Independent Audit

At Islandcasino we strive to give our customers the finest and most updated software to ensure statistical fairness in all our gaming venues.  The following is an independent audit on the xtech gaming software we currently use for our Bonus Casino.

Mathematical Audit Of The Following Software:
XTECH
May 25, 2003
Prepared By:

Michael Shackleford
The Wizard of Odds Consulting, Inc. 
Las Vegas, NV
Company Info

Xtech S.A.  is a Costa Rican corporation that develops online gaming software and markets this software in partnership with Digital Solutions S.A.  Under the terms of the arrangement between Xtech and Digital Solutions, Digital Solutions is responsible for product sales and customer support, and

  1. Represents and warrants that Xtech products will substantially achieve the functionality described in its program documentation;
  2. Indemnifies and holds harmless Digital Solutions customers against any damages (including reasonable legal costs) that may be awarded or agreed to be paid to any third party in respect of any claim or action that the normal operation, possession, or use of the Xtech products by Digital Solutions customers that infringes the Intellectual Property Rights of said third party.

Introduction
For all games except Craps, log files were made available to me by FTP for the period of July 1, 2002, to January 31, 2003.  The Craps log files are based on data from 01-01 to 05-20-2003.  Some files did not encompass this entire period of time, such as Blackjack, for example, due to the large file size.  This report shall show on a game-by-game basis the results of my audit.
Baccarat
Following are the linksall Baccarat results:

Number of bets

86709

Total bets

376253

Total win

-4864.65

Return

-0.01293

Following is the distribution of each card:

Card Player Banker Total

10c

4222

4056

8278

10d

4272

4026

8298

10h

4166

4116

8282

10s

4245

3966

8211

2c

4203

3973

8176

2d

4157

4197

8354

2h

4213

3988

8201

2s

4175

3982

8157

3c

4142

4139

8281

3d

4102

4126

8228

3h

4153

4082

8235

3s

4166

4064

8230

4c

4210

4057

8267

4d

4136

4164

8300

4h

4186

4096

8282

4s

4204

4004

8208

5c

4238

4004

8242

5d

4123

4169

8292

5h

4186

4065

8251

5s

4151

4059

8210

6c

4143

4070

8213

6d

4236

3872

8108

6h

4017

4174

8191

6s

4186

4136

8322

7c

4310

4047

8357

7d

4161

4107

8268

7h

4139

3921

8060

7s

4161

4086

8247

8c

4234

4029

8263

8d

4162

4170

8332

8h

4146

4053

8199

8s

4259

4057

8316

9c

4190

4035

8225

9d

4132

4062

8194

9h

4204

4052

8256

9s

4177

4100

8277

Ac

4107

4033

8140

Ad

4122

4108

8230

Ah

4243

4076

8319

As

4170

3996

8166

Jc

4111

4126

8237

Jd

4227

3969

8196

Jh

4118

4113

8231

Js

4115

4078

8193

Kc

4224

3935

8159

Kd

4049

4126

8175

Kh

4085

4131

8216

Ks

4133

4063

8196

Qc

4228

4091

8319

Qd

4222

4024

8246

Qh

4189

4114

8303

Qs

4237

3900

8137

Total

217087

211187

428274

Following are the results of a chi-squared test on each distribution.  Throughout this audit many chi- squared tests were performed.  To avoid repetition, I will explain how the test works here only. 

The chi- squared test is a measure of how skewed a distribution of random variables is compared to statistical expectations.  In the case of total cards in the Baccarat log files, the chi-squared statistic is 24.24903 with 51 degrees of freedom.  The lower the statistic, the less skewed the results are.  The probability that a fair test would be this skewed or more is 99.95%.

So this distribution is unusually unskewed.  The banker distribution is more skewed.  The probability of a fair test being more skewed is 9.32%.  Wherever a probability table appears, such as the one below, the probability refers to the probability that a fair test would show results more skewed than those observed.  Overall, these probabilities should fall uniformly from 0% to 100%.

Hand Probability

Banker

0.093221

Player

0.882165

Combined

0.999466

The next table shows the outcome of each hand, the theoretical probability (based on 6 decks), and the expected total based on the theoretical probabilities and the total sample size.

Winner Total Ex.  Prob Ex.  Total

Banker

39825

0.458653

39769

Player

38740

0.446279

38696

Tie

8144

0.095069

8243

Total

86709

1.000000

86709

The result of a chi-squared test on this table is a probability of 0.515909678.

