Dynamic Insurance

[k_c]

Hi,

Still don't get the subgroup thing. For instance, if I wanted an insurance index for AAvA, I'll remove
three aces from the pack. If I wanted an index for A2vA, I'll remove two aces. What am I missing?

Sincerely,
Cac

Start with cards remaining and running count.

My view is that the number of cards removed in a subgroup is input directly. That number remains absolute. (Subgroup cards present is determined from subgroup cards removed.) The number of variable cards is (input cards remaining - subgroup cards present). Since subgroup cards present are a given, probability of subset is independent of subgroup input.

The maximum number of subgroups is 10. In that case removals of all ten ranks would be input to define 1 possible subset. If what is input is not possible the probability of the subset will turn out to be 0. This dovetails into just inputting a shoe composition from scratch. That is the extreme.

A subgroup must be defined from 1 or more ranks in a main group. HiLo has 3 main groups: {2,3,4,5,6}, {7,8,9}, {T,A}

I'm trying to come up with an obvious example using HiLo. I've added notes to each of 4 inputs of 26 cards remaining of a single deck to try to help explain a subgroup defined as {7,8,9}. Basically number of {7,8,9} is being independently input.

26 cards remain, Subgroup {7,8,9}
RC, subgroup cards, and specific removals are input
There are 4 inputs with resultant rank probs p[1] through p[10]

Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: 0
Specific removals (1 - 10): {0,0,0,0,0,0,0,0,0,0}
Subgroup removals: {7,8,9}12

Number of subsets for above conditions: 1 (Note: subset is 13{2,3,4,5,6}, 0{7,8,9}, 13{T,A})
Prob of running count 0 with above removals from 1 deck: 0.25895

p[1] 0.1  p[2] 0.1  p[3] 0.1  p[4] 0.1  p[5] 0.1
p[6] 0.1  p[7] 0  p[8] 0  p[9] 0  p[10] 0.4


Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: -1
Specific removals (1 - 10): {0,0,0,0,0,0,0,0,0,0}
Subgroup removals: {7,8,9}12

Number of subsets for above conditions: 0 (Note: subset is impossible - need odd num cards for RC=-1 and 0 {7,8,9})
Prob of running count -1 with above removals from 1 deck: 0.00000e+000

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


Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: 0
Specific removals (1 - 10): {0,0,0,0,0,0,0,0,0,0}
Subgroup removals: {7,8,9}6

Number of subsets for above conditions: 1 (Note: subset is 10{2,3,4,5,6}, 6{7,8,9}, 10{T,A})
Prob of running count 0 with above removals from 1 deck: 0.24763

p[1] 0.076923  p[2] 0.076923  p[3] 0.076923  p[4] 0.076923  p[5] 0.076923
p[6] 0.076923  p[7] 0.076923  p[8] 0.076923  p[9] 0.076923  p[10] 0.30769


Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: 0
Specific removals (1 - 10): {0,0,0,0,0,0,1,0,0,0}
Subgroup removals: {7,8,9}6 (Note: range of input here is 1-12, since a 7 was specifically removed)

Number of subsets for above conditions: 1
Prob of running count 0 with above removals from 1 deck: 0.24763

p[1] 0.076923  p[2] 0.076923  p[3] 0.076923  p[4] 0.076923  p[5] 0.076923
p[6] 0.076923  p[7] 0.062937  p[8] 0.083916  p[9] 0.083916  p[10] 0.30769

Hope that helps in explaining what I'm up to.
k_c

[aceside]

Start with cards remaining and running count.

My view is that the number of cards removed in a subgroup is input directly. That number remains absolute. (Subgroup cards present is determined from subgroup cards removed.) The number of variable cards is (input cards remaining - subgroup cards present). Since subgroup cards present are a given, probability of subset is independent of subgroup input.

Let me take this opportunity to clear my doubts on the calculation of Bust Bonus. If 208 cards are randomly dealt out from a 6-deck shoe and the running count is +10, you calculated the dealer up card probability and then calculated the dealer bust rate when an ace is removed from the subgroup {T, A}. I do not understand the details in this second part, but feel there might be a tiny problem here.

