Question for Norm: Color of BJ

[k_c]

Aha, I see what your point is and I totally understand it. The "minimum TC" should be the starting point since we know that from there on down the EV is always negative. From there on up we could obtain the sum of the product between the EV and the probability and find at what point that average is maximum. I think that would be the best. It is likely that we will find some negative points along the way. Ideally, they should be the least.
Do you agree?

Anyway, I always check the indices by simulation with one, two and even three decimal places. I have also verified the TKO ones.
For SD the result is exact. For 8D it is +3.1. Later I will try the simulation with two decimal places. Stay online.

Sincerely,
Cac

I have to get up to go to work tomorrow so I have to go offline, but I think I see your point.

The probabilities of the subsets prior to the one with the minimum true count with an advantage are such that they only minimally change the best overall TC from the min TC with advantage?

k_c

[Cacarulo]

I have to get up to go to work tomorrow so I have to go offline, but I think I see your point.

The probabilities of the subsets prior to the one with the minimum true count with an advantage are such that they only minimally change the best overall TC from the min TC with advantage?

k_c

Prior to the minimum they are all negatives. At the minimum TC you have your first advantage but you can find sporadic negative advantages along the way up. I'll be back to you.

Sincerely,
Cac

[Cacarulo]

Well, after thinking about it a little more and based on what we discussed yesterday, I have modified my code and now I can tell you that I am 100% confident in the solution. On the other hand, I have verified it through simulations and the results fully coincide.
The following is an extract of an spreadsheet ordered by TC from lowest to highest, where the variation I was talking about is observed. You can see that the correct index is a bit higher than the minimum I was calculating.

	TC	Prob	EV	Prob * EV	Sum (Prob * EV)
( 11,185)	3,091892	0,01961051070365850	0,00105372979730678	0,00002066417946885	57,63264831624000000
( 22,370)	3,091892	0,00001119686997451	-0,52053179323200400	-0,00000582832680642	57,63262765206050000
( 16,269)	3,092937	0,00699640089763293	-0,04516980288349350	-0,00031602604943998	57,63263348038730000
( 21,353)	3,093484	0,00016383044305745	-0,31268406537960900	-0,00005122716896815	57,63294950643680000
( 10,168)	3,095238	0,02253357234170620	0,00980235711576238	0,00022088212318727	57,63300073360570000
( 15,252)	3,095238	0,00926736031581960	-0,02266881950064860	-0,00021008011824679	57,63277985148260000
( 20,336)	3,095238	0,00068818884366625	-0,20241674188064000	-0,00013930094353353	57,63298993160080000
( 19,319)	3,097179	0,00169310018565647	-0,13258038116458200	-0,00022447186796416	57,63312923254430000
( 14,235)	3,097872	0,01167981033096440	-0,00382770371993857	-0,00004470685345201	57,63335370441230000
( 23,386)	3,098446	0,00000001304565646	-1,06640724147185000	-0,00000001391198252	57,63339841126580000
( 9,151)	3,099338	0,02566602475060430	0,01750558208539750	0,00044929870307755	57,63339842517770000
( 18,302)	3,099338	0,00314652663647684	-0,08361742468006030	-0,00026310445402941	57,63294912647470000
( 22,369)	3,100271	0,00001335156194329	-0,48687067948827200	-0,00000650048403556	57,63321223092870000
( 13,218)	3,100917	0,01420788242596710	0,01245554529758500	0,00017696692313940	57,63321873141270000
( 17,285)	3,101754	0,00495703281637997	-0,04681633921127530	-0,00023207012981307	57,63304176448960000
( 21,352)	3,102273	0,00017669609004404	-0,28612575351597600	-0,00005055730190718	57,63327383461940000
( 8,134)	3,104478	0,02909105041479480	0,02438375912503510	0,00070934916600861	57,63332439192130000

My simulation results for 8D were: +3.1 with one digit precision and +3.09 with 2 digits and 104 cards remaining.
Tomorrow I will be posting the new indices from 1D to 8D. The results are not very different from what I had previously published but they are much more accurate.

Sincerely,
Cac

[k_c]

If the goal is to get the single most efficient TC index, my inclination to do this is to use a mid-shoe TC calculation as the index approximation. I wouldn't know the accuracy of this. I guess we'll see from Cac's sims.

Resulting TC insurance indexes for HiLo and true counted KO from mid-shoe:

          HiLo          KO

1 deck    1.33          -0.67
2 decks   2.41          -0.65
4 decks   2.86          -0.62
6 decks   3.02          -0.62
8 decks   3.10          -0.62

k_c

[Cacarulo]

If the goal is to get the single most efficient TC index, my inclination to do this is to use a mid-shoe TC calculation as the index approximation. I wouldn't know the accuracy of this. I guess we'll see from Cac's sims.

