P and B Events Statistics: A Comprehensive Comparison
Update: See the related post, Effect of Card Removal on Bias and Event Frequencies in Baccarat.
In this post, I comprehensively compare the Player (P) and Banker (B) events statistics in several different data sets, live and computer generated.
An “event” is a repeating sequence of one side. A 1s event is a single instance of that side with no repeats afterward, a 2s is two consecutive instances of that side (one repeat), a 3s is three on that side (two repeats), etc. For example, in the sequence – P BB PP B PPP BBBB P B P – there were for P three 1s (a single P), one 2s (PP), and one 3s (PPP), and for B, two 1s (B), one 2s (BB), no 3s (BBB), and one 4s (BBBB).
I originally examined the ratio of successive events in my 102,600 shoe data set in my post My Baccarat Shoe Factory. However, that analysis did not distinguish between P and B events, but considered the sum of both P and B events together. In the following analysis, P and B events are considered separately.
Moreover, as generously provided by a colleague, a new data set of 2,361 live shoes which were manually recorded at land-based casinos is now a part of my data sets. (We are in the process of collecting 10,000 live shoes.) In the following analysis, three sets of data are considered: 1) 2,361 Live Shoes, 2) Zumma 600+1000 Shoes, and 3) Virtuoid 1,000,000 Computer Generated Shoes.
Note: for the 2,361 live shoes, all were hand recorded during live play from B&M casinos which use the Shufflemaster shuffle machine, except for a few of the shoes which were hand shuffled. The B&M casinos are: Horseshoe, Imperial Palace, Barbary Coast, Emerald Queen, Orleans, Gold Coast, Sunset Station, Tulalip, Green Valley, Harrah’s, Mohegan Sun, and Foxwoods.
The tabulated results of the frequencies and percentages of P and B events are as follows:
1) 2361 Live Shoes:
| Events | B | P | Total | B % | P % |
| 1s | 21,020 | 21,440 | 42,460 | 25.083% | 25.584% |
| 2s | 10,462 | 10,544 | 21,006 | 12.484% | 12.582% |
| 3s | 5,153 | 5,041 | 10,194 | 6.149% | 6.015% |
| 4s | 2,614 | 2,539 | 5,153 | 3.119% | 3.030% |
| 5s | 1,294 | 1,164 | 2,458 | 1.544% | 1.389% |
| 6s | 668 | 603 | 1,271 | 0.797% | 0.720% |
| 7s | 368 | 301 | 669 | 0.439% | 0.359% |
| 8s | 161 | 123 | 284 | 0.192% | 0.147% |
| 9s | 88 | 63 | 151 | 0.105% | 0.075% |
| 10s | 52 | 35 | 87 | 0.062% | 0.042% |
| 11s | 25 | 14 | 39 | 0.030% | 0.017% |
| 12s | 10 | 7 | 17 | 0.012% | 0.008% |
| 13s | 2 | 4 | 6 | 0.002% | 0.005% |
| 14s | 3 | 0 | 3 | 0.004% | 0.000% |
| 15s | 2 | 0 | 2 | 0.002% | 0.000% |
| 16s | 3 | 0 | 3 | 0.004% | 0.000% |
| 17s | 0 | 0 | 0 | - | - |
| Total: | 41,925 | 41,878 | 83,803 | 50.028% | 49.972% |
2) Zumma 600+1000:
| Events | B | P | Total | B % | P % |
| 1s | 14,518 | 14,916 | 29,434 | 25.074% | 25.761% |
| 2s | 7,160 | 7,176 | 14,336 | 12.366% | 12.394% |
| 3s | 3,631 | 3,536 | 7,167 | 6.271% | 6.107% |
| 4s | 1,804 | 1,749 | 3,553 | 3.116% | 3.021% |
| 5s | 921 | 851 | 1,772 | 1.591% | 1.470% |
| 6s | 465 | 390 | 855 | 0.803% | 0.674% |
| 7s | 217 | 188 | 405 | 0.375% | 0.325% |
| 8s | 111 | 74 | 185 | 0.192% | 0.128% |
| 9s | 53 | 37 | 90 | 0.092% | 0.064% |
| 10s | 32 | 18 | 50 | 0.055% | 0.031% |
| 11s | 20 | 10 | 30 | 0.035% | 0.017% |
| 12s | 8 | 6 | 14 | 0.014% | 0.010% |
| 13s | 1 | 2 | 3 | 0.002% | 0.003% |
| 14s | 2 | 2 | 4 | 0.003% | 0.003% |
| 15s | 2 | 0 | 2 | 0.003% | 0.000% |
| 16s | 1 | 0 | 1 | 0.002% | 0.000% |
| 17s | 0 | 0 | 0 | - | - |
| Total: | 28,946 | 28,955 | 57,901 | 49.992% | 50.008% |
3) Virtuoid 1,000,000 Computer Generated Shoes:
| Events | B | P | Total | B % | P % |
| 1s | 9,598,170 | 9,846,180 | 19,444,350 | 25.002% | 25.648% |