I actually like another approach. Firstly we give an ace card to the dealer, then we randomly deal out 207 cards, and then we find the RC is +10 in the remaining 104 cards. Now we re-calculate the new dealer up card probability in these 104 cards and re-calculate the dealer bust rate. This means a lot of calculations, but is this better?

[Cacarulo]

Start with cards remaining and running count.

My view is that the number of cards removed in a subgroup is input directly. That number remains absolute. (Subgroup cards present is determined from subgroup cards removed.) The number of variable cards is (input cards remaining - subgroup cards present). Since subgroup cards present are a given, probability of subset is independent of subgroup input.

The maximum number of subgroups is 10. In that case removals of all ten ranks would be input to define 1 possible subset. If what is input is not possible the probability of the subset will turn out to be 0. This dovetails into just inputting a shoe composition from scratch. That is the extreme.

A subgroup must be defined from 1 or more ranks in a main group. HiLo has 3 main groups: {2,3,4,5,6}, {7,8,9}, {T,A}

I'm trying to come up with an obvious example using HiLo. I've added notes to each of 4 inputs of 26 cards remaining of a single deck to try to help explain a subgroup defined as {7,8,9}. Basically number of {7,8,9} is being independently input.

26 cards remain, Subgroup {7,8,9}
RC, subgroup cards, and specific removals are input
There are 4 inputs with resultant rank probs p[1] through p[10]

Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: 0
Specific removals (1 - 10): {0,0,0,0,0,0,0,0,0,0}
Subgroup removals: {7,8,9}12

Number of subsets for above conditions: 1 (Note: subset is 13{2,3,4,5,6}, 0{7,8,9}, 13{T,A})
Prob of running count 0 with above removals from 1 deck: 0.25895

p[1] 0.1  p[2] 0.1  p[3] 0.1  p[4] 0.1  p[5] 0.1
p[6] 0.1  p[7] 0  p[8] 0  p[9] 0  p[10] 0.4


Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: -1
Specific removals (1 - 10): {0,0,0,0,0,0,0,0,0,0}
Subgroup removals: {7,8,9}12

Number of subsets for above conditions: 0 (Note: subset is impossible - need odd num cards for RC=-1 and 0 {7,8,9})
Prob of running count -1 with above removals from 1 deck: 0.00000e+000

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


Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: 0
Specific removals (1 - 10): {0,0,0,0,0,0,0,0,0,0}
Subgroup removals: {7,8,9}6

Number of subsets for above conditions: 1 (Note: subset is 10{2,3,4,5,6}, 6{7,8,9}, 10{T,A})
Prob of running count 0 with above removals from 1 deck: 0.24763

p[1] 0.076923  p[2] 0.076923  p[3] 0.076923  p[4] 0.076923  p[5] 0.076923
p[6] 0.076923  p[7] 0.076923  p[8] 0.076923  p[9] 0.076923  p[10] 0.30769


Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Cards remaining: 26
Initial running count (full shoe): 0
Running count: 0
Specific removals (1 - 10): {0,0,0,0,0,0,1,0,0,0}
Subgroup removals: {7,8,9}6 (Note: range of input here is 1-12, since a 7 was specifically removed)

Number of subsets for above conditions: 1
Prob of running count 0 with above removals from 1 deck: 0.24763

p[1] 0.076923  p[2] 0.076923  p[3] 0.076923  p[4] 0.076923  p[5] 0.076923
p[6] 0.076923  p[7] 0.062937  p[8] 0.083916  p[9] 0.083916  p[10] 0.30769

Hope that helps in explaining what I'm up to.
k_c

I was trying to understand the subgroup issue and after some thought it occurred to me that maybe what you're trying to explain me is this, please correct me if I'm wrong.

Case 1) It would be all deck compositions in which at least 1, 2, 3 or 4 aces can be removed.
If I remove at least one ace it means that I can get compositions with 1, 2, 3 or 4 aces.
If I remove at least two aces it means that I can get compositions with 2, 3 or 4 aces.
If I remove at least three aces it means that I can get compositions with 3 or 4 aces.
If I remove four aces it means that I can only get compositions with 4 aces.

