Independent Audit

 

At Betanysports 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.