Casino Games and the Central Limit Theorem - Digital Scholarship ...

Casino Games and the Central Limit Theorem

1

Ashok Singh, Ph. D.* Rohan J. Dalpatadu, Ph.D. Dennis J. Murphy, Ph.D.

Abstract

Administration Email:

Introduction Administration

Department of Mathematical Sciences Email:

Senior Statistician Email:

UNLV Gaming Research & Review Journal Volume 17 Issue 2

Applications of the central limit theorem exist in the gaming literature as well.

experiments.

sample sizes that are meaningful to gaming operators. CLT and the Berry-Esseen Theorem

P(

x / n

x)

(x )

(x )

3

max| F x / n

(x )

(x )

3

n

,

3

E(| X - |3 )

onstant.

UNLV Gaming Research & Review Journal Volume 17 Issue 2

Slot game 1 2 3

1.33 1.33 1.33 1.33

100000 is of no practical value in the present situation. Methodology

programmed using the ggplot2 package in R.

these two areas of the casino. Slot Games

x

x1 x2

xN UNLV Gaming Research & Review Journal Volume 17 Issue 2

approximate formula:

hands.

Baccarat

x1 x2

xN

UNLV Gaming Research & Review Journal Volume 17 Issue 2

sample size. Results summarized in this section. Slot Games

o

that:

approximate formula.

UNLV Gaming Research & Review Journal Volume 17 Issue 2

n 100 250 500 1000 2000

n 100 250 500 1000 2000

Minimum

Maximum Mean

Median

0.10

sd

Skewness

Kurtosis

Skewness

Kurtosis

1.21

0.53

Minimum

Maximum Mean

Median

0.10 0.32 0.53

sd 1.21 0.55 0.3 0.25

12.21

222.11

n 100 250 500 1000 2000

n 100 250 500 1000 2000

50

Minimum 0.10

Maximum Mean

Median

112.30

sd

Skewness

Kurtosis

1.25

20.25

Minimum

Maximum Mean

0.13 0.50

Median

sd 1.25

23.32 12.22

0.50

UNLV Gaming Research & Review Journal Volume 17 Issue 2

Skewness

Kurtosis

:

Game 1 n

Game 2

SIM

Game 3

SIM

SIM

SIM

100 3.11 250

3.15

0.53 2.30

500 1.32 1000

0.20 1.12

1.15

1.21

2000 1.1 0.53 1.23

0.55 1.25

1.02

1.05

1.13 1.32

Baccarat

Conclusions and Limitations

UNLV Gaming Research & Review Journal Volume 17 Issue 2

51

n

Minimum

100 250 500 1000 2000

Maximum

Mean

Median

Skewness

Kurtosis

0.0010

0.0100 0.0200

0.2500 0.1100 0.0500

0.0100

n

SIM

100 250 500 1000 2000

52

0.1302

UNLV Gaming Research & Review Journal Volume 17 Issue 2

0.1000 0.0500 0.0300 0.0200

UNLV Gaming Research & Review Journal Volume 17 Issue 2

53

UNLV Gaming Research & Review Journal Volume 17 Issue 2

UNLV Gaming Research & Review Journal Volume 17 Issue 2

55

UNLV Gaming Research & Review Journal Volume 17 Issue 2

simulations. UNLV Gaming Research & Review Journal Volume 17 Issue 2

UNLV Gaming Research & Review Journal Volume 17 Issue 2

Appendix A

1 2 3

1 1 1 3

1 2 2 3

15 15 10 10 20 20

5 5 5 5 5 11 12 10 20 20

5 5 5 5 5 5 15 11 10 20 20

10 10 20 20

100 100 250 11000 2 2 2 2 5

100 100 300 11000 2 2 2 2 5

100 100 250 11000 2 2 2 2 5

100 100 250 11000 3 2 2 2 5

10 20

10 20

10 20

10 20

5

10 11 12 13 15

0.000013

0.001315 0.000110 0.000053 0.000051

20 21 22 23 25

0.000010 0.000001

30 31 32 33 35

1 2 3

2 3 3 5

100 100

0.000013

0.001315 0.000110 0.000051

UNLV Gaming Research & Review Journal Volume 17 Issue 2

Appendix A

0.000003

50 51 52 53 55

0.000003 0.000010 0.000001

0 50 50 150 250 2 2 2 2

50 50 150 250 2 2 2 2

0.001232

0.000012

0.000115

50 50 150 250 2 2 2 2 2 3 3 3 3

50 50 150 250 1 2 2 2 5 5

15 15 10 10 20 20

10 20 20

10 10 20 20

10 10 20 20

100 100 250 250 0

100 100 250 250 0

100 100 250 250 0

100 100 250 250 0

0.000051 0.000003 0.000003 0.000010 0.000001

References

UNLV Gaming Research & Review Journal Volume 17 Issue 2

pdf

553.

York.

UNLV Gaming Research & Review Journal Volume 17 Issue 2