in demonstrating an interest-
ing and practical use of the Normal approximation
in the calculation of binomial probabilities.
In Australia, the UK and Europe, roulette in-
volves a wheel with 37 numbered slots consisting
of 18 red slots, 18 black slots and 1 green slot. (In
the USA the wheel has 38 numbered slots consist-
ing of 18 red slots, 18 black slots and 2 green slots.
This type of wheel is not considered here.) In each
game the wheel is spun by a dealer in one direction
while at the same time a small metal ball is rolled
in the opposite direction around the rim of the
wheel. After a few laps of the wheel, gravity forces
the ball to drop to the bottom of the wheel and it
lands on a number (and colour) that is declared
the winner.
Although there are many types of bet that a player
can make in roulette, the ones of interest here are
the popular ones that pay even money if success-
ful. That is, you get double your money back if
you win or nothing if you lose. These include:
Red or black
(betting whether the number spun will be a
red
one
or a
black
one)
Odd or even
(betting whether the number spun will be an
odd
one
or an
even
one)
Low or high
(betting whether the number will be in the
low
range
1–18 or
high
range 1936).
All of these bets lose if the number spun is the
green zero. Since there are 37 equally likely pos-
sibilities, it is easy to see that the probability that
any one of the above bets is successful on a single
spin is 18/37 or 0.486486. An important issue for
gamblers is just how they can most effectively
invest their money. Given that they have decided
to play one or a combination of the above options,
what is their chance of winning a speciﬁed amount
and how should they go about doing it?
EXACT BINOMIAL PROBABILITIES
When a player bets at roulette on a particular spin,
the result is either ‘win’ or ‘lose’. That is, since
there are only two possible outcomes, it is ideal to
represent the situation by a binomial distribution.
The scenario considered here is the following.
•A player starts the evening with a total of
n
units to invest playing roulette. (A
unit
is a sum
of money such as \$5, \$10, \$25, \$50 etc.)
The player wishes to make a minimum proﬁt of
t
units at the end of play.
Each bet consists of
c
units.
It follows that the maximum number of bets that
the player can make is [
n
/
c
], that is, the integer
part of
n
/
c
. To simplify matters, it can be assumed
that
n
is a multiple of
c
, so that the player simply
makes a total of
n
/
c
bets. In a general binomial
Teaching Statistics. Volume 26, Number 3, Autumn 2004
21
situation, the probability of exactly
r
successes in
a series of
n
independent binomial trials where the
constant probability of success at any one trial is
p
is given by
(1)
where
r
=
0, 1, 2, 3, . . . ,
n
.
Roulette certainly satisﬁes the necessary assump-
tions to use this formula, since wheel spins are
independent and there is the same chance of being
successful on any spin.
If the player wishes to be at least
t
after a total of
n
/
c
bets each of one unit, then a
total of at least (
n
+
t
) units is required at the
end. Since a successful bet will return precisely
2 units in any of the above gambling options, the
minimum number of successful bets required is
(
n
+
t
)/2. (If the value (
n
+
t
)/2 is not an integer
then it must be raised to the next highest integer.)
Using equation (1), the
exact
chance of this hap-
pening is
(2)
In the case of the gambling options being con-
sidered, the value of
p
is 18/37. The arithmetic
involved in performing the calculations in equa-
tion (2) is extremely tedious and time consuming,
requiring the aid of a statistical computer program.
This makes it useless for the average gambler.
Fortunately, there is a much easier way of arriving
NORMAL APPROXIMATION
TO THE BINOMIAL
It is well known that in certain circumstances rea-
sonable approximations to binomial probabilities
can be found by using the Normal distribution. A
rule of thumb here is that the values of both
np
and
n
(1
– p
) should be at least 5, but even if this
is not true good approximations can nevertheless
still be found. To make the approximation even
more accurate, a
continuity correction
of 0.5 is
subtracted from the numerator of the appropriate
Z
quantity. The required formula is
(3)
In the problem at hand,
r
=
(
n
+
t
)/2 and
p
=
0.4865. Substituting into equation (3), the required
approximating formula is
(4)
To test the accuracy of this approximation, the
following example compares the exact solution
with that given by equation (4) in a speciﬁc case.