Now let's go to what I think you call subgroup:

Case 2) It would be all deck compositions in which "at most" I can remove 1, 2, 3 or 4 aces.
If I remove at most one ace as it means that I can only get compositions with one ace, no more and no less.
If I remove at most two aces it means that I can only get compositions with two aces, no more and no less.
If I remove at most three aces it means that I can only get compositions with three aces, no more and no less.
If I remove four aces it means that I can only get compositions with four aces.

Is this what you mean?

Sincerely,
Cac

[k_c]

I was trying to understand the subgroup issue and after some thought it occurred to me that maybe what you're trying to explain me is this, please correct me if I'm wrong.

Case 1) It would be all deck compositions in which at least 1, 2, 3 or 4 aces can be removed.
If I remove at least one ace it means that I can get compositions with 1, 2, 3 or 4 aces.
If I remove at least two aces it means that I can get compositions with 2, 3 or 4 aces.
If I remove at least three aces it means that I can get compositions with 3 or 4 aces.
If I remove four aces it means that I can only get compositions with 4 aces.

Now let's go to what I think you call subgroup:

Case 2) It would be all deck compositions in which "at most" I can remove 1, 2, 3 or 4 aces.
If I remove at most one ace as it means that I can only get compositions with one ace, no more and no less.
If I remove at most two aces it means that I can only get compositions with two aces, no more and no less.
If I remove at most three aces it means that I can only get compositions with three aces, no more and no less.
If I remove four aces it means that I can only get compositions with four aces.

Is this what you mean?

Sincerely,
Cac

It sounds like you are seeing what I am trying to do.

re:subgroup{ace} HiLo single deck
re:Case 1)
If 0 aces are specifically removed, user has option to input 0,1,2,3,4 subgroup removals
If 1 ace is specifically removed, user has option to input 1,2,3,4 subgroup removals
If 2 aces are specifically removed, user has option to input 2,3,4 subgroup removals
If 3 aces are specifically removed, user has option to input 3,4 subgroup removals
If 4 aces are specifically removed, user input has to be 4 subgroup removals

re:Case 2) Once user has made input it fixes the number of aces in subsets depending upon input

I am trying to allow for all possible side counts of subgroups within the main count groups. As I mentioned an extreme case would be to define a subgroup for each rank in which case the exact number of each rank present in the subset would be the input. There could only be 1 subset in this case and it would have to conform to to number of cards and running count that was input. The idea is to build a bridge between direct input and count input.

Anything not in a subgroup is handled normally.

Side counting subgroups is optional but once user has made decision to side count he must be prepared to follow through with input for all defined subgroups.

k_c

[Cacarulo]

It sounds like you are seeing what I am trying to do.

re:subgroup{ace} HiLo single deck
re:Case 1)
If 0 aces are specifically removed, user has option to input 0,1,2,3,4 subgroup removals
If 1 ace is specifically removed, user has option to input 1,2,3,4 subgroup removals
If 2 aces are specifically removed, user has option to input 2,3,4 subgroup removals
If 3 aces are specifically removed, user has option to input 3,4 subgroup removals
If 4 aces are specifically removed, user input has to be 4 subgroup removals

re:Case 2) Once user has made input it fixes the number of aces in subsets depending upon input

I am trying to allow for all possible side counts of subgroups within the main count groups. As I mentioned an extreme case would be to define a subgroup for each rank in which case the exact number of each rank present in the subset would be the input. There could only be 1 subset in this case and it would have to conform to to number of cards and running count that was input. The idea is to build a bridge between direct input and count input.

Anything not in a subgroup is handled normally.

Side counting subgroups is optional but once user has made decision to side count he must be prepared to follow through with input for all defined subgroups.

k_c

But you are not telling me if what I think is the same as what you are doing.
The first case is what I do to calculate an index such as AAvA (at least I must remove three aces). I guess you should do the same, right?
The second case I could use it for combinatorial analysis of dynamic indices. It's just an idea.

Sincerely,
Cac

[k_c]

But you are not telling me if what I think is the same as what you are doing.
The first case is what I do to calculate an index such as AAvA (at least I must remove three aces). I guess you should do the same, right?
The second case I could use it for combinatorial analysis of dynamic indices. It's just an idea.