| 2s | 4,798,752 | 4,797,125 | 9,595,877 | 12.500% | 12.496% |
| 3s | 2,402,247 | 2,333,094 | 4,735,341 | 6.257% | 6.077% |
| 4s | 1,201,685 | 1,136,427 | 2,338,112 | 3.130% | 2.960% |
| 5s | 600,221 | 553,952 | 1,154,173 | 1.563% | 1.443% |
| 6s | 300,328 | 268,645 | 568,973 | 0.782% | 0.700% |
| 7s | 150,073 | 130,592 | 280,665 | 0.391% | 0.340% |
| 8s | 75,015 | 63,567 | 138,582 | 0.195% | 0.166% |
| 9s | 37,016 | 30,728 | 67,744 | 0.096% | 0.080% |
| 10s | 18,461 | 14,884 | 33,345 | 0.048% | 0.039% |
| 11s | 9,448 | 7,346 | 16,794 | 0.025% | 0.019% |
| 12s | 4,666 | 3,648 | 8,314 | 0.012% | 0.010% |
| 13s | 2,407 | 1,740 | 4,147 | 0.006% | 0.005% |
| 14s | 1,140 | 862 | 2,002 | 0.003% | 0.002% |
| 15s | 582 | 388 | 970 | 0.002% | 0.001% |
| 16s | 280 | 188 | 468 | 0.001% | 0.000% |
| 17s | 141 | 106 | 247 | 0.000% | 0.000% |
| 18s | 72 | 41 | 113 | 0.000% | 0.000% |
| 19s | 47 | 26 | 73 | 0.000% | 0.000% |
| 20s | 15 | 9 | 24 | 0.000% | 0.000% |
| 21s | 9 | 6 | 15 | 0.000% | 0.000% |
| 22s | 8 | 2 | 10 | 0.000% | 0.000% |
| 23s | 0 | 3 | 3 | 0.000% | 0.000% |
| 24s | 3 | 0 | 3 | 0.000% | 0.000% |
| 25s | 3 | 1 | 4 | 0.000% | 0.000% |
| 26s | 0 | 0 | 0 | - | - |
| Total: | 19,200,789 | 19,189,560 | 38,390,349 | 50.015% | 49.985% |
For ease of comparison, the B and P Events Percentages for the three data sets are summarized in the next two tables:
| B Events | 2361 Live | Zumma 600+1000 | Virtuoid 1,000,000 |
| 1s | 25.083% | 25.074% | 25.002% |
| 2s | 12.484% | 12.366% | 12.500% |
| 3s | 6.149% | 6.271% | 6.257% |
| 4s | 3.119% | 3.116% | 3.130% |
| 5s | 1.544% | 1.591% | 1.563% |
| 6s | 0.797% | 0.803% | 0.782% |
| 7s | 0.439% | 0.375% | 0.391% |
| 8s | 0.192% | 0.192% | 0.195% |
| 9s | 0.105% | 0.092% | 0.096% |
| 10s | 0.062% | 0.055% | 0.048% |
| 11s | 0.030% | 0.035% | 0.025% |
| 12s | 0.012% | 0.014% | 0.012% |
| 13s | 0.002% | 0.002% | 0.006% |
| 14s | 0.004% | 0.003% | 0.003% |
| 15s | 0.002% | 0.003% | 0.002% |
| 16s | 0.004% | 0.002% | 0.001% |
| P Events | 2361 Live | Zumma 600+1000 | Virtuoid 1,000,000 |
| 1s | 25.584% | 25.761% | 25.648% |
| 2s | 12.582% | 12.394% | 12.496% |
| 3s | 6.015% | 6.107% | 6.077% |
| 4s | 3.030% | 3.021% | 2.960% |
| 5s | 1.389% | 1.470% | 1.443% |
| 6s | 0.720% | 0.674% | 0.700% |
| 7s | 0.359% | 0.325% | 0.340% |
| 8s | 0.147% | 0.128% | 0.166% |
| 9s | 0.075% | 0.064% | 0.080% |
| 10s | 0.042% | 0.031% | 0.039% |
| 11s | 0.017% | 0.017% | 0.019% |
| 12s | 0.008% | 0.010% | 0.010% |
| 13s | 0.005% | 0.003% | 0.005% |
| 14s | 0.000% | 0.003% | 0.002% |
| 15s | 0.000% | 0.000% | 0.001% |
| 16s | 0.000% | 0.000% | 0.000% |
Graphical summary plots of the above results are presented in the following charts at various zoom levels.
Overall, the above results verify the following:
1. Because the B outcome is more likely than the P outcome, longer B events occur more frequently than longer P events, while shorter P events (especially 1s P) occur more frequently than shorter B events.
2. All data sets, 2361 Live, Zumma 600+1000, and Virtuoid 1,000,000 show significant agreement in event statistics. This suggests that baccarat shoes played in live land-based casinos are statistically random. In terms of events statistics, there is no evidence of persistent bias or other unique signature present. Thus, randomly generated baccarat shoes (with realistic drawing rules) are completely statistically comparable to live shoes in terms of event frequencies. Hence, simulations performed with either live or realistically generated shoes should yield the same overall results, and either kind of data set is trustworthy when used as the basis of testing strategies.
