Surrender Question

[k_c]

Take your time, I'm not in a hurry.
We both agree that there are 12 subsets that make up -13. Regarding the difference between 4.054326 and 4.028648, I would have to check my code since I haven't touched it for more than 20 years.
As I told you before, I am not 100% sure that my number is correct. The algorithm was never compared to other researchers as no one was interested in subset generation at the time.
What my algorithm does is to generate a subset given the following data: RC, remaining cards, and a card counting system. Before generating it I have the possibility to remove the cards that I want. Although the algorithm works perfectly, it does not mean that what it generates is correct.

For checking purposes and whenever you have time, could you pass me your generated subset with an Ace removed for an RC of +1 and 38 cards remaining? This would be SD.

Thank you.

Sincerely,
Cac

Here are the rank probs I get for single deck, RC=+1, 38 cards remaining, 1 ace specifically removed
If you want number of rank just multiply each rank prob by 38.

Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 38
Initial running count (full shoe): 0
Running count: 1
Specific removals (1 - 10): {1,0,0,0,0,0,0,0,0,0}

Subgroup removals: None

Number of subsets for above conditions: 6
Prob of running count 1 with above removals from 1 deck: 0.11848

p[1] 0.062504  p[2] 0.073909  p[3] 0.073909  p[4] 0.073909  p[5] 0.073909
p[6] 0.073909  p[7] 0.078199  p[8] 0.078199  p[9] 0.078199  p[10] 0.33336

Press any key to continue:

It helps me to solve a simple problem and then work to apply to other problems. Below is data for the simplest subset (1 card) dealt from 2 decks with various specific removals:

2 decks, HiLo

Cards           Specific
Remaining   RC  Removals  Rank Probs                                                       Prob of Subset

1           -1  none      p[1] 0  p[2] 0.2  p[3] 0.2  p[4] 0.2  p[5] 0.2                   0.38462
                          p[6] 0.2  p[7] 0  p[8] 0  p[9] 0  p[10] 0

1           -1  A         p[1] 0  p[2] 0.2  p[3] 0.2  p[4] 0.2  p[5] 0.2                   0.38835
                          p[6] 0.2  p[7] 0  p[8] 0  p[9] 0  p[10] 0

1           -1  A,7       p[1] 0  p[2] 0.2  p[3] 0.2  p[4] 0.2  p[5] 0.2                   0.39216
                          p[6] 0.2  p[7] 0  p[8] 0  p[9] 0  p[10] 0

1           -1  A,7,T     p[1] 0  p[2] 0.2  p[3] 0.2  p[4] 0.2  p[5] 0.2                   0.39604
                          p[6] 0.2  p[7] 0  p[8] 0  p[9] 0  p[10] 0

1           -1  7,T       p[1] 0  p[2] 0.2  p[3] 0.2  p[4] 0.2  p[5] 0.2                   0.39216
                          p[6] 0.2  p[7] 0  p[8] 0  p[9] 0  p[10] 0

1           -1  2         p[1] 0  p[2] 0.17949  p[3] 0.20513  p[4] 0.20513  p[5] 0.20513   0.37864
                          p[6] 0.20513  p[7] 0  p[8] 0  p[9] 0  p[10] 0
______________________________________________________________________________________________________

1           0   none      p[1] 0  p[2] 0  p[3] 0  p[4] 0  p[5] 0                           0.23077
                          p[6] 0  p[7] 0.33333  p[8] 0.33333  p[9] 0.33333  p[10] 0

1           0   A         p[1] 0  p[2] 0  p[3] 0  p[4] 0  p[5] 0                           0.23301
                          p[6] 0  p[7] 0.33333  p[8] 0.33333  p[9] 0.33333  p[10] 0

1           0   A,7       p[1] 0  p[2] 0  p[3] 0  p[4] 0  p[5] 0                           0.22549
                          p[6] 0  p[7] 0.30435  p[8] 0.34783  p[9] 0.34783  p[10] 0

1           0   A,7,T     p[1] 0  p[2] 0  p[3] 0  p[4] 0  p[5] 0                           0.22772
                          p[6] 0  p[7] 0.30435  p[8] 0.34783  p[9] 0.34783  p[10] 0

1           0   7,T       p[1] 0  p[2] 0  p[3] 0  p[4] 0  p[5] 0                           0.22549
                          p[6] 0  p[7] 0.30435  p[8] 0.34783  p[9] 0.34783  p[10] 0