Sincerely,
Cac

I think we are in agreement. It's a matter of terminology. If I define a subgroup it requires a direct input. If no subgroups at all are defined, ranks are specifically removed as required. So in that case for A-A v A three aces are specifically removed. I'm sure we agree when there are no subgroups.

Subgroup removals are absolute and override specific removals. That is why there is a minimum when a specific removal rank happens to coincide with a subgroup rank. Also every possible subset (for a given cards remaining and RC) must contain the same exact number of subgroup cards depending upon the input.

Anyway I think we have basic agreement. I thought that subgroup input may at least help clarify your dynamic indexes when I first posted. It seems you are able to arrive at the single best TC index out of the RC index jungle. It looks to me that subgroup input may make that RC index jungle even more dense. Also I still have to look out for possible additional errors.

k_c

[Cacarulo]

I think we are in agreement. It's a matter of terminology. If I define a subgroup it requires a direct input. If no subgroups at all are defined, ranks are specifically removed as required. So in that case for A-A v A three aces are specifically removed. I'm sure we agree when there are no subgroups.

Subgroup removals are absolute and override specific removals. That is why there is a minimum when a specific removal rank happens to coincide with a subgroup rank. Also every possible subset (for a given cards remaining and RC) must contain the same exact number of subgroup cards depending upon the input.

Anyway I think we have basic agreement. I thought that subgroup input may at least help clarify your dynamic indexes when I first posted. It seems you are able to arrive at the single best TC index out of the RC index jungle. It looks to me that subgroup input may make that RC index jungle even more dense. Also I still have to look out for possible additional errors.

k_c

Yes, you said it, it's a matter of terminology. I'm glad we agree.
Just to be sure: in my first case if I remove four aces obviously
it has to be the same as in the second case since we would be at the maximum possible. Do you agree?

Sincerely,
Cac

[k_c]

Yes, you said it, it's a matter of terminology. I'm glad we agree.
Just to be sure: in my first case if I remove four aces obviously
it has to be the same as in the second case since we would be at the maximum possible. Do you agree?

Sincerely,
Cac

Yes, agree.

k_c

[k_c]

I have a minor change in the edited data from post #20. For insurance I did not want to compute all possible running counts because it's obviously a waste of time for many of them so I compute an approximate minimum RC for each number of cards. Because of rounding the minimum wasn't low enough and needed to be decreased by 1 in some of the initial cases of cards remaining for generic single deck insurance with 4 aces removed as a subgroup. Specifically removing 4 aces for each cards remaining value would arrive at the same data points.

Count tags {1,-1,-1,-1,-1,-1,0,0,0,1}
Decks: 1
Insurance Data (without regard to hand comp)
Side counted subgroup removals (no input defaults to minimum):
{1} (1 to 4): 4

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

48      -4      -4.33
46      -3      -3.39
45      -4      -4.62
44      -3      -3.55
35      -2      -2.97
23      -1      -2.26
12      0       0.00
3       1       17.33
2       0       0.00
1       1       52.00

k_c

[aceside]

Specifically removing 4 aces for each cards remaining value would arrive at the same data points.

Let me try to understand your results. Firstly, we fix the dealing depth at 26 of 52 cards (thus 0.5 deck remaining). Secondly, we calculate the deficit number of aces per deck in the remaining half deck:

(4-2)/0.5=4.

Thirdly, we find the insurance index drop because of this much deficit of aces:

1.4-(-2/0.5)=5.4.

Finally, we find that, for every one deficit of aces per remaining deck, the insurance index drops:
5.4/4=1.4.

[Freightman]

I’ve taken another look at this - maybe I’m wrong, however, the conclusion I’ve come to is that I’ve looked at this backwards.

Proper proportions on dd indicate an index of 2.4 for insurance. As your table shows an increase of index for a decrease of aces, then it would seem that your revised index is not counter intuitive and is based on the entirety of the remaining cards prior to hand being dealt. For insurance purposes the only thing that’s relevant is the number of 10 value cards once dealer shows an ace.