Addendum:
The results in the tables of the above analysis counted all events in each shoe and considered each shoe as a closed, separate entity. For the sake of comparison, the results in the following tables omit the count of the final event in the shoe (ref: see comment below by Rick):
1) 2361 Live Shoes, Omit Final Event:
| Events | B | P | Total | B % | P % |
| 1s | 20,402 | 20,869 | 41,271 | 25.052% | 25.625% |
| 2s | 10,143 | 10,252 | 20,395 | 12.455% | 12.588% |
| 3s | 5,023 | 4,904 | 9,927 | 6.168% | 6.022% |
| 4s | 2,536 | 2,460 | 4,996 | 3.114% | 3.021% |
| 5s | 1,254 | 1,132 | 2,386 | 1.540% | 1.390% |
| 6s | 651 | 586 | 1,237 | 0.799% | 0.720% |
| 7s | 359 | 293 | 652 | 0.441% | 0.360% |
| 8s | 158 | 120 | 278 | 0.194% | 0.147% |
| 9s | 86 | 62 | 148 | 0.106% | 0.076% |
| 10s | 49 | 34 | 83 | 0.060% | 0.042% |
| 11s | 24 | 14 | 38 | 0.029% | 0.017% |
| 12s | 9 | 7 | 16 | 0.011% | 0.009% |
| 13s | 2 | 4 | 6 | 0.002% | 0.005% |
| 14s | 2 | 0 | 2 | 0.002% | 0.000% |
| 15s | 2 | 0 | 2 | 0.002% | 0.000% |
| 16s | 3 | 0 | 3 | 0.004% | 0.000% |
| 17s | 0 | 0 | 0 | - | - |
| Total: | 40,703 | 40,737 | 81,440 | 49.979% | 50.021% |
2) Zumma 600+1000, Omit Final Event:
| Events | B | P | Total | B % | P % |
| 1s | 14,128 | 14,521 | 28,649 | 25.094% | 25.792% |
| 2s | 6,946 | 6,972 | 13,918 | 12.337% | 12.383% |
| 3s | 3,527 | 3,422 | 6,949 | 6.265% | 6.078% |
| 4s | 1,767 | 1,699 | 3,466 | 3.138% | 3.018% |
| 5s | 896 | 831 | 1,727 | 1.591% | 1.476% |
| 6s | 453 | 379 | 832 | 0.805% | 0.673% |
| 7s | 208 | 184 | 392 | 0.369% | 0.327% |
| 8s | 108 | 74 | 182 | 0.192% | 0.131% |
| 9s | 52 | 35 | 87 | 0.092% | 0.062% |
| 10s | 29 | 17 | 46 | 0.052% | 0.030% |
| 11s | 20 | 10 | 30 | 0.036% | 0.018% |
| 12s | 8 | 5 | 13 | 0.014% | 0.009% |
| 13s | 1 | 2 | 3 | 0.002% | 0.004% |
| 14s | 2 | 2 | 4 | 0.004% | 0.004% |
| 15s | 2 | 0 | 2 | 0.004% | 0.000% |
| 16s | 1 | 0 | 1 | 0.002% | 0.000% |
| 17s | 0 | 0 | 0 | - | - |
| Total: | 28,148 | 28,153 | 56,301 | 49.996% | 50.004% |
3) Virtuoid 1,000,000 Computer Generated Shoes, Omit Final Event:
| Events | B | P | Total | B % | P % |
| 1s | 9,349,069 | 9,595,152 | 18,944,221 | 25.004% | 25.662% |
| 2s | 4,672,490 | 4,673,356 | 9,345,846 | 12.497% | 12.499% |
| 3s | 2,338,669 | 2,271,830 | 4,610,499 | 6.255% | 6.076% |
| 4s | 1,169,408 | 1,106,186 | 2,275,594 | 3.128% | 2.958% |
| 5s | 583,793 | 539,006 | 1,122,799 | 1.561% | 1.442% |
| 6s | 292,108 | 261,323 | 553,431 | 0.781% | 0.699% |
| 7s | 145,824 | 127,100 | 272,924 | 0.390% | 0.340% |
| 8s | 72,908 | 61,811 | 134,719 | 0.195% | 0.165% |
| 9s | 35,900 | 29,842 | 65,742 | 0.096% | 0.080% |
| 10s | 17,936 | 14,435 | 32,371 | 0.048% | 0.039% |
| 11s | 9,160 | 7,147 | 16,307 | 0.024% | 0.019% |
| 12s | 4,500 | 3,549 | 8,049 | 0.012% | 0.009% |
| 13s | 2,340 | 1,697 | 4,037 | 0.006% | 0.005% |
| 14s | 1,106 | 833 | 1,939 | 0.003% | 0.002% |
| 15s | 561 | 377 | 938 | 0.002% | 0.001% |
| 16s | 270 | 185 | 455 | 0.001% | 0.000% |
| 17s | 139 | 102 | 241 | 0.000% | 0.000% |
| 18s | 70 | 40 | 110 | 0.000% | 0.000% |
| 19s | 46 | 26 | 72 | 0.000% | 0.000% |
| 20s | 14 | 9 | 23 | 0.000% | 0.000% |
| 21s | 9 | 4 | 13 | 0.000% | 0.000% |
| 22s | 8 | 2 | 10 | 0.000% | 0.000% |
| 23s | 0 | 2 | 2 | 0.000% | 0.000% |
| 24s | 3 | 0 | 3 | 0.000% | 0.000% |
| 25s | 3 | 1 | 4 | 0.000% | 0.000% |
| 26s | 0 | 0 | 0 | - | - |
| Total: | 18,696,334 | 18,694,015 | 37,390,349 | 50.003% | 49.997% |