Blackjack
Following are the linksall Blackjack results:

Total number of bets

1256917

Total initial bets

31406562

Total win

-517035

Return

-0.01646

The next table shows the card distribution:

Card Total Player Total Dealer Total

10c

62732

66903

129635

10d

62852

67076

129928

10h

62677

66759

129436

10s

62670

66957

129627

2c

63077

67150

130227

2d

62711

67383

130094

2h

63009

67737

130746

2s

62903

67586

130489

3c

63208

67205

130413

3d

63099

67331

130430

3h

62738

67040

129778

3s

62792

67767

130559

4c

62706

67310

130016

4d

62607

67784

130391

4h

63061

67127

130188

4s

63128

67229

130357

5c

63205

67307

130512

5d

62939

67677

130616

5h

62819

67415

130234

5s

62768

67282

130050

6c

63519

66929

130448

6d

63126

66962

130088

6h

62780

67727

130507

6s

62903

67932

130835

7c

62805

66829

129634

7d

63267

67122

130389

7h

63006

66857

129863

7s

62819

67172

129991

8c

63051

66646

129697

8d

62910

67017

129927

8h

62690

66856

129546

8s

62344

67564

129908

9c

63102

66789

129891

9d

62831

67370

130201

9h

62771

67360

130131

9s

62587

66875

129462

Ac

62805

66559

129364

Ad

62806

66800

129606

Ah

62930

66686

129616

As

62893

66641

129534

Jc

62859

66580

129439

Jd

63435

67008

130443

Jh

62944

66596

129540

Js

63065

66798

129863

Kc

62646

66935

129581

Kd

63332

66411

129743

Kh

63135

66768

129903

Ks

63215

66966

130181

Qc

62742

67199

129941

Qd

63030

66665

129695

Qh

63103

66624

129727

Qs

62962

67158

130120

Total

3272114

3488426

6760540

Next are the results of a chi-squared test on each distribution:

Test

Probability

Player

0.809737

Dealer

0.827439

Combined

0.829907

Craps
The initial Craps files I looked at from August 1, 2002, to January 1, 2003, showed an unusually high number of fives thrown on each die, although not enough to skew the odds by more than a negligible amount.  However, data from January 1 to May 20, 2003, looks much better as follows:

Face Actual Expected

1

67869

67713.67

2

67673

67713.67

3

67765

67713.67

4

67846

67713.67

5

67337

67713.67

6

67792

67713.67

Total

406282

406282.00

This distribution has a chi-squared statistic of 2.864 with 5 degrees of freedom.  The probability of a fair die producing a distribution this skewed or more is 0.720917.  So these recent results are well within the normal range.  The nature of the log files made it difficult to look at the entire game linksall.  However, I studied the result of player pass line bets.

Total amount bet

242189

Total amount won

-2311

Number of wins

17756

Number of losses

18323

Return

-0.01509

The house edge links this sample is not far from the mathematical expectation of 1.41%.  The percentage of winning pass line bets is 49.21%.  The expected winning percentage is 49.29%, again very close to expectation.

Roulette
Following are the linksall Roulette results:

Number of bets

10042

Total bets

240522

Total win

-13598

Return

-0.05654

The next table shows the frequency of each outcome:

Balls Count

00

262

0

282

1

271

2

275

3

239

4

243

5

256

6

246

7

268

8

264

9

258

10

308

11

261

12

262

13

259

14

294

15

257

16

235

17

281

18

260

19

264

20

286

21

277

22

269

23

263

24

273

25

238

26

259

27

289

28

278

29

268

30

243

31

269

32

258

33

254

34

269

35

244

36

260

Total

10042

A result of a chi-squared test on this distribution is a probability of 0.534021.

Caribbean Stud Poker
Following are the linksall results for Caribbean Stud Poker:

Total hands

187945

Total initial bets

2612373

Total final bets

5418495

Total win

-199830

House edge

-0.07649

The next table shows the card distribution:

Card Player Dealer Total

10c

18231

18033

36264

10d

18007

18226

36233

10h

18275

18091

36366

10s

18406

18060

36466

2c

18079

18099

36178

2d

18192

18262

36454

2h

17910

18127

36037

2s

18052

18143

36195

3c

17982

17958

35940

3d

17944

18127

36071

3h

18067

17998

36065

3s

17921

18040

35961

4c

18156

17944

36100

4d

18086

18225

36311

4h

18196

18160

36356

4s

18087

18181

36268

5c

17878

18007

35885

5d

18161

18174

36335

5h

18271

17925

36196

5s

17839

18178

36017

6c

18287

18081

36368

6d

18122

18117

36239

6h

18229

17998

36227

6s

18178

17750

35928

7c

18153

17852

36005

7d

18045

18243

36288

7h

18122

18135

36257

7s

18147

18260

36407

8c

18156

17771

35927

8d

17865

18111

35976

8h

18144

17944

36088

8s

18105

17966

36071

9c

18220

18206

36426

9d

18144

18149

36293

9h

17948

18220

36168

9s

17946

17964

35910

Ac

18063

18063

36126

Ad

18168

18052

36220

Ah

18375

17853

36228

As

18077

17966

36043

Jc

17955

18243

36198

Jd

18196

17823

36019

Jh

18145

18052

36197

Js

17799

17959

35758

Kc

17857

18323

36180

Kd

17961

18184

36145

Kh

18046

18213

36259

Ks

17762

18059

35821

Qc

17813

18043

35856

Qd

17845

18103

35948

Qh

18058

18006

36064

Qs

18054

18058

36112

Total

939725

939725

1879450

The next table shows the results of a chi-squared test on each card distribution:

Hand Probability

Banker

0.127167

Player

0.154424

Combined

0.823958

Following is the frequency of each poker hand, the theoretical probability, and the expected total based on the sample size:

Hand Player Dealer Probability Expected

Royal flush

0

0

0.000002

0.29

Straight flush

2

2

0.000014

2.60

Four of a kind

38

40

0.000240

45.12

Full house

279

278

0.001441

270.75

Flush

368

386

0.001965

369.39

Straight

770

717

0.003925

737.62

Three of a kind

4004

4026

0.021128

3970.99

Two pair

8722

8945

0.047539

8934.72

Pair

79423

79503

0.422569

79419.74

Nothing

94339

94048

0.501177

94193.79

Total

187945

187945

1

187945

A chi-squared test on the hand distribution above results in the following probabilities:

Hand Probability

Player

0.456381

Dealer

0.484061

Combined

0.669520

Slots
Following are the linksall results for both types of slot machines:

Item Jurassic TD fever Total

Number of bets

280983

341644

622627

Total amount bet

157309

822851

980160

Total return

-5085

-23696

-28781

Expected value

-0.03232

-0.0288

-0.02936

Without knowing the expected outcome of the games, there is nothing to compare the actual results to.

Video Poker
Following are the linksall outcomes of the three video poker games:

Game Hands Total Bet Total Win Return

Jacks or Better

8933

78272.25

-1234

-1.58%

Deuces Wild

2031

4988.75

-549.75

-11.02%

Joker Wild

905

2153.25

-191.25

-8.88%

Total

11869

85414.25

-1975

-2.31%

The next table shows the frequency each card occurred in each game:

Card Jacks or Better Deuces Wild Joker Poker Total

10c

890

209

80

1179

10d

831

177

94

1102

10h

885

190

71

1146

10s

840

174

78

1092

2c

867

203

79

1149

2d

844

201

96

1141

2h

879

194

96

1169

2s

812

176

91

1079

3c

844

186

76

1106

3d

830

192

84

1106

3h

831

189

84

1104

3s

857

176

74

1107

4c

841

216

81

1138

4d

835

233

96

1164

4h

894

204

88

1186

4s

855

196

83

1134

5c

826

212

103

1141

5d

876

170

86

1132

5h

871

181

79

1131

5s

841

171

80

1092

6c

927

210

87

1224

6d

875

210

89

1174

6h

819

204

95

1118

6s

859

181

85

1125

7c

874

189

89

1152

7d

884

208

98

1190

7h

814

207

69

1090

7s

851

195

81

1127

8c

891

204

88

1183

8d

907

198

83

1188

8h

859

170

98

1127

8s

907

208

94

1209

9c

857

196

86

1139

9d

835

185

90

1110

9h

801

202

80

1083

9s

844

209

94

1147

Ac

861

194

81

1136

Ad

840

192

94

1126

Ah

898

192

85

1175

As

837

202

83

1122

Jc

847

190

105

1142

Jd

909

199

86

1194

Jh

859

185

89

1133

Js

900

206

85

1191

Kc

841

192

74

1107

Kd

865

188

75

1128

Kh

801

232

90

1123

Ks

841

202

85

1128

Qc

850

227

62

1139

Qd

883

158

89

1130

Qh

847

161

87

1095

Qs

933

209

66

1208

Joker

0

0

84

84

Total

44665

10155

4525

59345

The next table shows the results of a chi-squared test on each game:

Game Probability

Jacks or Better

0.30557321

Deuces Wild

0.04871101

Joker Poker

0.65621183

Combined

0.34847908

Next, the poker value of the initial hand was scored in each game.  In joker poker, hands with a joker were not scored.  In deuces wild, the deuces were not considered wild.  Following shows the distribution compared to statistical expectations:

Hand Actual Combinations Probability Expected

Royal flush

0

4

0.000002

0.02

Str flush

1

36

0.000014

0.16

Four of a kind

3

624

0.000240

2.83

Full house

21

3744

0.001441

16.98

Flush

27

5108

0.001965

23.16

Straight

46

10200

0.003925

46.25

Three of a kind

273

54912

0.021128

249.00

Two pair

586

123552

0.047539

560.25

Pair

4989

1098240

0.422569

4979.98

Nothing

5839

1302540

0.501177

5906.38

Total

11785

2598960

1.000000

11785.00

A chi-squared test on the actual distribution results in a probability of 33.53%.

Conclusion
Aside from the Craps data from 2002, all the results of this audit fall within statistical norms according to my analysis.  The Craps data from 2003 look fine and well within the normal range of statistical variation.  Overall, for the most recent data submitted, I can say that XTECH is offering fair games of chance, and the outcomes fall within statistical expectations.

If you have any questions about this audit, please contact me by one of the methods below.

"Contact information removed, per the request of Mr.  Shackleford."
Thank you for choosing me to audit your software, and I hope to continue to do so on a regular basis.
Michael Shackleford, A.S.A.