There are some unproven concepts and unverified conclusions floating around this thread and also this one:
https://www.blackjacktheforum.com/sh...CSM-simulation
I've run a few sims in an effort to confirm the facts about what is being discussed.
1 - Last Card of a Round Bias?
The idea from which most of the rest of discussion stemmed was that the running count during the first round of a shoe tends toward the negative. We'll deal with that soon, but this idea was supported by the suggestion that a round of basic strategy blackjack tends to end with a high card and, in particular with a 10-valued card.
Norm correctly pointed out that there are more 10s than any other rank (4/13 vs 1/13), so this makes some sense. But if we're using this notion to support the idea that the running count tends toward the negative during the first round, what we really want to know is whether the negatively tagged ranks tend to end a round more than one would expect by random chance. The cards are dealt randomly (or at least unpredictably) and so the final card, which is the only one subject to a stopping decision, can be the only source of such a bias.
Because it's the most commonly used counting system and because it's probably the one being assumed here, let's use Hi-Lo. The negatively tagged ranks are Ace and the 10-valued cards. All sims were run using 6D S17 DOA DAS SPL3 SA1 (one card on split aces) and were run for 2 billion rounds using my own software.
I first wanted to know whether this bias exists in general, so I ran a sim with 7 players heads up vs the dealer and 4.5/6 decks in play. I used 7 players because I wanted to isolate the dealer's role in this issue. With 7 players at the table, the dealer almost always plays out his hand. Here is the resulting distribution of the final card dealt during the simulation. For reference 1/13 = 7.6923% and 4/13 = 30.7692%:
Code:
A 2 3 4 5 6 7 8 9 10
------------------------------------------------------------------------------------------------------------
5.9235% 3.3775% 4.8635% 6.2381% 7.5187% 7.8666% 8.1897% 8.4965% 8.7871% 38.7389%
We can see that there is a general trend for which high value cards end the round more frequently and the low value cards less so. The Ace is a special case because it is both a high and low card. In particular though, we see that 10-valued cards end the round much more often than randomness suggests (38.7389% vs 30.7692%). Also, while the ace finishes a round less often than if it were random (it is used as a low card more of than than as a high card), the 10s and aces together finish a round 44.6624% of the time which is more than the sum of the probabilities for 2 through 6 which is 29.8644%.
So the final card of a round does tend toward the negative significantly in general while the remaining cards dealt during the round should come randomly. This suggests a potential bias toward the negative in general. Here are the final card distributions for a few more scenarios:
4.5/6 decks, one player. The trend persists even when the dealer finishes his hand less often:
Code:
A 2 3 4 5 6 7 8 9 10
------------------------------------------------------------------------------------------------------------
6.3118% 2.9461% 4.0992% 5.1693% 6.1641% 6.8743% 8.0664% 8.7357% 9.3868% 42.2462%
And since we're going to move on and talk about the initial round of the shoe next, here are the distributions for 7 and players and 1 player respectively for the initial round only:
Code:
7 Players:
A 2 3 4 5 6 7 8 9 10
------------------------------------------------------------------------------------------------------------
5.9236% 3.3716% 4.8576% 6.2340% 7.5149% 7.8641% 8.1863% 8.4963% 8.7878% 38.7638%
1 Player:
A 2 3 4 5 6 7 8 9 10
------------------------------------------------------------------------------------------------------------
6.3141% 2.9400% 4.0921% 5.1624% 6.1585% 6.8682% 8.0632% 8.7352% 9.3904% 42.2760%
We can see that the frequencies for 1 round are very close to those for 4.5/6
Running Count Bias for the First Round of the Shoe?
