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  1. 10-02-2022, 08:29 PM #1
    k_c
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    Question for Cac

    I don't know if I will explain this very well.

    I try to test my methods by trying them on a problem with an obvious answer. Use insurance count (tag non-tens as -1 and tens as +2) (sign of tag relative to what remains in shoe) as the basis for a question.

    For this count this is data listing min RC values for single deck where Prob(ten) >= 1/3, which we should agree on.
    Code:
    Count tags {-1,-1,-1,-1,-1,-1,-1,-1,-1,2}
Decks: 1
Insurance Data (without regard to hand comp)
No subgroup (removals) are defined

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

48      0       0.00
47      1       1.11
46      2       2.26
45      0       0.00
44      1       1.18
43      2       2.42
42      0       0.00
41      1       1.27
40      2       2.60
39      0       0.00
38      1       1.37
37      2       2.81
36      0       0.00
35      1       1.49
34      2       3.06
33      0       0.00
32      1       1.63
31      2       3.35
30      0       0.00
29      1       1.79
28      2       3.71
27      0       0.00
26      1       2.00
25      2       4.16
24      0       0.00
23      1       2.26
22      2       4.73
21      0       0.00
20      1       2.60
19      2       5.47
18      0       0.00
17      1       3.06
16      2       6.50
15      0       0.00
14      1       3.71
13      2       8.00
12      0       0.00
11      1       4.73
10      2       10.40
9       0       0.00
8       1       6.50
7       2       14.86
6       0       0.00
5       1       10.40
4       2       26.00
3       0       0.00
2       1       26.00
1       2       104.00
    Using TC as the index metric, obvious answer is buy insurance when TC > 0. Also acceptable, buy insurance when TC >= 0

    For any of the above data points I can get this answer by interpolating to find RC when prob(ten) = 1/3, using 27, 26, 25 cards remaining as example.

    27 cards (no interpolation necessary since prob(ten) = 1/3
    RC = 0, prob(ten) = 1/3
    TC = 52*0/27 = 0

    26 cards single deck insurance (insurance count)
    RC = +1, prob(ten) = 9/26
    RC = -2, prob(ten) = 8/26
    interpolate: RC = 0, prob(ten) = (8 2/3)/26
    TC = 52*0/26 = 0

    25 cards single deck insurance (insurance count)
    RC = +2, prob(ten) = 9/25
    RC = -1, prob(ten) = 8/25
    interpolate: RC = 0, prob(ten) = (8 1/3)/25
    TC = 52*0/25 = 0

    This is my question:
    How would your method of finding TC index work on this data set?

    Thanks,
    k_c
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  2. 10-02-2022, 10:22 PM #2
    Cacarulo
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    Quote Originally Posted by k_c View Post
    I don't know if I will explain this very well.

    I try to test my methods by trying them on a problem with an obvious answer. Use insurance count (tag non-tens as -1 and tens as +2) (sign of tag relative to what remains in shoe) as the basis for a question.

    For this count this is data listing min RC values for single deck where Prob(ten) >= 1/3, which we should agree on.
    Code:
    Count tags {-1,-1,-1,-1,-1,-1,-1,-1,-1,2}
Decks: 1
Insurance Data (without regard to hand comp)
No subgroup (removals) are defined

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

48      0       0.00
47      1       1.11
46      2       2.26
45      0       0.00
44      1       1.18
43      2       2.42
42      0       0.00
41      1       1.27
40      2       2.60
39      0       0.00
38      1       1.37
37      2       2.81
36      0       0.00
35      1       1.49
34      2       3.06
33      0       0.00
32      1       1.63
31      2       3.35
30      0       0.00
29      1       1.79
28      2       3.71
27      0       0.00
26      1       2.00
25      2       4.16
24      0       0.00
23      1       2.26
22      2       4.73
21      0       0.00
20      1       2.60
19      2       5.47
18      0       0.00
17      1       3.06
16      2       6.50
15      0       0.00
14      1       3.71
13      2       8.00
12      0       0.00
11      1       4.73
10      2       10.40
9       0       0.00
8       1       6.50
7       2       14.86
6       0       0.00
5       1       10.40
4       2       26.00
3       0       0.00
2       1       26.00
1       2       104.00
    Using TC as the index metric, obvious answer is buy insurance when TC > 0. Also acceptable, buy insurance when TC >= 0

    For any of the above data points I can get this answer by interpolating to find RC when prob(ten) = 1/3, using 27, 26, 25 cards remaining as example.