1           0   2         p[1] 0  p[2] 0  p[3] 0  p[4] 0  p[5] 0                           0.23301
                          p[6] 0  p[7] 0.33333  p[8] 0.33333  p[9] 0.33333  p[10] 0
_______________________________________________________________________________________________________

1           1   none     p[1] 0.2  p[2] 0  p[3] 0  p[4] 0  p[5] 0                          0.38462
                         p[6] 0  p[7] 0  p[8] 0  p[9] 0  p[10] 0.8

1           1   A        p[1] 0.17949  p[2] 0  p[3] 0  p[4] 0  p[5] 0                      0.37864
                         p[6] 0  p[7] 0  p[8] 0  p[9] 0  p[10] 0.82051

1           1   A,7      p[1] 0.17949  p[2] 0  p[3] 0  p[4] 0  p[5] 0                      0.38235
                         p[6] 0  p[7] 0  p[8] 0  p[9] 0  p[10] 0.82051

1           1   A,7,T    p[1] 0.18421  p[2] 0  p[3] 0  p[4] 0  p[5] 0                      0.37624
                         p[6] 0  p[7] 0  p[8] 0  p[9] 0  p[10] 0.81579

1           1   7,T      p[1] 0.2  p[2] 0  p[3] 0  p[4] 0  p[5] 0                          0.38235
                         p[6] 0  p[7] 0  p[8] 0  p[9] 0  p[10] 0.8

1           1   2        p[1] 0.2  p[2] 0  p[3] 0  p[4] 0  p[5] 0                          0.38835
                         p[6] 0  p[7] 0  p[8] 0  p[9] 0  p[10] 0.8

Finally you asked why I do not include up card as a specific removal when generating indexes. I agree that would be better but my CA computes EVs for an input hand comp for all up cards. When I went into generating indexes I adapted to my CA as is. I do include an ace up card as a specific removal as well as hand comp when generating insurance indexes, however.

Basically what I do is to weight each of the possible count subsets by their probabilities. There was a previous thread that asked about the probability of a running count at a given pen. I added this page to my website and it includes how I compute subset probs. I use these to compute rank probs. http://www.bjstrat.net/RC_prob.html

It would be nice to somehow simplify this method somehow though. Maybe yours could.

k_c

[aceside]

Basically what I do is to weight each of the possible count subsets by their probabilities. There was a previous thread that asked about the probability of a running count at a given pen. I added this page to my website and it includes how I compute subset probs. I use these to compute rank probs. http://www.bjstrat.net/RC_prob.html

It would be nice to somehow simplify this method somehow though. Maybe yours could.

k_c

Wonderful work! I’ve read the details you posted. Let me just do a little rough math to estimate the whole thing.

The probability of the subgroup 26(2-6) 13(7-9) 13(T-A) is

[C(40, 26) xC(24,13) xC(40,13)] /C(104,52)
= [(2.3x10^10) x(2.5x10^6) x(1.2x10^10)] /(1.6x10^30)
=4.3x10^(-4).

The probability of the RC=-13 can be approximated as
2 x4.3x10^(-4)= 8.6x10^(-4)
These numbers sound good for me. Thank you!

[aceside]

Based on all these considerations, I propose these two HiLo surrender/stand indices for the hand 17vsA:
For the hand T-7vsA, TC=+1;
For the hand 9-8vsA, TC=+2.

[Cacarulo]

After reviewing my code I realized that the method I was using was a Gaussian approximation. Back in the days when computers weren't that fast, doing it under strict combinatorial analysis took a lot of processing time, especially in shoes. In fact, to obtain the probability of a certain RC with N cards remaining, it was much faster to use a Normal approximation (there is an explanation of how to do it in TOB). My idea is to try to improve the results using the Gaussian approach.
Anyway, I can also do it using combinatorial analysis. Here are my numbers to 12 digits of precision so we can see if our programs match:

1) RC =  +1 | CR = 38 | A removed (1D)

|   0.118480844761 |   0.062504209740   0.073908841110   0.073908841110   0.073908841110   0.073908841110   0.073908841110   0.078198599809   0.078198599809   0.078198599809    0.333355785282 |
|   0.118480844761 |   2.375159970136   2.808535962172   2.808535962172   2.808535962172   2.808535962172   2.808535962172   2.971546792759   2.971546792759   2.971546792759  12.667519840725 |