Resulting TC insurance indexes for HiLo and true counted KO from mid-shoe:

          HiLo          KO

1 deck    1.33          -0.67
2 decks   2.41          -0.65
4 decks   2.86          -0.62
6 decks   3.02          -0.62
8 decks   3.10          -0.62

k_c

Why in the middle of the shoe? The goal is to find a single index that maximizes the expected value given the following parameters: number of decks, cards remaining, cards removed, and a card counting system. This index can be integer or decimal, floored or rounded. Below is the table that I calculated with my program for Hi-Lo and for TKO.
An important fact is that numbers for integer or decimal values with one digit of precision can be generated with the new version of CVDATA. Maybe Norman can help with the verification.

a) Hi-Lo

+----------+-------+-----+----------+-----------+-----+----------------------+
|   Play   | Decks |  CR | TC Index |  RC Index | IRC | Total EV >= TC Index |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   1   |  20 |      2   |      1    |   0 |  0.58171980020663483 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   2   |  26 |      2   |      3    |   0 |  0.97386472171957084 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   3   |  39 |      3   |      5    |   0 |  0.90293467695143381 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   4   |  52 |      3   |      6    |   0 |  0.83027321442020385 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   5   |  65 |      3   |      8    |   0 |  0.75983020092833753 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   6   |  78 |      3   |      9    |   0 |  0.69413408108466679 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   7   |  91 |      3   |     11    |   0 |  0.63388417844782019 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   8   | 104 |      3   |     13    |   0 |  0.57910212499303571 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|   Play   | Decks |  CR | TC Index |  RC Index | IRC | Total EV >= TC Index |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   1   |  20 |    1.4   |      1    |   0 |  0.58948461336093605 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   2   |  26 |    2.4   |      3    |   0 |  0.98281795951625006 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   3   |  39 |    2.7   |      5    |   0 |  0.90816953949120360 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   4   |  52 |    2.9   |      6    |   0 |  0.83159288103950568 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   5   |  65 |    3.0   |      8    |   0 |  0.75983020092833709 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   6   |  78 |    3.0   |      9    |   0 |  0.69413408108466634 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   7   |  91 |    3.1   |     11    |   0 |  0.63441572887679987 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   8   | 104 |    3.1   |     13    |   0 |  0.58054269141619064 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|   Play   | Decks |  CR | TC Index |  RC Index | IRC | Total EV >= TC Index |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   1   |  20 |   1.36   |      1    |   0 |  0.58949259373855689 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   2   |  26 |   2.40   |      3    |   0 |  0.98281795951624784 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   3   |  39 |   2.73   |      5    |   0 |  0.90817947158494905 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   4   |  52 |   2.88   |      6    |   0 |  0.83169540860503743 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   5   |  65 |   2.97   |      8    |   0 |  0.75994376091156390 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   6   |  78 |   3.03   |      9    |   0 |  0.69421662302300535 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   7   |  91 |   3.07   |     11    |   0 |  0.63455736758509551 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   8   | 104 |   3.10   |     13    |   0 |  0.58054269141618986 |
+----------+-------+-----+----------+-----------+-----+----------------------+

b) TKO

+----------+-------+-----+----------+-----------+-----+----------------------+
|   Play   | Decks |  CR | TC Index |  RC Index | IRC | Total EV >= TC Index |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   1   |  20 |     -3   |     -1    |  -4 |  0.62207920068091205 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   2   |  26 |     -2   |     -1    |  -8 |  1.03126899891434554 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   3   |  39 |     -1   |     -1    | -12 |  0.96589993725524714 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   4   |  52 |     -1   |     -1    | -16 |  0.89567516597613084 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   5   |  65 |     -1   |     -2    | -20 |  0.82406581226090381 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   6   |  78 |     -1   |     -2    | -24 |  0.75620947896421276 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   7   |  91 |     -1   |     -2    | -28 |  0.69325066039581429 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   8   | 104 |     -1   |     -2    | -32 |  0.63555057036254503 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|   Play   | Decks |  CR | TC Index |  RC Index | IRC | Total EV >= TC Index |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   1   |  20 |   -2.7   |     -1    |  -4 |  0.62411302793561507 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   2   |  26 |   -1.6   |     -1    |  -8 |  1.04217654345964661 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   3   |  39 |   -1.3   |     -1    | -12 |  0.97122453130055963 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   4   |  52 |   -1.1   |     -1    | -16 |  0.89657584952559255 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   5   |  65 |   -1.0   |     -2    | -20 |  0.82406581226090447 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   6   |  78 |   -0.9   |     -2    | -24 |  0.75632462454726679 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   7   |  91 |   -0.9   |     -2    | -28 |  0.69476873014568830 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   8   | 104 |   -0.9   |     -2    | -32 |  0.63837723117899026 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|   Play   | Decks |  CR | TC Index |  RC Index | IRC | Total EV >= TC Index |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   1   |  20 |  -2.67   |     -1    |  -4 |  0.62411302793561529 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   2   |  26 |  -1.60   |     -1    |  -8 |  1.04217654345964506 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   3   |  39 |  -1.26   |     -1    | -12 |  0.97135441954627266 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   4   |  52 |  -1.11   |     -1    | -16 |  0.89658797329402284 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   5   |  65 |  -1.01   |     -2    | -20 |  0.82407138460075480 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   6   |  78 |  -0.95   |     -2    | -24 |  0.75658940195274205 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   7   |  91 |  -0.90   |     -2    | -28 |  0.69476873014568685 |
+----------+-------+-----+----------+-----------+-----+----------------------+
|    Ins   |   8   | 104 |  -0.87   |     -2    | -32 |  0.63854885119060589 |
+----------+-------+-----+----------+-----------+-----+----------------------+

Sincerely,
Cac