    27 cards (no interpolation necessary since prob(ten) = 1/3
    RC = 0, prob(ten) = 1/3
    TC = 52*0/27 = 0

    26 cards single deck insurance (insurance count)
    RC = +1, prob(ten) = 9/26
    RC = -2, prob(ten) = 8/26
    interpolate: RC = 0, prob(ten) = (8 2/3)/26
    TC = 52*0/26 = 0

    25 cards single deck insurance (insurance count)
    RC = +2, prob(ten) = 9/25
    RC = -1, prob(ten) = 8/25
    interpolate: RC = 0, prob(ten) = (8 1/3)/25
    TC = 52*0/25 = 0

    This is my question:
    How would your method of finding TC index work on this data set?

    Thanks,
    k_c
    I'll give you an idea of ??what I do. Maybe this can help you.
    Let's use as an example a deck of 52 cards and the unbalanced count of tens (1 1 1 1 1 1 1 1 1 -2).
    1) Remove an ace from the pack (which is the ace the dealer receives)
    2) Let's assume that we are going to play until there are 10 cards left (CL)
    3) Two nested loops are needed, one loop that goes through all the possible RC from a minimum to a maximum.
    It doesn't matter if you don't know the limits, it can be between -50 and +50. Then the ones that do not correspond
    will be discarded (they are the ones whose probability is equal to zero).
    The other loop will go through the different depths between 51 (52 -1) cards and 11 cards (10 have been left out).
    4) Within these two nested loops you are going to calculate, through a combinatorial analysis, the probability and the
    expected value for each combination of RC and depth and the values ??obtained are going to be accumulated into two arrays.
    The index of the array is going to be the TC which in turn you get from doing (RC / depth * 52).
    array1 (TC) += probability * EV
    array2 (TC) += probability
    5) You are going to get the index from array1. To do this, you have to go through that array from the lowest value of TC
    until you find the TC in which the expected value becomes positive. That point corresponds to the searched index.
    It's actually a little more elaborate but for now I don't want to complicate it for you.
    The other array is used to know the frequency of each TC.

    Hope this helps.

    Sincerely,
    Cac
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  3. 10-03-2022, 07:30 AM #3
    k_c
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    Quote Originally Posted by Cacarulo View Post
    I'll give you an idea of ??what I do. Maybe this can help you.
    Let's use as an example a deck of 52 cards and the unbalanced count of tens (1 1 1 1 1 1 1 1 1 -2).
    1) Remove an ace from the pack (which is the ace the dealer receives)

    The other array is used to know the frequency of each TC.
    Cac
    I store my data in a list.

    If I construct a list with a specific pen and rc for the insurance count it will contain either 0 or 1 entries (elems) depending upon rc. If elems = 0, probRC = 0.

    This is what I presently do:
    Code:
    list = new subsetList (inputCount, decks, pen, rc, specificRem);
list->getProbRC(rc, probRC, pRank);
long elems = list->elems;
delete list;
list = NULL;
    What I used to do was construct a list with all rc possible to use. For more complicated counts too much data crashes program.
    For the insurance count where pen = 26 there are 17 entries in the list and sum of probRC of each entry = 1. For any count sum of probRC of entries in list = 1 regardless of number of entries.

    What I could do is something like this:
    Code:
    list = new subsetList (inputCount, decks, pen, specificRem);

for (rc = minValue; rc <= maxValue; ++rc)
     list->getProbRC(rc, probRC, pRank);

long elems = list->elems;
    I can relate to probRC, but probTC? (relative to simple insurance count)

    k_c
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  4. 10-03-2022, 10:59 AM #4
    Cacarulo
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    Quote Originally Posted by k_c View Post
    I store my data in a list.

    If I construct a list with a specific pen and rc for the insurance count it will contain either 0 or 1 entries (elems) depending upon rc. If elems = 0, probRC = 0.

    This is what I presently do:
    Code:
    list = new subsetList (inputCount, decks, pen, rc, specificRem);
list->getProbRC(rc, probRC, pRank);
long elems = list->elems;
delete list;
list = NULL;
    What I used to do was construct a list with all rc possible to use. For more complicated counts too much data crashes program.
    For the insurance count where pen = 26 there are 17 entries in the list and sum of probRC of each entry = 1. For any count sum of probRC of entries in list = 1 regardless of number of entries.