2) RC = -13 | CR = 52 | No cards removed (2D)

|   0.001319524919 |   0.051923076923   0.101923076923   0.101923076923   0.101923076923   0.101923076923   0.101923076923   0.076923076923   0.076923076923   0.076923076923    0.207692307692 |
|   0.001319524919 |   2.700000000000   5.300000000000   5.300000000000   5.300000000000   5.300000000000   5.300000000000   4.000000000000   4.000000000000   4.000000000000  10.800000000000 |

3) RC = -13 | CR = 52 | A removed (2D)

|   0.001748370518 |   0.046449210406   0.101757691595   0.101757691595   0.101757691595   0.101757691595   0.101757691595   0.077474361349   0.077474361349   0.077474361349    0.212339247570 |
|   0.001748370518 |   2.415358941111   5.291399962953   5.291399962953   5.291399962953   5.291399962953   5.291399962953   4.028666790158   4.028666790158   4.028666790158  11.041640873651 |

4) RC = -13 | CR = 52 | A,T,7 removed (2D)

|   0.002324475807 |   0.048194174755   0.102325104020   0.102325104020   0.102325104020   0.102325104020   0.102325104020   0.069010552983   0.078869203409   0.078869203409    0.213431345344 |
|   0.002324475807 |   2.506097087263   5.320905409029   5.320905409029   5.320905409029   5.320905409029   5.320905409029   3.588548755129   4.101198577291   4.101198577291  11.098429957881 |

5) RC = -13 | CR = 52 | T,7 removed (2D)

|   0.001752858519 |   0.053838176772   0.102492222353   0.102492222353   0.102492222353   0.102492222353   0.102492222353   0.068501931969   0.078287922250   0.078287922250    0.208622934993 |
|   0.001752858519 |   2.799585192160   5.329595562356   5.329595562356   5.329595562356   5.329595562356   5.329595562356   3.562100462394   4.070971957022   4.070971957022  10.848392619621 |

Sincerely,
Cac

PS: These days I am going to publish the exact insurance indices for Hi-Lo taking into account different penetrations.

[k_c]

After reviewing my code I realized that the method I was using was a Gaussian approximation. Back in the days when computers weren't that fast, doing it under strict combinatorial analysis took a lot of processing time, especially in shoes. In fact, to obtain the probability of a certain RC with N cards remaining, it was much faster to use a Normal approximation (there is an explanation of how to do it in TOB). My idea is to try to improve the results using the Gaussian approach.
Anyway, I can also do it using combinatorial analysis. Here are my numbers to 12 digits of precision so we can see if our programs match:

1) RC =  +1 | CR = 38 | A removed (1D)

|   0.118480844761 |   0.062504209740   0.073908841110   0.073908841110   0.073908841110   0.073908841110   0.073908841110   0.078198599809   0.078198599809   0.078198599809    0.333355785282 |
|   0.118480844761 |   2.375159970136   2.808535962172   2.808535962172   2.808535962172   2.808535962172   2.808535962172   2.971546792759   2.971546792759   2.971546792759  12.667519840725 |

2) RC = -13 | CR = 52 | No cards removed (2D)

|   0.001319524919 |   0.051923076923   0.101923076923   0.101923076923   0.101923076923   0.101923076923   0.101923076923   0.076923076923   0.076923076923   0.076923076923    0.207692307692 |
|   0.001319524919 |   2.700000000000   5.300000000000   5.300000000000   5.300000000000   5.300000000000   5.300000000000   4.000000000000   4.000000000000   4.000000000000  10.800000000000 |

3) RC = -13 | CR = 52 | A removed (2D)

|   0.001748370518 |   0.046449210406   0.101757691595   0.101757691595   0.101757691595   0.101757691595   0.101757691595   0.077474361349   0.077474361349   0.077474361349    0.212339247570 |
|   0.001748370518 |   2.415358941111   5.291399962953   5.291399962953   5.291399962953   5.291399962953   5.291399962953   4.028666790158   4.028666790158   4.028666790158  11.041640873651 |

4) RC = -13 | CR = 52 | A,T,7 removed (2D)

|   0.002324475807 |   0.048194174755   0.102325104020   0.102325104020   0.102325104020   0.102325104020   0.102325104020   0.069010552983   0.078869203409   0.078869203409    0.213431345344 |
|   0.002324475807 |   2.506097087263   5.320905409029   5.320905409029   5.320905409029   5.320905409029   5.320905409029   3.588548755129   4.101198577291   4.101198577291  11.098429957881 |

5) RC = -13 | CR = 52 | T,7 removed (2D)

|   0.001752858519 |   0.053838176772   0.102492222353   0.102492222353   0.102492222353   0.102492222353   0.102492222353   0.068501931969   0.078287922250   0.078287922250    0.208622934993 |
|   0.001752858519 |   2.799585192160   5.329595562356   5.329595562356   5.329595562356   5.329595562356   5.329595562356   3.562100462394   4.070971957022   4.070971957022  10.848392619621 |

Sincerely,
Cac

PS: These days I am going to publish the exact insurance indices for Hi-Lo taking into account different penetrations.