The differences of the percentages of B, P and Total Events between the two cases are given below (Delta = percentages counting all events minus percentages omitting last event):
1) 2361 Live Shoes, Delta Percentages:
| Events | B Delta % | P Detla % | Total Delta % |
| 1s | 0.0311% | -0.0412% | -0.0101% |
| 2s | 0.0295% | -0.0065% | 0.0230% |
| 3s | -0.0188% | -0.0063% | -0.0251% |
| 4s | 0.0053% | 0.0091% | 0.0144% |
| 5s | 0.0043% | -0.0010% | 0.0033% |
| 6s | -0.0023% | 0.0000% | -0.0023% |
| 7s | -0.0017% | -0.0006% | -0.0023% |
| 8s | -0.0019% | -0.0006% | -0.0025% |
| 9s | -0.0006% | -0.0010% | -0.0015% |
| 10s | 0.0019% | 0.0000% | 0.0019% |
| 11s | 0.0004% | -0.0005% | -0.0001% |
| 12s | 0.0009% | -0.0002% | 0.0006% |
| 13s | -0.0001% | -0.0001% | -0.0002% |
| 14s | 0.0011% | 0.0000% | 0.0011% |
| 15s | -0.0001% | 0.0000% | -0.0001% |
| 16s | -0.0001% | 0.0000% | -0.0001% |
| 17s | - | - | - |
| Total: | 0.0489% | -0.0489% | 0.0000% |
2) Zumma 600+1000, Delta Percentages:
| Events | B Delta % | P Detla % | Total Delta % |
| 1s | -0.0199% | -0.0305% | -0.0504% |
| 2s | 0.0287% | 0.0101% | 0.0388% |
| 3s | 0.0065% | 0.0289% | 0.0354% |
| 4s | -0.0228% | 0.0030% | -0.0199% |
| 5s | -0.0008% | -0.0062% | -0.0070% |
| 6s | -0.0015% | 0.0004% | -0.0011% |
| 7s | 0.0053% | -0.0021% | 0.0032% |
| 8s | -0.0001% | -0.0036% | -0.0038% |
| 9s | -0.0008% | 0.0017% | 0.0009% |
| 10s | 0.0038% | 0.0009% | 0.0047% |
| 11s | -0.0010% | -0.0005% | -0.0015% |
| 12s | -0.0004% | 0.0015% | 0.0011% |
| 13s | 0.0000% | -0.0001% | -0.0001% |
| 14s | -0.0001% | -0.0001% | -0.0002% |
| 15s | -0.0001% | 0.0000% | -0.0001% |
| 16s | 0.0000% | 0.0000% | 0.0000% |
| 17s | - | - | - |
| Total: | -0.0033% | 0.0033% | 0.0000% |
3) Virtuoid 1,000,000 Computer Generated Shoes, Delta Percentages:
| Events | B Delta % | P Detla % | Total Delta % |
| 1s | -0.0024% | -0.0146% | -0.0170% |
| 2s | 0.0034% | -0.0032% | 0.0002% |
| 3s | 0.0027% | 0.0013% | 0.0040% |
| 4s | 0.0026% | 0.0017% | 0.0043% |
| 5s | 0.0021% | 0.0014% | 0.0035% |
| 6s | 0.0011% | 0.0009% | 0.0019% |
| 7s | 0.0009% | 0.0002% | 0.0012% |
| 8s | 0.0004% | 0.0003% | 0.0007% |
| 9s | 0.0004% | 0.0002% | 0.0006% |
| 10s | 0.0001% | 0.0002% | 0.0003% |
| 11s | 0.0001% | 0.0000% | 0.0001% |
| 12s | 0.0001% | 0.0000% | 0.0001% |
| 13s | 0.0000% | 0.0000% | 0.0000% |
| 14s | 0.0000% | 0.0000% | 0.0000% |
| 15s | 0.0000% | 0.0000% | 0.0000% |
| 16s | 0.0000% | 0.0000% | 0.0000% |
| 17s | 0.0000% | 0.0000% | 0.0000% |
| 18s | 0.0000% | 0.0000% | 0.0000% |
| 19s | 0.0000% | 0.0000% | 0.0000% |
| 20s | 0.0000% | 0.0000% | 0.0000% |
| 21s | 0.0000% | 0.0000% | 0.0000% |
| 22s | 0.0000% | 0.0000% | 0.0000% |
| 23s | 0.0000% | 0.0000% | 0.0000% |
| 24s | 0.0000% | 0.0000% | 0.0000% |
| 25s | 0.0000% | 0.0000% | 0.0000% |
| 26s | - | - | - |
| Total: | 0.0115% | -0.0115% | 0.0000% |
Addendum:
Following are the statistics for chop events.
Whereas the above events are based on the number of consecutive repeats, the chop event is based on the number of consecutive opposites. For example, in the sequence BBB PPP, there is one chop event of length 1, since there is one occurrence of an opposite. As another example, in the sequence BB PP B P B PPP B PP, there is one chop event of length 1, one of 2, and one of 4.
Note: if a chop event at the end of a shoe has not fully completed, it is not counted in the statistics.
