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Teaching Statistics. Volume 26, Number 3, Autumn 2004

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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

This article discusses some strategies for playing

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 19–36).

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

units ahead

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

at the answer.

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

bets made

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

BEING AHEAD

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

made (n)

Minimum no.

of units needed

to be ahead (t)

Minimum no.

of wins needed

Probability of being at least

40 units ahead

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

made (n)

Minimum no.

of wins needed to

at least break even

Probability

of at least

breaking even

Minimum no.

of wins needed

to be ahead

Probability of

being ahead

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

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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

Note by the Editor

“Teaching Statistics” is a refereed journal, and

owes a great debt of gratitude to the many people

who act as referees. To give an idea of what they

have to do, I can do no better than repeat a para-

graph from the last occasion when a note of this

nature was published. Here it is.

“Their duties are not so much to check the technical

correctness of submitted articles, though this is

of course important, but to form a professional

judgement as to whether the article is suitable

overall for our journal. Thus the questions that

arise include whether the article is to do with

teaching in some sense (we do of course publish

some articles on statistical education research,

provided there is an immediate relevance to a

teaching situation), whether it is at the right level

for our target audience, whether it is (as it says in

the Notes for Guidance) “light and readable”, and

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I really cannot overemphasise the importance of

this work. The standards of the journal to a large

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Anyone who would like to be added to the list is

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Since the last publication of a note of this nature,

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Geoffrey Clarke, Steve Crowther, Neville Davies,

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Scrimshaw, Neil Sheldon, Graham Smith. Again,

our thanks.

Gerald Goodall

Editor

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