It appears we are in complete agreement.

This is how I output insurance indexes:

Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 8
Insurance Data (without regard to hand comp)
No subgroup (removals) are defined

**** Player hand: x-x ****
Cards   RC      TC ref

384     29      3.93
383     28      3.80
382     27      3.68
379     26      3.57
376     25      3.46
370     24      3.37
362     23      3.30
352     22      3.25
341     21      3.20
327     20      3.18
313     19      3.16
298     18      3.14
283     17      3.12
267     16      3.12
251     15      3.11
234     14      3.11
218     13      3.10
201     12      3.10
185     11      3.09
168     10      3.10
151     9       3.10
134     8       3.10
117     7       3.11
100     6       3.12
83      5       3.13
66      4       3.15
49      3       3.18
32      2       3.25
15      1       3.47
2       0       0.00
1       1       52.00

Press any key to continue

Above outputs in about a second or so. I can generate indexes for HiLo fairly efficiently. It takes about 3 minutes to generate HiLo RC indexes for a hand of 2-2 for 208 cards remaining from 8 decks for all up cards. T-6 for 8 decks takes only a second or so.

It's a different story for Wong Halves. For 8 decks with 208 cards remaining HiLo has 49 count subsets whereas Wong Halves has 582905. Insurance indexes are somewhat attainable but others are more of a problem.

8 decks is about the worst case to worry about. I wonder if efficiency can be further improved.

k_c

[Cacarulo]

It appears we are in complete agreement.

This is how I output insurance indexes:

Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 8
Insurance Data (without regard to hand comp)
No subgroup (removals) are defined

**** Player hand: x-x ****
Cards   RC      TC ref

384     29      3.93
383     28      3.80
382     27      3.68
379     26      3.57
376     25      3.46
370     24      3.37
362     23      3.30
352     22      3.25
341     21      3.20
327     20      3.18
313     19      3.16
298     18      3.14
283     17      3.12
267     16      3.12
251     15      3.11
234     14      3.11
218     13      3.10
201     12      3.10
185     11      3.09
168     10      3.10
151     9       3.10
134     8       3.10
117     7       3.11
100     6       3.12
83      5       3.13
66      4       3.15
49      3       3.18
32      2       3.25
15      1       3.47
2       0       0.00
1       1       52.00

Press any key to continue

Above outputs in about a second or so. I can generate indexes for HiLo fairly efficiently. It takes about 3 minutes to generate HiLo RC indexes for a hand of 2-2 for 208 cards remaining from 8 decks for all up cards. T-6 for 8 decks takes only a second or so.

It's a different story for Wong Halves. For 8 decks with 208 cards remaining HiLo has 49 count subsets whereas Wong Halves has 582905. Insurance indexes are somewhat attainable but others are more of a problem.

8 decks is about the worst case to worry about. I wonder if efficiency can be further improved.

k_c

Excellent!

Here is my approach for SD:

+----------+----------------------------+-----+-----+---------------------+
|   Play   |              TC            |  RC | IRC |          EV         |
+----------+--------------+-------------+-----+-----+---------------------+
|    Ins   |   1.368421   |     1/ 38   |   1 |   0 | 0.00673558467365609 |
+----------+--------------+-------------+-----+-----+---------------------+

This is the precise index for SD. The first TC where a positive advantage exists occurs exactly when the RC equals +1 and there are 38 cards left in the deck. Index = 1/38*52 = 1.368421
Notice that the EV at this point is equal to 0.00673558467365609%
In the same way, and for a penetration of 32/52 cards or 20 cards remaining, the average RC index between 51 and 20 cards turns out to be equal to +1.

