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Independent Audit

 

At 5Dimes 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

 

Represents and warrants that Xtech products will substantially achieve the functionality described in its program documentation;

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 overall 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
10d4272 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 overall 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 overall.  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 over 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 overall 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 overall 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 overall 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 overall 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.

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