Chop event counts:
| Chop Events | 2361 Live Shoes | Zumma 600 + 1000 | Virtuoid 1M Computer Generated Shoes |
| 1 | 20,370 | 13,938 | 9,346,755 |
| 2 | 10,081 | 6,848 | 4,613,026 |
| 3 | 4,970 | 3,487 | 2,274,146 |
| 4 | 2,381 | 1,710 | 1,122,426 |
| 5 | 1,164 | 828 | 552,224 |
| 6 | 594 | 450 | 273,034 |
| 7 | 290 | 199 | 133,941 |
| 8 | 145 | 96 | 66,148 |
| 9 | 77 | 46 | 32,359 |
| 10 | 36 | 20 | 16,228 |
| 11 | 16 | 6 | 7,785 |
| 12 | 13 | 3 | 3,929 |
| 13 | 2 | 3 | 1,975 |
| 14 | 3 | 3 | 910 |
| 15 | 3 | 0 | 460 |
| 16 | 2 | 0 | 212 |
| 17 | 0 | 0 | 119 |
| 18 | 0 | 0 | 47 |
| 19 | 0 | 0 | 25 |
| 20 | 0 | 0 | 16 |
| 21 | 0 | 0 | 7 |
| 22 | 0 | 0 | 5 |
| 23 | 0 | 0 | 1 |
| 24 | 0 | 0 | 0 |
| Total | 40,147 | 27,637 | 18,445,778 |
Chop event percentages:
| Chop Events | 2361 Live Shoes | Zumma 600 + 1000 | Virtuoid 1M Computer Generated Shoes |
| 1 | 50.7385% | 50.4324% | 50.6715% |
| 2 | 25.1102% | 24.7784% | 25.0086% |
| 3 | 12.3795% | 12.6171% | 12.3288% |
| 4 | 5.9307% | 6.1874% | 6.0850% |
| 5 | 2.8993% | 2.9960% | 2.9938% |
| 6 | 1.4796% | 1.6283% | 1.4802% |
| 7 | 0.7223% | 0.7200% | 0.7261% |
| 8 | 0.3612% | 0.3474% | 0.3586% |
| 9 | 0.1918% | 0.1664% | 0.1754% |
| 10 | 0.0897% | 0.0724% | 0.0880% |
| 11 | 0.0399% | 0.0217% | 0.0422% |
| 12 | 0.0324% | 0.0109% | 0.0213% |
| 13 | 0.0050% | 0.0109% | 0.0107% |
| 14 | 0.0075% | 0.0109% | 0.0049% |
| 15 | 0.0075% | 0.0000% | 0.0025% |
| 16 | 0.0050% | 0.0000% | 0.0011% |
| 17 | 0.0000% | 0.0000% | 0.0006% |
| 18 | 0.0000% | 0.0000% | 0.0003% |
| 19 | 0.0000% | 0.0000% | 0.0001% |
| 20 | 0.0000% | 0.0000% | 0.0001% |
| 21 | 0.0000% | 0.0000% | 0.0000% |
| 22 | 0.0000% | 0.0000% | 0.0000% |
| 23 | 0.0000% | 0.0000% | 0.0000% |
| 24 | 0.0000% | 0.0000% | 0.0000% |
Follow-opposite set wins are next examined. For every set of 6 player-banker decisions, there are 5 follows-opposites. (Follows is synonymous with repeats.) Either the number of follows or opposites will dominate in each set. The following table tabulates the number of sets in which follows or opposites dominate (wins).
Statistics for follow-opposite set wins:
| Data Set | Follows Set Wins | Opposites Set Wins | Total | Follows Set Wins Percentage | Opposites Set Wins Percentage |
| Zumma 600 | 3,288 | 3,282 | 6,570 | 50.05% | 49.95% |
| Zumma 1000 | 5,872 | 5,882 | 11,754 | 49.96% | 50.04% |
| Live 2361 | 13,337 | 13,297 | 26,634 | 50.08% | 49.92% |
| Virtuoid 1M | 6,124,421 | 6,119,716 | 12,244,137 | 50.02% | 49.98% |
Disclaimer: The betting strategies and results presented are for educational and entertainment purposes only. Gambling involves substantial risks, and the odds are not in the player’s favor by design. The author does not state nor imply any system, method, or approach offers users any advantage, and he shall not be held liable under any circumstances for any losses whatsoever.
February 26, 2011 at 8:02 pm
[...] 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 [...]
March 4, 2011 at 2:23 pm
Hello again,
Just wondering how you tabulate the very last “run” of the shoe? For example if a shoe ends with these 5 hands: PPP BB do you count that as a run of 2 B’s?
Reason I ask is: in this example it’s obvious that the run of 3 P’s came to a definite end when the B showed up. Whereas, the run of 2 B’s didn’t really get a chance to come to a definite end, or continue with another B. I think I read else-where (Liar’s) that when doing your tests you consider each shoe as a seperate entity so it wouldn’t be determined by the first hand(s) of the next shoe, which I totally agree with. However, I believe counting those 2 B’s as a run of 2 slightly skews the data (percentage-wise) to the upside for runs of 1 & to the downside for runs of 2 or more. The way I eliminate this slight error is to totally eliminate the very last run of the shoe, whether it’s a run of 1 player or it’s a run of 10 Bankers. If I am correct & you were to re-run your trials with the last run of the shoe eliminated as if it didn’t happen then I would expect the percentages for runs of one to go down slightly & the percentages for runs of 3 & more to go up slightly. (2′s might be about the same?)
Hope this makes sense. It’s kind of a nebulous concept. The only reason I mention it at all is that I can tell that you are very careful about getting everything as exact as possible. I realize that this is a very small difference in percentages, & is more or less N/A unless you’ve got OCD like me
Still seeking zero, Rick
March 4, 2011 at 4:43 pm
@Rick
Thanks for your suggestion. My original analysis included all events in the shoe, including the final one. For the sake of comparison, I amended the posting to include the statistics which interest you, omitting the count of the final event in the shoe. To help you compare the differences in the percentages of the total events, I provide a table of deltas of B %, P %, and Total % at the very end. While the smaller live data sets didn’t consistently show what you had expected, the much larger computer generated data set of 1 million shoes did. Hopefully the expected difference helps you in your search for zero
By the way, with the same reasoning that the last event never really “ends,” so it could be argued that the first event never really “begins.” Thus, someone could argue that both the first and last events in a shoe should be omitted. But that would be creating even more work for me, so we’ll just pretend I never pointed that out, LOL.