EXAMPLE
Consider a gambler who has 200 units to spend
playing the
red or black
option on roulette. Sup-
pose the gambler wishes to devise a strategy that
will maximize the chances of winning at least 40
units (i.e. a proﬁt of at least 20%) and wants to
compare the probabilities of doing so for each of
the following seven strategies:
betting 1 unit on each of 200 spins
betting 2 units on each of 100 spins
betting 4 units on each of 50 spins
betting 5 units on each of 40 spins
betting 10 units on each of 20 spins
betting 20 units on each of 10 spins
betting 40 units on each of 5 spins.
For each case, the relevant values of
n
and
t
are
shown in table 1. The ﬁnal column shows the
approximated values, using equation (4), of the
probabilities that the gambler will be ahead by at
least 40 units. These require only the calculation of
the appropriate
z
-value and looking up a single
value from standard Normal distribution tables.
The second-last column, found using the statistical
package Minitab, shows the
exact
probabilities
calculated using equation (2). There is virtually no
difference between these two columns to three
decimal places, so that the approximation is
extremely accurate in this case.
The probabilities in table 1 provide interesting
reading and also serve to illustrate an important
Pr n
rpp
rnr
( ) ( )exactly successes =
1
Pnt n
n
rpp
rnr
rnt
n
( ( )/ )
( )
()/
at least successes in spins+
=
≥+
2
1
2
Pr n
PZ rnp
np p
( )
.
( )
at least successes in trials
≈>
−−
05
1
Ptn
PZ tn
n
( )
. . .
.
player will be at least units ahead after spins
≈>
+−
05 00135 0 5
0 2498
22
Teaching Statistics. Volume 26, Number 3, Autumn 2004
statistical principle of gambling. As the number of
decreases
(with a corresponding increase
in bet size), the probability that the gambler will
reach the target of at least 40 units ahead
increases
dramatically. There is essentially
no chance
of getting
there by betting one unit at a time.
On the other hand, the bolder options represent
far better propositions, such that 5 bets each of
40 units gives a 47% chance of success. Having said
that, what of the boldest strategy of all, namely
placing the entire 200 units on one spin? The
chance of success here is 18/37 or 0.4865, a ﬁgure
higher than for any of the above strategies!
So if it is optimal to place the entire stake on one
spin, why doesn’t every player do this? The reasons
may vary, but most likely a major factor is that
there is no ‘entertainment factor’ in doing so. That
is, if you lose all your money on the ﬁrst spin then
your gambling night is ﬁnished, and most people
will not take that risk to spoil the evening. Another
is that gamblers almost certainly do not know
about the relevant probabilities involved anyway.
THE PROBABILITY OF
Many gamblers who do try to have an evening’s
entertainment would also like to go home at least
breaking even. That is, go home with their original
stake or better. The chance of at least breaking even
can easily be calculated by setting
t
=
0 in equation
(4). The probability of actually being ahead can be
found by adjusting the number of wins required by
adding one win to the break-even point.
The relevant calculations are shown in table 2,
from which it is once again clear that betting in
larger amounts increases the chance of at least
breaking even. In fact, ten bets each of twenty units
yields a 60% chance, while betting one unit at a
time drops it to less than 40%. A signiﬁcant part
of these probabilities is assigned to exactly break-
ing even, as can be seen in the ﬁnal column which
gives the probabilities of actually ending up a
winner. There is not a huge difference here, although
it is interesting to note that the worst chance still
comes from betting one unit at a time.
REMARKS
The probability approximations discussed in this
paper apply to a wide range of gambling oppor-
tunities including craps, two up, Keno and other
games where the outcomes are independent and
the probability of success on each play is the same.
They can also be used to compare other betting
options within roulette itself, such as column bets,
street bets, dozen bets and straight-up bets.
Table 1. The probability of being at least 40 units ahead for various betting options with a 200 unit stake
Size of bet
(units)
No. of bets
Minimum no.
of units needed
Minimum no.
of wins needed
Probability of being at least
Binomial
exact
Normal
approximation
1 200 40 120 0.0008 0.0008
2 100 20 60 0.0148 0.0150
45010300.0714 0.0715
5408 24 0.1005 0.1006
10 20 4 12 0.2144 0.2142
20 10 2 6 0.3444 0.3439
40 5 1 3 0.4747 0.4759
Table 2. The probability of at least breaking even and being ahead for various betting options with a 200 unit stake
*It is not possible to exactly break even with this option. At least three wins are required from ﬁve plays.