    What I could do is something like this:
    Code:
    list = new subsetList (inputCount, decks, pen, specificRem);

for (rc = minValue; rc <= maxValue; ++rc)
     list->getProbRC(rc, probRC, pRank);

long elems = list->elems;
    I can relate to probRC, but probTC? (relative to simple insurance count)

    k_c
    I don't understand where you're stuck. You have all the data and algorithms correct. The only thing left for you is to calculate the TC as TC = floor (RC / Cards_Left * 52).
    Here it is worth a clarification: the previous formula calculates the exact TC where the remaining decks (Cards_Left / 52) are calculated exactly without rounding or truncating.
    In my program I can calculate the remaining decks in various ways.

    Perhaps viewing the following RC distributions as a function of Cards_Left will help:

    1) HiLo

    Code:
     51 |  -1 
 50 |  -2  -1   0 
 49 |  -3  -2  -1   0   1 
 48 |  -4  -3  -2  -1   0   1   2 
 47 |  -5  -4  -3  -2  -1   0   1   2   3 
 46 |  -6  -5  -4  -3  -2  -1   0   1   2   3   4 
 45 |  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5 
 44 |  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6 
 43 |  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7 
 42 | -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8 
 41 | -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9 
 40 | -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10 
 39 | -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11 
 38 | -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12 
 37 | -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13 
 36 | -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14 
 35 | -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15 
 34 | -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16 
 33 | -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17 
 32 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18 
 31 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 30 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 29 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 28 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 27 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 26 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 25 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 24 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 23 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 22 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 21 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 20 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 19 | -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 18 | -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18 
 17 | -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17 
 16 | -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16 
 15 | -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15 
 14 | -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14 
 13 | -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13 
 12 | -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12 
 11 | -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11 
 10 | -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10 
  9 |  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9 
  8 |  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8 
  7 |  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7 
  6 |  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6 
  5 |  -5  -4  -3  -2  -1   0   1   2   3   4   5 
  4 |  -4  -3  -2  -1   0   1   2   3   4 
  3 |  -3  -2  -1   0   1   2   3
  2 |  -2  -1   0   1   2 
  1 |  -1   0   1
    2) UnbalancedTen

    Code:
     51 |  -3 
 50 |  -5  -2 
 49 |  -7  -4  -1 
 48 |  -9  -6  -3   0 
 47 | -11  -8  -5  -2   1 
 46 | -13 -10  -7  -4  -1   2 
 45 | -15 -12  -9  -6  -3   0   3 
 44 | -17 -14 -11  -8  -5  -2   1   4 
 43 | -19 -16 -13 -10  -7  -4  -1   2   5 
 42 | -21 -18 -15 -12  -9  -6  -3   0   3   6 
 41 | -23 -20 -17 -14 -11  -8  -5  -2   1   4   7 
 40 | -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8 
 39 | -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9 
 38 | -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10 
 37 | -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11 
 36 | -33 -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12 
 35 | -35 -32 -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13 
 34 | -34 -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14 
 33 | -33 -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15 
 32 | -32 -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16 
 31 | -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17 
 30 | -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18 
 29 | -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19 
 28 | -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20 
 27 | -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21 
 26 | -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22 
 25 | -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23 
 24 | -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24 
 23 | -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25 
 22 | -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26 
 21 | -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27 
 20 | -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28 
 19 | -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26  29 
 18 | -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27  30 
 17 | -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28  31 
 16 | -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26  29  32 
 15 | -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27  30 
 14 | -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28 
 13 | -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26 
 12 | -12  -9  -6  -3   0   3   6   9  12  15  18  21  24 
 11 | -11  -8  -5  -2   1   4   7  10  13  16  19  22 
 10 | -10  -7  -4  -1   2   5   8  11  14  17  20 
  9 |  -9  -6  -3   0   3   6   9  12  15  18 
  8 |  -8  -5  -2   1   4   7  10  13  16 
  7 |  -7  -4  -1   2   5   8  11  14 
  6 |  -6  -3   0   3   6   9  12 
  5 |  -5  -2   1   4   7  10 
  4 |  -4  -1   2   5   8 
  3 |  -3   0   3   6 
  2 |  -2   1   4 
  1 |  -1   2

    Sincerely,
    Cac
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  5. 10-03-2022, 11:37 AM #5
    k_c
    k_c is offline
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    Quote Originally Posted by Cacarulo View Post
    I don't understand where you're stuck. You have all the data and algorithms correct. The only thing left for you is to calculate the TC as TC = floor (RC / Cards_Left * 52).
    Here it is worth a clarification: the previous formula calculates the exact TC where the remaining decks (Cards_Left / 52) are calculated exactly without rounding or truncating.
    In my program I can calculate the remaining decks in various ways.

