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
  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 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.
E-mail: mail@gamingmath.com
Phone: 702-804-0015
Postal mail: 9200 Sienna Vista Dr.
Las Vegas, NV 89117
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.