March 4, 2011 at 5:44 pm
You are exactally correct about the first event. That very thought has been bugging me for years, because I have always counted it in charting my shoes for b runs & p runs.
In addition to p runs & b runs I also chart chop runs. When charting chop runs I have always eliminated first & last events because it is much more obviously a problem. IE: if a shoe starts out P B P B PP is that a chop run of 4 or 5? Answer is neither, & I wait for one side to repeat b/4 I start counting chops. When charting chop runs I count PPP BB as a chop run of one. PPP B PP is a chop run of two. PPP B P BB is a chop run of three. Etc.
If you ran 1 million shoes the results would probably fall right in between your percentage results for P runs & B runs. OR would it?
March 4, 2011 at 8:28 pm
@Rick
Interestingly, you count chop runs the same way as my friend Archer does. We have performed analysis for these kinds of chop runs, and the statistics come out to be as you expected. Unfortunately we have been unable as of yet to find reliable and consistent indicators which would allow us to exploit these types of statistics, since everything turns out to be essentially 50/50 in the long run, and any deviation from being exactly 50/50 as expected by the drawing rules is neutralized by commissions.
March 11, 2011 at 4:25 pm
Do you think you could post (or send me via e-mail) the data from your analysis of chop runs? Would greatly appreciate it.
Thanks for all your work, Rick
March 11, 2011 at 4:29 pm
@Rick
Sure thing. I have it buried somewhere, but I think I can dig it up. It’s exactly the same ratios as the P/B events. We’ve looked at them in other ways, too, and everything comes out evenly, such that there are no exploitable biases, at least not that we could find.
March 11, 2011 at 10:30 pm
@Rick
I’ve updated the post to include chop event statistics as well as the results of analyzing follow-opposite set wins, which is a test based on sets of chop events.
March 13, 2011 at 12:58 am
Thank You for posting the chop data. Looks like everything turns out as it should and I couldn’t find anything earth-shaking in there either. I tend to think about Bac in terms of runs — either B or P or chop. Unfortunately, as one would expect, it seems to turn out that the numbers of runs of various lengths all fall in line, especially when looking at the larger sample sizes. It is, however, very useful to me to look at results of very large samples because it reminds me that just because the last 3 shoes I played had a larger than expected number of long runs, doesn’t mean that the trend will continue, or end.
Thanks again for your help, Rick
PS. Saved me alot of time because charting chop runs is a real pain in the A**.
March 13, 2011 at 8:48 am
@Rick
You’re quite welcome. Yeah, some of the methods I’ve tested in the rest of the blog do some insane tracking and derivative calculations, much more involved than chop and run event counts like these, but they are all unexploitable and not worth the effort tracking or calculating.
March 5, 2011 at 6:45 pm
When you run the million shoe program does it always turn out very close to 50.68% Bank & 49.32% Player?
March 5, 2011 at 7:00 pm
@Rick
Yes, out of 75,793,090 total non-tie decisions, 38,406,392 (50.673%) are Banker, and 37,386,698 (49.327%) are Player.
March 12, 2011 at 9:36 am
I also do not count first events in m play. Last events don’t come into play too much because I generally will never see the last event since I will have a stop at end of shoe.
March 29, 2011 at 8:02 am
[...] However, as I performed statistical tests of live and my computer generated shoes, I could find nothing to distinguish one from the other. Please refer to the following studies in which I performed detailed analysis and comparisons of Zumma 600+1000 (supposedly gathered from physical casinos), 2361 Live Shoes (all of which were personally collected by my friend and most of which were shuffled by SM), and my computer generated shoes:
P and B Events Statistics: A Comprehensive Comparison
My Baccarat Shoe Factory
I’ve performed many other tests, too, and I’m too lazy to dig up the results right now, lol. But in all my studies, I could find no signature in live SM or hand-shuffled shoes which would indicate they are not random and not for all intents and purposes identical to my computer generated, completely random shoes. [...]
June 15, 2011 at 9:36 pm
So what is your final take on live shoes? Are they not biased at all then? Or maybe there is shuffle control involved to create bias but you took a large enough amount of sample (2,361 shoes) to average the statistic out? What’s your thought? Thanks.
P.S. What is/are your source(s) for the 2,361 live shoes?
June 16, 2011 at 8:35 am
@S
Based on my analysis, I find no evidence that live shoes contain any persistent, artificial bias.
In any small enough sample, normal variance in randomness can apparently exhibit a bias, but that occurs naturally in any random set. Over a large enough sample, the live data set yields overall statistical properties of a random sample.
Someone might then argue that it’s impossible to really determine whether casinos were artificially controlling the shuffle, because if half the time they’re producing a streaky bias, and the other half of the time, they’re producing a choppy bias, then how would that differ statistically from a “truly” random distribution?
My response to that would be: That may be true, but there are many reasons why shuffle control is bad for the House (ref. Shuffle Control: Why It’s Bad for the House), so why would they want or need to do it?
