Independent Audit
At Lucky99 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 Classic 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 |
| 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 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.