Preparing to perform large scale modeling of baccarat methods, I wrote programs to first analyze existing baccarat shoe data, as well as generate my own.
The existing baccarat shoe data are from the popular Zumma 600 (600 live shoes) and Zumma 1000 (1000 live shoes) books, as well as the Wizard of Odds (1000 simulated shoes using a virtual 8-deck shoe).
In addition, I wrote my own program to simulate an 8-deck shoe, allowing me to generate a practically limitless number of realistic baccarat shoes. In my program, after a shuffle procedure thoroughly randomizes the shoe, cards are dealt according to baccarat drawing rules. In the same manner as the Wizard of Odds procedure, cards are dealt until less than 6 cards remain in the shoe. Extensive checking of the resulting output verifies my program produces realistic baccarat shoes. To form a substantial, preliminary data set, I used my program to generate 100,000 unique baccarat shoes.
To provide an extra degree of confidence that my generated shoes were realistically simulating what one might encounter when playing a live baccarat game at a casino, I analyzed the ratios of successive SAP and FOE event frequencies in all of the data sample sets (Zumma, Wizard of Odds, and my simulated shoes). SAP events are normal Player/Banker events, while FOE events are derivatives of SAP events. Because they are derivatives, FOE events offer an extra layer of testing sensitivity for the data sample.
Based on the probabilities of event occurrences in a random distribution, the ratio of each successive event should be 1/2.
the number of 1s should be 1/2 the total number of events,
the number of 2s should be 1/2 the 1s,
the number of 3s should be 1/2 the 2s,
the number of 4s should be 1/2 the 3s,
My analysis shows that in all data sets (Zumma 600, Zumma 1000, Wizard of Odds 1000, and my simulated shoes Virtuoid 1000), the actual ratios of successive events agrees with what is expected in a random distribution.
Note: The scatter at the higher numbered events is due to a relative scarcity of occurrences. Moreover, I limited the highest number of events graphed to 12, even though the highest event number in the data set is 16. Events 13-16 occurred too infrequently to form a statistically sufficient set.
These results suggest the following:
– Zumma live shoes exhibit event frequency distributions expected in a random data set.
– Wizard and Virtuoid simulated shoes using a virtual shoe exhibit event frequency distributions expected in a random data set.
– There is no evidence of shoe shuffle control in the Zumma 600 or Zumma 1000 data sets. Intentional shuffle control would bias the event frequencies and show up as significant departures from what is theoretically expected.
Thus, one of the following two statements must be true:
1) Zumma recorded shoes from casinos which did not artificially control the shuffle.
2) Zumma did not record shoes from real, physical casinos.
To elaborate, my analysis shows that both Zumma data sets exhibit characteristics which are consistent with a random distribution.
So, one can conclude either one of two things:
1. If one believes Zumma is telling the truth that its sample was collected from real casinos, then the results suggest that those casinos were not intentionally controlling the shuffle, but offering a truly fair and random game.
2. If one doubts Zumma is telling the truth that its sample was collected from real casinos, and one believes that casino shuffles are controlled to be biased and not random, then since Zumma data sets actually exhibit random characteristics, one can use the results to suggest that Zumma did not collect its data from real casinos.
My results in and of themselves cannot confirm which of the above two is true. But one statement is true, and the other is false, and it all depends on your assumptions about casino shuffle control.
If you believe Statement 1 above, then the simulated, virtual shoes by Wizard and myself are just as good as live shoes from a statistical standpoint, since their event frequencies are consistent with what is expected in a random distribution.
If you believe Statement 2 above, then live shoes would be expected to have more biases than simulated shoes, and the biases should be revealed in an analysis of the ratio of successive event frequencies.
Of course, even if Zumma did collect its data from real casinos, my analysis does not conclusively say whether or not other casinos intentionally control the shuffle. The SAP and FOE events of a particular casino would have be analyzed on a case-by-case basis to quantitatively determine whether it is offering a fair, random game.
Legally speaking, all casinos are supposed to offer perfectly fair games, but there are some who insist they do not. (I had written about the idea of shuffle control in these posts: Shuffle Control: Why It’s Bad for the House and Beating Random.)
With my new Baccarat Shoe Factory, I generated 100,000 unique baccarat shoes in preparation for large-scale testing of baccarat methods.
I also performed tests of the ratios of SAP and FOE event frequencies to verify my generated shoes conformed statistically to what is expected in a random distribution. Because of the significantly larger sample, I plot events up to 20, compared to only 12 for Zumma and Wizard of Odds.
As before, scatter at the higher events is due to relatively fewer occurrences. I limit the analysis to events of 20 or less, even though there were a few 21-25 events in both the SAP and FOE, as shown below in the numerical data table.
Numerical statistics from the 100,000 shoe data set:
(P=Player, B=Banker, T=Ties, R and A are derivatives of P and B)
|Total P:||3,738,579||44.6207%||(44.6274% theoretical)|
|Total B:||3,841,096||45.8443%||(45.8597% theoretical)|
|Total T:||798,901||9.5350%||( 9.5156% theoretical)|
|SAP Events||SAP Count||SAP Ratios|
|Total SAP Events:||3,842,804|
|FOE Events||FOE Count||FOE Ratios|
|Total FOE Events:||3,791,930|
My analysis verifies that my virtual 8-deck shoes are being sufficiently shuffled to produce characteristically random baccarat decisions with averages agreeing with theoretically calculated expectancies, making them statistically and practically equivalent to the Zumma and Wizard of Odds data sets. Thus, the results of testing baccarat methods when using Zumma, Wizard of Odds, or my own data should realistically reflect the results one might get when playing at physical casinos offering fair games in the real-world.
Follow-up Shoe Disparity Analysis: Disparity Data.
Follow-up: Separate P and B events analysis over 2361 live shoes, Zumma 600+1000 live shoes, and one million computer generated shoes: P and B Events Statistics: A Comprehensive Comparison