Moreover, the key question is: Can you consistently exploit a bias to your advantage when playing? What indicators will tell you the bias is one way or the other, so that you can consistently profit? If you can accurately detect the bias in live play, whether the bias is artificially or naturally occurring, then you will have the key to winning baccarat.
If you do have such an indicator, we should be able to computationally simulate it, and thus objectively demonstrate its effectiveness in winning. So far, though, I have not yet been able to find such an indicator which consistently and accurately determines the present bias.
Regarding the sources for the 2,361 live shoes, all were hand recorded during live play from B&M casinos which use the Shufflemaster shuffle machine, except for a few of the shoes which were hand shuffled. The B&M casinos are: Horseshoe, Imperial Palace, Barbary Coast, Emerald Queen, Orleans, Gold Coast, Sunset Station, Tulalip, Green Valley, Harrah’s, Mohegan Sun, and Foxwoods.
June 16, 2011 at 3:36 pm
Thank you for your reply, but more importantly, all your effort and hard work.
Ellis and DiMenosCor (baccaratforums.com) also claims from first hand experience that hand shuffles create streakier shoes more often then not. Would you have a big enough sample size – of hand shuffled shoes – to test?
I tend to trust the man with the stats to back it up than their (Ellis, DiMenosCor) perceived reality.
June 17, 2011 at 8:29 am
@S
Thanks!
Unfortunately, I don’t have hundreds or thousands of specifically hand shuffled shoes to test Ellis & DiMenosCor’s observations.
I did once ask a baccarat dealer (who hand shuffled) whether he believed some dealers could manipulate the bias of a shoe during a hand shuffle. He affirmed it was possible, because he had personally witnessed someone who demonstrated the skill. But he said that if someone applied that kind of skill to fix a real game, it would definitely be illegal.
I suppose it’s the same kind of skill that some craps shooters have to control the outcome of a throw, or that some roulette dealers have to control the outcome of a spin. Because these are mechanical events, one can imagine that the fine motor control needed to master the shuffle, throw, and spin can be honed through experience and practice.
June 16, 2011 at 3:24 pm
Thanks for your reply.
I also find it fascinating how the P dominates specifically with the “1″ events and about breaks even with the “2″ events and then B starts to dominate with the “3″ events and up ward, as it should. My rationale would tell me that B should dominate from the very first event of “1″ onto upward. Why is that?
S
June 17, 2011 at 8:17 am
@S
Good observation. It makes sense that P has more 1 events than B, since B is more likely due to the drawing rules. That is, because B is the more likely result, there is a greater chance that a B will follow a P (hence making the P runs shorter), as well as that a B will follow a B (hence making B runs longer). So, you would expect P will dominate the shorter events, and at some point B will dominate the longer events. According to my analysis, P clearly dominates the 1s, and B clearly dominates the 3+s, and the cross-over event where they are essentially neck-and-neck appears to be the 2s.
June 23, 2011 at 1:16 am
Thanks, that clears it up pretty darn good! One last query for me for awhile:
[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,
etc …]
Does the above axiom still hold true for a 6 deck shoe – or say a 10 deck or 12 deck shoe for that matter (not that the last two exists in baccarat)?
It may be an inept question to a possible obvious answer I may know already but for the sake of my peace of mind, I am troubling you for the last time in asking you. Thanks.
S
June 23, 2011 at 7:44 am
@S
Thanks for your question. The relative frequency of events should follow the “50% rule” for any number of decks.
The “shoe” is purely an illusory entity with completely artificial boundaries. The only reason why casino use more than one deck in a shoe is to make it harder for card counters. In terms of events in the long run, the relative frequencies will always follow the “50% rule,” because the probability of getting a Banker or Player is basically equal regardless how many decks you use.
For example, the following are the probabilities of getting Banker and Player for 1, 6, and 8 decks in the shoe, not counting Ties:
1 deck:
B: 50.7096%
P: 49.2904%
6 decks:
B: 50.6837%
P: 49.3163%
8 decks:
B: 50.6824%
P: 49.3176%.
So, just like flipping a coin, it’s always basically 50/50 (with a slight edge to Banker, which is why commissions is charged for Banker wins), and thus, the “50% rule” for event frequencies applies in all cases.
June 24, 2011 at 6:42 pm
Do you think there is a correlation for all three of the Total Delta Percentage of the 1 events – B and P combined – being in the negative? Or it doesn’t mean anything and I am looking to much into it. My OCD is killing me and I need your opinion;-)
June 25, 2011 at 6:03 am
@S
Thanks for the interesting observation. All three of the Total Delta Percentage of the 1 events being negative means that the last event has a slightly greater chance of being a 2+ event than a 1 event, and hence the tiny excess of 1 events when the last event is not counted. So, in these data sets, if you wait until the very last hand to make a bet, you would have a slightly greater chance of winning that bet if you bet Repeat (to catch the run of a 2+ event) than Opposite (to catch a 1 event chop). The tiny edge is on the order of hundredths of a percent, though, so it does not appear very practical.
July 7, 2011 at 4:48 pm
Thanks, I have learn so much Baccarat from this perspective.
July 7, 2011 at 4:40 pm
Hi Imspirit,
For better or worse – or both, I have been obsessed with this particular chart and statistic page of your wordpress blog and as I have told you my OCD won’t let up so I am writing again to ask you one final question before I move on to other interesting topics on your site.