Based on the bias confirmed above, it was suggested that the running count during the first round of a shoe should tend toward the negative. I once again rans sims using 7 players and one player. Here is the distribution of the Hi-Lo running count after one round has been played off the top of the shoe with 7 players:
Code:
-19.0: 4 = 0.00%
-18.0: 54 = 0.00%
-17.0: 390 = 0.00%
-16.0: 2,787 = 0.00%
-15.0: 16,178 = 0.00%
-14.0: 79,986 = 0.00%
-13.0: 315,037 = 0.02%
-12.0: 1,014,843 = 0.05%
-11.0: 2,761,620 = 0.14%
-10.0: 6,500,571 = 0.33%
-9.0: 13,532,477 = 0.68%
-8.0: 25,288,297 = 1.26%
-7.0: 42,962,343 = 2.15%
-6.0: 66,837,622 = 3.34%
-5.0: 95,921,440 = 4.80%
-4.0: 127,625,071 = 6.38%
-3.0: 158,002,569 = 7.90%
-2.0: 182,605,477 = 9.13%
-1.0: 197,475,935 = 9.87%
0.0: 200,250,564 = 10.01%
1.0: 190,742,782 = 9.54%
2.0: 171,053,267 = 8.55%
3.0: 144,616,397 = 7.23%
4.0: 115,489,675 = 5.77%
5.0: 87,247,180 = 4.36%
6.0: 62,435,664 = 3.12%
7.0: 42,386,408 = 2.12%
8.0: 27,326,377 = 1.37%
9.0: 16,739,413 = 0.84%
10.0: 9,760,852 = 0.49%
11.0: 5,422,912 = 0.27%
12.0: 2,868,135 = 0.14%
13.0: 1,450,538 = 0.07%
14.0: 699,716 = 0.03%
15.0: 322,818 = 0.02%
16.0: 143,470 = 0.01%
17.0: 60,947 = 0.00%
18.0: 24,541 = 0.00%
19.0: 9,769 = 0.00%
20.0: 3,772 = 0.00%
21.0: 1,359 = 0.00%
22.0: 493 = 0.00%
23.0: 161 = 0.00%
24.0: 60 = 0.00%
25.0: 21 = 0.00%
26.0: 4 = 0.00%
27.0: 4 = 0.00%
Positive = 878,806,735 = 43.94%
Zero = 200,250,564 = 10.01%
Negative = 920,942,701 = 46.05%
The running count does indeed end up negative more often than positive. However, since RC=0 is part of the TC=0 bin, the true count remains non-negative more often than it goes negative.
A similar trend is shown with a single player in action:
Code:
-10.0: 24 = 0.00%
-9.0: 320 = 0.00%
-8.0: 3,335 = 0.00%
-7.0: 26,437 = 0.00%
-6.0: 373,720 = 0.02%
-5.0: 1,581,197 = 0.08%
-4.0: 49,153,892 = 2.46%
-3.0: 155,829,745 = 7.79%
-2.0: 292,373,869 = 14.62%
-1.0: 344,666,114 = 17.23%
0.0: 397,224,920 = 19.86%
1.0: 311,213,292 = 15.56%
2.0: 219,965,408 = 11.00%
3.0: 124,904,784 = 6.25%
4.0: 63,080,489 = 3.15%
5.0: 26,309,655 = 1.32%
6.0: 9,326,538 = 0.47%
7.0: 2,910,546 = 0.15%
8.0: 799,753 = 0.04%
9.0: 190,590 = 0.01%
10.0: 45,864 = 0.00%
11.0: 13,323 = 0.00%
12.0: 4,292 = 0.00%
13.0: 1,359 = 0.00%
14.0: 379 = 0.00%
15.0: 109 = 0.00%
16.0: 35 = 0.00%
17.0: 9 = 0.00%
18.0: 2 = 0.00%
Positive = 758,766,427 = 37.94%
Zero = 397,224,920 = 19.86%
Negative = 844,008,653 = 42.20%
This is where Don referred us to Thorp's theorum stating that EV is invariant for all basic strategy rounds throughout the shoe and asked: How can this be if the second round of the shoe is biased toward beginning with a negative running count? I can't say whether these results violate Thorp's theorum, mainly because I don't have access to the proof in order to evaluate the situation mathematically and I don't want to speculate.
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