So, with 26 cards remaining prior to round, 3 aces already removed, dealer is dealt the 4th ace. With 32 - 10 value cards to begin with, (heads up) and 22 cards remaining prior to insurance decision, remaining deck composition is a max of 22 and a minimum of 6 - 10 value cards.

So, dealer BJ probability is now somewhere between 6/22 to 16/22. This equates to a range of 27.27% to 72.7% for dealer BJ. Obviously, the extremes are non probable. So, assuming no 10 value cards are visible when round is dealt, strike point for insurance is 7.3 10 value cards or 33(.18)%.

The inescapable conclusion is that the theory is potentially valid for play decisions only. The theory though does lead credence to the theory of QTC.

[k_c]

Let me try to understand your results. Firstly, we fix the dealing depth at 26 of 52 cards (thus 0.5 deck remaining). Secondly, we calculate the deficit number of aces per deck in the remaining half deck:

(4-2)/0.5=4.

Thirdly, we find the insurance index drop because of this much deficit of aces:

1.4-(-2/0.5)=5.4.

Finally, we find that, for every one deficit of aces per remaining deck, the insurance index drops:
5.4/4=1.4.

The only thing I'm doing is providing below data for HiLo single deck where the high card group consists of (0 aces + tens):

Cards

47-48     insure if RC >= -4
46        insure if RC >= -3
45        insure if RC >= -4
36-44     insure if RC >= -3
24-35     insure if RC >= -2
13-23     insure if RC >= -1
4-12      insure if RC >= 0
3         insure if RC >= 1
2         insure if RC >= 0
1         insure if RC >= 1 (can't be > 1)

Cacarulo is out to provide the statistically best single true count (= 52*RC/(cards remaining) for each data point) and to further adapt that to aces removed as a reasonable simplification that would be easy to remember.

k_c

[Cacarulo]

The only thing I'm doing is providing below data for HiLo single deck where the high card group consists of (0 aces + tens):

Cards

47-48     insure if RC >= -4
46        insure if RC >= -3
45        insure if RC >= -4
36-44     insure if RC >= -3
24-35     insure if RC >= -2
13-23     insure if RC >= -1
4-12      insure if RC >= 0
3         insure if RC >= 1
2         insure if RC >= 0
1         insure if RC >= 1 (can't be > 1)

Cacarulo is out to provide the statistically best single true count (= 52*RC/(cards remaining) for each data point) and to further adapt that to aces removed as a reasonable simplification that would be easy to remember.

k_c

This table is much clearer than the previous one. The important thing is that our results agree 100%. The only detail is that I only take EV > 0 and you take EV >= 0. There are only two results where EV = 0: -4/48 and -2/24.
What we still do not agree on is the way of presenting the data. It is very complicated to carry out a control that depends on each depth. For me, the ideal is, given a penetration, to calculate the minimum TC from which the expected value is maximum.
In this particular case and with a penetration of 32/52 (20 cards left) the best index is -4.425532. In other words, we will buy insurance when the TC is greater than or equal to -4.425532.
The choice of this index arises from a table ordered by TC from lowest to highest. In it we look for the index that maximizes the expected value.
Here is an extract of the table:

| RC |  -2 | CR |  22 | FQ | 0.13152983652263245 | EV |  -0.01178311361355877 | TC |  -4.727273 | TCF |      -4.73 |
| RC |  -3 | CR |  33 | FQ | 0.14053631374905032 | EV |  -0.00891545881695388 | TC |  -4.727273 | TCF |      -4.73 |
| RC |  -4 | CR |  44 | FQ | 0.22826086956521735 | EV |  -0.01151330938564987 | TC |  -4.727273 | TCF |      -4.73 |
| RC |  -4 | CR |  45 | FQ | 0.23473635522664191 | EV |   0.00361247947454846 | TC |  -4.622222 | TCF |      -4.63 |
| RC |  -3 | CR |  34 | FQ | 0.14352121975417773 | EV |  -0.00482779202185191 | TC |  -4.588235 | TCF |      -4.59 |
| RC |  -2 | CR |  23 | FQ | 0.13133542597315229 | EV |  -0.00563441233306816 | TC |  -4.521739 | TCF |      -4.53 |
| RC |  -4 | CR |  46 | FQ | 0.34219858156028365 | EV |  -0.01058797026357294 | TC |  -4.521739 | TCF |      -4.53 |
| RC |  -3 | CR |  35 | FQ | 0.14687994217230230 | EV |  -0.00098852793468573 | TC |  -4.457143 | TCF |      -4.46 |
| RC |  -4 | CR |  47 | FQ | 0.25000000000000006 | EV |   0.02127659574468077 | TC |  -4.425532 | TCF |      -4.43 |
| RC |  -2 | CR |  24 | FQ | 0.13127110336382861 | EV |   0.00000000000000000 | TC |  -4.333333 | TCF |      -4.34 |
| RC |  -3 | CR |  36 | FQ | 0.15067512012645540 | EV |   0.00266662480199220 | TC |  -4.333333 | TCF |      -4.34 |
| RC |  -4 | CR |  48 | FQ | 1.00000000000000000 | EV |   0.00000000000000000 | TC |  -4.333333 | TCF |      -4.34 |
| RC |  -3 | CR |  37 | FQ | 0.15501466784779966 | EV |   0.00610036826646621 | TC |  -4.216216 | TCF |      -4.22 |
| RC |  -2 | CR |  25 | FQ | 0.13133542597315226 | EV |   0.00518365934642273 | TC |  -4.160000 | TCF |      -4.16 |
| RC |  -3 | CR |  38 | FQ | 0.15996246536863803 | EV |   0.00930548216808247 | TC |  -4.105263 | TCF |      -4.11 |
| RC |  -2 | CR |  26 | FQ | 0.13152983652263245 | EV |   0.00997032690378052 | TC |  -4.000000 | TCF |      -4.00 |
| RC |  -3 | CR |  39 | FQ | 0.16565633536577018 | EV |   0.01250001308246951 | TC |  -4.000000 | TCF |      -4.00 |
| RC |  -3 | CR |  40 | FQ | 0.17246082813195468 | EV |   0.01528887217046893 | TC |  -3.900000 | TCF |      -3.90 |
| RC |  -2 | CR |  27 | FQ | 0.13185563157124838 | EV |   0.01439701560539941 | TC |  -3.851852 | TCF |      -3.86 |
| RC |  -3 | CR |  41 | FQ | 0.18016361797959371 | EV |   0.01792862967709130 | TC |  -3.804878 | TCF |      -3.81 |
| RC |  -2 | CR |  28 | FQ | 0.13231356309311726 | EV |   0.01850395693034845 | TC |  -3.714286 | TCF |      -3.72 |
| RC |  -3 | CR |  42 | FQ | 0.18956425255502335 | EV |   0.02150865160700777 | TC |  -3.714286 | TCF |      -3.72 |
| RC |  -3 | CR |  43 | FQ | 0.20284949401508143 | EV |   0.02102222740000070 | TC |  -3.627907 | TCF |      -3.63 |
| RC |  -2 | CR |  29 | FQ | 0.13290768032282205 | EV |   0.02232653847771005 | TC |  -3.586207 | TCF |      -3.59 |
| RC |  -3 | CR |  44 | FQ | 0.21009353479288723 | EV |   0.03006582458637275 | TC |  -3.545455 | TCF |      -3.55 |
| RC |  -2 | CR |  30 | FQ | 0.13364183949005423 | EV |   0.02588394140253358 | TC |  -3.466667 | TCF |      -3.47 |
| RC |  -3 | CR |  45 | FQ | 0.25208140610545793 | EV |   0.02018348623853217 | TC |  -3.466667 | TCF |      -3.47 |
| RC |  -3 | CR |  46 | FQ | 0.21276595744680851 | EV |   0.04347826086956519 | TC |  -3.391304 | TCF |      -3.40 |

If you look closely, -4.622222 would be the first index candidate since the expected value is positive. The problem is that between this candidate and the next one (-4.425532) we find four that are negative. Therefore the best index is -4.425532 since the sum of all the products between the expected value and the frequency from the index upwards is the best.

Sincerely,
Cac