For 2D and 26 cards left:

+----------+----------------------------+-----+-----+---------------------+
|   Play   |              TC            |  RC | IRC |          EV         |
+----------+--------------+-------------+-----+-----+---------------------+
|    Ins   |   2.400000   |     3/ 65   |   3 |   0 | 0.07726304005333251 |
+----------+--------------+-------------+-----+-----+---------------------+

For 3D and 39 cards left:

+----------+----------------------------+-----+-----+---------------------+
|   Play   |              TC            |  RC | IRC |          EV         |
+----------+--------------+-------------+-----+-----+---------------------+
|    Ins   |   2.701299   |     4/ 77   |   5 |   0 | 0.01586588260364952 |
+----------+--------------+-------------+-----+-----+---------------------+

For 4D and 52 cards left:

+----------+----------------------------+-----+-----+---------------------+
|   Play   |              TC            |  RC | IRC |          EV         |
+----------+--------------+-------------+-----+-----+---------------------+
|    Ins   |   2.857143   |     5/ 91   |   6 |   0 | 0.00225658769383852 |
+----------+--------------+-------------+-----+-----+---------------------+

For 5D and 65 cards left:

+----------+----------------------------+-----+-----+---------------------+
|   Play   |              TC            |  RC | IRC |          EV         |
+----------+--------------+-------------+-----+-----+---------------------+
|    Ins   |   2.959350   |     7/123   |   8 |   0 | 0.02248153833293021 |
+----------+--------------+-------------+-----+-----+---------------------+

For 6D and 78 cards left:

+----------+----------------------------+-----+-----+---------------------+
|   Play   |              TC            |  RC | IRC |          EV         |
+----------+--------------+-------------+-----+-----+---------------------+
|    Ins   |   3.014493   |     8/138   |   9 |   0 | 0.00370493169572494 |
+----------+--------------+-------------+-----+-----+---------------------+

Sincerely,
Cac

[aceside]

Wonderful! However, there is not very much meat in 6 or 8 deck games, and also there are no available single deck games. I greatly hope to see these numbers on 2-deck games.

[Cacarulo]

Wonderful! However, there is not very much meat in 6 or 8 deck games, and also there are no available single deck games. I greatly hope to see these numbers on 2-deck games.

I've posted a 2D index.

Sincerely,
Cac

[aceside]

I've posted a 2D index.

Sincerely,
Cac

Nice, I just saw it, but can you dig a little more into this? Post the 2-deck results like what K_C just did for 8-decks, like this:
100 6 3.12
83 5 3.13
66 4 3.15
49 3 3.18
32 2 3.25
15 1 3.47

[Cacarulo]

Nice, I just saw it, but can you dig a little more into this? Post the 2-deck results like what K_C just did for 8-decks, like this:
100 6 3.12
83 5 3.13
66 4 3.15
49 3 3.18
32 2 3.25
15 1 3.47

Sorry, that wouldn't be an index. The index should be the minimum TC at which one should buy insurance. That would include any TC greater or equal than the index.

Sincerely,
Cac

[aceside]

Sorry, that wouldn't be an index. The index should be the minimum TC at which one should buy insurance. That would include any TC greater or equal than the index.

Sincerely,
Cac

One main point here is to find out how the index varies with the dealing depth. Some indices increase while others decrease. K_C has showed that the insurance index mostly decreases with the dealing depth. It does not strictly decrease all the way though. I guess this is true for both 8-decks and 2-decks.

[Cacarulo]

One main point here is to find out how the index varies with the dealing depth. Some indices increase while others decrease. K_C has showed that the insurance index mostly decreases with the dealing depth. It does not strictly decrease all the way though. I guess this is true for both 8-decks and 2-decks.

It is one thing to see how an index varies according to penetration and another thing is the index itself.
K_C showed in his data that the lowest TC at which there is an advantage is found when there are 185 cards remaining and the RC is equal to +11. That corresponds to the index we need to find and the value is 3.091892. Any TC above or equal to that index will have a positive advantage and therefore we must buy insurance.
Suppose we know in advance that the index is 3.09 but we have to play with horrible penetration. When the TC is greater than or equal to 3.09 we will buy insurance. If you look at the data, that will always happen.

Sincerely,
Cac

[Cacarulo]

For 7D and 52 cards remaining:

+----------+----------------------------+-----+-----+---------------------+
|   Play   |              TC            |  RC | IRC |          EV         |
+----------+--------------+-------------+-----+-----+---------------------+
|    Ins   |   3.058824   |     8/136   |   9 |   0 | 0.00005756832546222 |
+----------+--------------+-------------+-----+-----+---------------------+

For 8D and 52 cards remaining:

+----------+----------------------------+-----+-----+---------------------+
|   Play   |              TC            |  RC | IRC |          EV         |
+----------+--------------+-------------+-----+-----+---------------------+
|    Ins   |   3.091892   |    11/185   |  10 |   0 | 0.00105372979730678 |
+----------+--------------+-------------+-----+-----+---------------------+

Sincerely,
Cac