    Perhaps viewing the following RC distributions as a function of Cards_Left will help:


    2) UnbalancedTen

    Code:
     51 |  -3 
 50 |  -5  -2 
 49 |  -7  -4  -1 
 48 |  -9  -6  -3   0 
 47 | -11  -8  -5  -2   1 
 46 | -13 -10  -7  -4  -1   2 
 45 | -15 -12  -9  -6  -3   0   3 
 44 | -17 -14 -11  -8  -5  -2   1   4 
 43 | -19 -16 -13 -10  -7  -4  -1   2   5 
 42 | -21 -18 -15 -12  -9  -6  -3   0   3   6 
 41 | -23 -20 -17 -14 -11  -8  -5  -2   1   4   7 
 40 | -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8 
 39 | -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9 
 38 | -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10 
 37 | -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11 
 36 | -33 -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12 
 35 | -35 -32 -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13 
 34 | -34 -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14 
 33 | -33 -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15 
 32 | -32 -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16 
 31 | -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17 
 30 | -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18 
 29 | -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19 
 28 | -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20 
 27 | -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21 
 26 | -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22 
 25 | -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23 
 24 | -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24 
 23 | -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25 
 22 | -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26 
 21 | -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27 
 20 | -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28 
 19 | -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26  29 
 18 | -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27  30 
 17 | -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28  31 
 16 | -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26  29  32 
 15 | -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27  30 
 14 | -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28 
 13 | -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26 
 12 | -12  -9  -6  -3   0   3   6   9  12  15  18  21  24 
 11 | -11  -8  -5  -2   1   4   7  10  13  16  19  22 
 10 | -10  -7  -4  -1   2   5   8  11  14  17  20 
  9 |  -9  -6  -3   0   3   6   9  12  15  18 
  8 |  -8  -5  -2   1   4   7  10  13  16 
  7 |  -7  -4  -1   2   5   8  11  14 
  6 |  -6  -3   0   3   6   9  12 
  5 |  -5  -2   1   4   7  10 
  4 |  -4  -1   2   5   8 
  3 |  -3   0   3   6 
  2 |  -2   1   4 
  1 |  -1   2

    Sincerely,
    Cac
    I have previously never used floor funtion.

    When I change output of TC using floor function and change nothing else this is output:
    Code:
    Count tags {-1,-1,-1,-1,-1,-1,-1,-1,-1,2}
Decks: 1
Insurance Data (without regard to hand comp)
No subgroup (removals) are defined

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

48      0       0.00
47      1       1.00
46      2       2.00
45      0       0.00
44      1       1.00
43      2       2.00
42      0       0.00
41      1       1.00
40      2       2.00
39      0       0.00
38      1       1.00
37      2       2.00
36      0       0.00
35      1       1.00
34      2       3.00
33      0       0.00
32      1       1.00
31      2       3.00
30      0       0.00
29      1       1.00
28      2       3.00
27      0       0.00
26      1       2.00
25      2       4.00
24      0       0.00
23      1       2.00
22      2       4.00
21      0       0.00
20      1       2.00
19      2       5.00
18      0       0.00
17      1       3.00
16      2       6.00
15      0       0.00
14      1       3.00
13      2       8.00
12      0       0.00
11      1       4.00
10      2       10.00
9       0       0.00
8       1       6.00
7       2       14.00
6       0       0.00
5       1       10.00
4       2       26.00
3       0       0.00
2       1       26.00
1       2       104.00
    How can we transition from this and simply say TC index = 0.00 in all cases?

    k_c
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  6. 10-03-2022, 11:57 AM #6
    Cacarulo
    Cacarulo is offline
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    How can we transition from this and simply say TC index = 0.00 in all cases?
    Read again the methodology that I sent you earlier. The answer is there.
    Once you assemble the arrays you will see it more clearly. This is what I get from the arrays:

    Code:
    |       104 |  0.00871477100434657 |  2.00000000000000000 |
|        94 |  0.00000000000157696 |  1.81282495667244370 |
|        93 |  0.00000000002582721 |  1.79999999999999982 |
|        92 |  0.00000000291990962 |  1.77090909090909099 |
|        91 |  0.00000001888279004 |  1.75000000000000000 |
|        89 |  0.00000011539482804 |  1.72727272727272685 |
|        88 |  0.00000061443999346 |  1.70000000000000018 |
|        86 |  0.00000290322938725 |  1.66666666666666630 |
|        85 |  0.00000000001263757 |  1.64705882352941169 |
|        84 |  0.00001232995767432 |  1.62500000000000000 |
|        83 |  0.00000000204895869 |  1.60000000000000075 |
|        81 |  0.00004748601158011 |  1.57142857142857140 |
|        80 |  0.00000010700247691 |  1.53846153846153832 |
|        79 |  0.00000000000264831 |  1.52631578947368407 |
|        78 |  0.00016704038144613 |  1.50000000000000044 |
|        76 |  0.00000000104259939 |  1.47058823529411731 |
|        75 |  0.00000280244582382 |  1.45454545454545459 |
|        74 |  0.00000001001713137 |  1.43750000000000000 |
|        72 |  0.00054354543772864 |  1.40000000000000013 |
|        71 |  0.00000000033898419 |  1.36842105263157876 |
|        70 |  0.00000043364161695 |  1.35714285714285721 |
|        69 |  0.00004387543124798 |  1.33333333333333348 |
|        68 |  0.00000215788328434 |  1.30769230769230749 |
|        67 |  0.00000003892371046 |  1.29411764705882293 |
|        66 |  0.00000000005561459 |  1.28571428571428559 |
|        65 |  0.00169413476984208 |  1.25000000000000000 |
|        62 |  0.00000142216800880 |  1.20011666964613362 |
|        61 |  0.00003468047098998 |  1.18181818181818166 |
|        60 |  0.00000012776135551 |  1.16666666666666696 |
|        59 |  0.00044653704872974 |  1.14285714285714279 |
|        58 |  0.00000078431276579 |  1.11764705882352944 |
|        57 |  0.00011494953741622 |  1.10000000000000009 |
|        56 |  0.00002466219327307 |  1.07692334951209756 |
|        55 |  0.00000387542778391 |  1.06250000000000000 |
|        54 |  0.00000038131483604 |  1.05237716471420817 |
|        52 |  0.00557287748507175 |  1.00000000000000000 |
|        49 |  0.00000098985500915 |  0.95024721217439700 |
|        48 |  0.00006574386419129 |  0.93037593984962408 |
|        47 |  0.00024697563712821 |  0.90909115571936860 |
|        46 |  0.00000495213522480 |  0.89473684210526305 |
|        45 |  0.00091596628308714 |  0.87500000000000000 |
|        44 |  0.00017428659431586 |  0.84629621318248338 |
|        43 |  0.00002064258472653 |  0.83333333333333326 |
|        42 |  0.00000084422952156 |  0.82608695652173880 |
|        41 |  0.00355000323823723 |  0.79999999999999993 |
|        40 |  0.00000500312850994 |  0.77276324321463064 |
|        39 |  0.00053207026460195 |  0.75197575923828264 |
|        38 |  0.00004212583313604 |  0.73676600341560983 |
|        37 |  0.00235642369969060 |  0.71428571428571419 |
|        36 |  0.00001024331819497 |  0.69565217391304324 |
|        35 |  0.00021420777749855 |  0.68736137655442942 |
|        34 |  0.00160784843995766 |  0.66666534073775563 |
|        33 |  0.00119028157534114 |  0.63725719265324676 |
|        32 |  0.00079238019541892 |  0.61562675968216229 |
|        31 |  0.00053494269277241 |  0.60003142672513032 |
|        30 |  0.00058621117922467 |  0.58461036843821323 |
|        29 |  0.00024998750897103 |  0.56785934865976717 |
|        28 |  0.00002061509815479 |  0.55440322652129603 |
|        26 |  0.02831763219996664 |  0.50000000000000000 |
|        23 |  0.00017953786589658 |  0.44632471376367122 |
|        22 |  0.00114804178525575 |  0.43265146754851591 |
|        21 |  0.00209148765373934 |  0.41546099791878849 |
|        20 |  0.00447136480366181 |  0.39093339124937432 |
|        19 |  0.00056184058125404 |  0.37492946689727213 |