Size of bet
(units)
No. of bets
Minimum no.
of wins needed to
at least break even
Probability
of at least
breaking even
Minimum no.
of wins needed
Probability of
1 200 100 0.3775 101 0.3251
2 100 50 0.4321 51 0.3553
450250.4797 26 0.3695
540200.4943 21 0.3708
10 20 10 0.5499 11 0.3650
20 10 5 0.5994 6 0.3441
40 5 3 0.4747* 3 0.4747*
1
Teaching Statistics. Volume 26, Number 3, Autumn 2004
23
As stated previously, the formulae provided apply
to the Australian and European versions of roulette
where there is only
one
green zero slot on the wheel.
In the case of the US version where there are
two
such zeros (called 0 and 00), the only change required
is that now
p
=
18/38
=
0.473684 in equation (3),
with a corresponding adjustment to equation (4).
It is therefore clear that players have a smaller
chance of winning in this version, whatever option
they choose to play.
Acknowledgement
The author acknowledges the support of this
research that was undertaken while a Visiting
Professor in the School of Economics, Mathem-
atics and Statistics at Birkbeck College, University
of London.
References
Croucher, J.S. (2003). Gambling and Sport: A
Statistical Approach. Sydney: Macquarie
Lighthouse Press.
Epstein, R. (1977). The Theory of Gambling
and Statistical Logic. New York: Academic
Press.
Malmuth, R. (1994). Gambling Theory and
Other Topics (4th edn). Two Plus Two.
THANKS TO THE
REFEREES
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Editor
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... Schilling (2012) provides approximation formulas for the longest run of red or black. Other simulations of a roulette wheel can be found, e.g., in Turner (1998), Croucher (2005 or Kendall (2018). The probability of a negative total profit is 80 percent. ...
... We will investigate how well a player using a doubling strategy would compete compared to a player betting a comparable amount on a simple chance and a player betting a comparable amount on single number each time (plein). See also Turner (1998) or Croucher (2005). Assumptions: a) Martingale player: Initial bet is one unit, doubling after each loss up to a table limit of 512 (a maximum of 9 doublings). ...
Conference Paper
Full-text available
Some gamblers use a martingale or doubling strategy as a way of improving their chances of winning. This paper derives important formulas for the martingale strategy, such as the distribution, the expected value, the standard deviation of the profit, the risk of a loss or the expected bet of one or multiple martingale rounds. A computer simulation study with R of the doubling strategy is presented. The results of doubling to gambling with a constant sized bet on simple chances (red or black numbers, even or odd numbers, and low (1-18) or high (19-36) numbers) and on single numbers (straight bets) are compared. In the long run, a loss is inevitable because of the negative expected value. The martingale strategy and the constant bet strategy on a single number are riskier than the constant bet strategy on a simple chance. This higher risk leads, however, to a higher chance of a positive profit in the short term. But on the other hand, higher risk means that the losses suffered by doublers and by single number bettors are much greater than that suffered by constant bettors.
... Under the assumption that wins on different lines played in the same spin are independent of one another, the two formulations are identical and the above results hold. If we also allow the bet per line B to vary, then The probability of achieving any winning play is independent of the amount bet and is therefore given by L hit hit B L 1 1 , so that the standard deviation of a particular game's payout is therefore linear in the bet per line B and follows the square root of the number of lines played L. This shows that if one wishes to increase the volatility of payouts (a desirable strategy when expected return is negative), one should increase the bet per line rather than the number of lines played, a similar message to the one identified in Croucher (2005). The probability of success is obviously not changed by the amount bet per line. ...
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This article illustrates the concept of statistical independence using the example of slot machines that may be played on multiple lines.
Article
This article describes how roulette can be used to teach basic concepts of probability. Various bets are used to illustrate the computation of expected value. A betting system shows variations in patterns that often appear in random events.
Gambling Theory and Other Topics
• R Malmuth
Malmuth, R. (1994). Gambling Theory and Other Topics (4th edn). Two Plus Two.
Gambling and Sport: A Statistical Approach
• J S Croucher
Croucher, J.S. (2003). Gambling and Sport: A Statistical Approach. Sydney: Macquarie Lighthouse Press.