I now know B side events are all theoretically and mathematically closed to 50%, based on my calculations, from the prior event down the line, starting from the 1 events and downward to the 2 events, 3 events, etc., at least up until we reach the erratic and volatile higher numbered events which we should ignore.
While the P side events is skewed just slightly with the 1 events being around 51% from all total events (thankfully from your explanation); while the 2 events and on downward are around 48% from each subsequent prior event, again, at least up until we reach the erratic and volatile higher numbered events which we should ignore.
The interesting part for me is that when all was said and done, the total amount of events on the B and the P side were practically dead even with mere hundredths or thousandths of a difference in percentage.
I also know that with an average of say 75+ hands – not including ties – there is at least one banker decision over player decision per shoe on average; so with over 1 million shoes, there should be around 1 million more banker decisions then player.
So my question is: does the sheer amount of 1 events on the P side
balance out, as evident on your charts, the top heavy B side decision over P side decision mentioned one paragraph above? (Because logic would tell you that the B side would create more overall events down the line due to more B side decisions except the 1 events on the P side.)
Or do you think if we were to do another 1 million shoe, god forbid, would the B side have a slight advantage in overall event count – at least over the 50% mark versus the P side?
Thank you once again.
S
July 7, 2011 at 9:05 pm
@S – Thanks for your comments. I don’t understand your questions, though. Please rephrase. Thanks.
July 7, 2011 at 10:59 pm
Sorry I’ll keep it short,
The total count for B side and P side events are darn near equal.
Before omitting final event:
2,361 Live Shoes:
B P Total B % P %
41,878 41,925 83,803 49.972% 50.028%
Zumma 600+1000:
B P Total B % P %
28,951 28,950 57,901 50.001% 49.999%
Virtuoid 1,000,000 Computer Generated Shoes:
B P Total B %
19,202,716 19,187,633 38,390,349 50.020%
P %
49.980%
After omitting final event:
2,361 Live Shoes:
B P Total B % P %
40,703 40,737 81,440 49.979% 50.021%
Zumma 600+1000:
B P Total B % P %
28,148 28,153 56,301 49.996% 50.004%
Virtuoid 1,000,000 Computer Generated Shoes:
B P Total B %
18,696,334 18,694,015 37,390,349 50.003%
P %
49.997%
So that tells me that B side and P side event count totals are practically dead even. The miniscule difference is in the hundredths or thousandths of a percent.
My question was, are the B and P sides event count total theoretically supposed to be equal even though there is at least one more Banker decision than Player per shoe on average?
July 8, 2011 at 11:00 am
@S
That’s a good question. I’ll have to think about it more, but I think it’s going to be 50/50 regardless of the odds of P/B. For example, imagine an extreme situation where one decision is 1,000 times more likely to appear than the other. That is, you can imagine a game where 1 P is expected for every 1,000 Bs. Then, the patterns you would get would be something like … P-(950 Bs)-P-(1100 Bs)-P-(1000 Bs)- … etc … (of course, there would probably be more variance in an actual sequence). Even with such an imbalance in the odds, the actual number of P events should come out to be roughly equal to the number of B events, and the only difference would be the average length of the events. Generalizing, it would seem that regardless of the odds for the individual decisions themselves, the total count of events for each should be about the same: 50/50.
I’ll think about it some more, but I think the above reasoning is probably on track. Let me know if you have additional thoughts about this.
July 8, 2011 at 2:30 pm
Thanks Imspirit,
I think that is the case as well; 1 million shoes with over 75 million decisions would be more than enough data to prove any stats that show up to be mathematical theory, right?
I am curious – not asking you in any way to assist me – is it completely automated doing that many Virtuoid shoes and coming up with all that statistics?
Because it would be a pain in the arse to manually count B and P side events and event counts by hand, I would think.
S.
July 8, 2011 at 7:13 pm
@S
You’re welcome –
Yes, all of the work I’ve posted in this blog is the result of computational simulations and analysis.
Indeed, only computerized studies can reveal the whole forest instead of just the trees.
I’m always open to new ideas regarding testing, so if you have any, feel free to propose them.
July 20, 2011 at 1:35 pm
” …but I think it’s going to be 50/50 regardless of the odds of P/B. For example, imagine an extreme situation where one decision is 1,000 times more likely to appear than the other. That is, you can imagine a game where 1 P is expected for every 1,000 Bs. Then, the patterns you would get would be something like … P-(950 Bs)-P-(1100 Bs)-P-(1000 Bs)- … etc … (of course, there would probably be more variance in an actual sequence). Even with such an imbalance in the odds, the actual number of P events should come out to be roughly equal to the number of B events, and the only difference would be the average length of the events. Generalizing, it would seem that regardless of the odds for the individual decisions themselves, the total count of events for each should be about the same: 50/50.”
To me that was pretty profound and obvious at the same time – only after your explanation. You know those times when would say to yourself, “Oh, yeah why didn’t I think of that”
Great observation. Some people have a knack to rationalize statistics and make sense as well.
July 22, 2011 at 11:55 am
@S Thanks much. If only we could rationalize a consistent way to beat the casinos.
October 28, 2011 at 6:59 pm
[...] In the same spirit as my previous analysis of card value removal from 100,000 8-deck shoes and its effect on Banker (B), Player (P), and Tie (T) expectancies in baccarat, in this analysis, I examine its effect on bias and B and P event frequencies over the same data sets. The definition of an event is given in my prior analysis of P and B event statistics in 1 million shoes. [...]