|        18 |  0.00515434319090128 |  0.36020785934513228 |
|        17 |  0.00627889499237681 |  0.33385621431956813 |
|        16 |  0.00331551428946029 |  0.31459958099531204 |
|        15 |  0.00164849153076835 |  0.30304918220143839 |
|        14 |  0.01180284911857132 |  0.28377075011933750 |
|        13 |  0.00831900989971149 |  0.25415565796320805 |
|        12 |  0.00444586248330500 |  0.23660929562690006 |
|        11 |  0.00491463032112485 |  0.22331489610256541 |
|        10 |  0.02168642985412893 |  0.19999999999999990 |
|         9 |  0.00644723933782312 |  0.17778520825362779 |
|         8 |  0.01422232357584844 |  0.15916454008745787 |
|         7 |  0.00944368618281479 |  0.14108362235274344 |
|         6 |  0.01711522515242123 |  0.12463070776752874 |
|         5 |  0.01605857014446175 |  0.10583558658162687 |
|         4 |  0.02216349669607403 |  0.08640204440221574 |
|         3 |  0.02408297229015970 |  0.06586101585896612 |
|         2 |  0.02559654723463824 |  0.04650772469795619 |
|         1 |  0.02940911552579521 |  0.02693192982359423 |
|        -2 |  0.04680045940086009 | -0.02585514096402246 |
|        -3 |  0.05839132204279256 | -0.04592230359816294 |
|        -4 |  0.07090358504550298 | -0.06614055023135565 |
|        -5 |  0.03502125088674130 | -0.08663693722547899 |
|        -6 |  0.04592331313814468 | -0.10415894876732154 |
|        -7 |  0.03468944744545854 | -0.12489370469198741 |
|        -8 |  0.03636714909623130 | -0.14466592458066274 |
|        -9 |  0.02420005793126845 | -0.16451936561462860 |
|       -10 |  0.02486080551606965 | -0.18353544056261387 |
|       -11 |  0.02038495612294155 | -0.20354595308019077 |
|       -12 |  0.01532869795078525 | -0.22259503687233592 |
|       -13 |  0.03894686613946247 | -0.24954723803196077 |
|       -14 |  0.00080989335537749 | -0.26795849902205870 |
|       -15 |  0.01093248186377699 | -0.27882703196169500 |
|       -16 |  0.01406786328230392 | -0.30114621791639995 |
|       -17 |  0.00481905158117618 | -0.31896326089162186 |
|       -18 |  0.01126052264892294 | -0.33420239515465161 |
|       -19 |  0.00762527797492671 | -0.35497267983234110 |
|       -20 |  0.00537093486559662 | -0.37316482762760189 |
|       -21 |  0.01801042182419816 | -0.39999853302049365 |
|       -22 |  0.00129906745042194 | -0.42137186852973024 |
|       -23 |  0.00396274715883349 | -0.43423415635535390 |
|       -24 |  0.00524302171262204 | -0.45346298679295921 |
|       -25 |  0.00289832384058324 | -0.47175178783928612 |
|       -26 |  0.01044042271488793 | -0.49957257348688450 |
|       -27 |  0.00022173003543806 | -0.50606119544374928 |
|       -28 |  0.00357199862289537 | -0.53351642333846871 |
|       -29 |  0.00079307750523909 | -0.55079027561422045 |
|       -30 |  0.00644165281717133 | -0.57141013454565792 |
|       -31 |  0.00032944637150629 | -0.59028248638169123 |
|       -32 |  0.00130362128163072 | -0.60131673732001456 |
|       -33 |  0.00418993054904777 | -0.62500000000000011 |
|       -34 |  0.00062257914666534 | -0.64668475755092125 |
|       -35 |  0.00278975680418676 | -0.66666577754524736 |
|       -36 |  0.00023555528866450 | -0.68407953715517900 |
|       -37 |  0.00187454179945584 | -0.70000491647215080 |
|       -38 |  0.00134223589377089 | -0.72646282678620899 |
|       -39 |  0.00086745082199328 | -0.74965167535242383 |
|       -40 |  0.00056156501838238 | -0.76910060208855813 |
|       -41 |  0.00036102366584841 | -0.78567292568131819 |
|       -42 |  0.00022880167207248 | -0.79999069392982158 |
|       -43 |  0.00022828057142051 | -0.81666643961004726 |
|       -44 |  0.00008039959467130 | -0.83653844120138621 |
|       -45 |  0.00002999390339164 | -0.85424028268551233 |
|       -46 |  0.00000425403966710 | -0.87313860252004571 |
|       -47 |  0.00000015328441153 | -0.89098134070490687 |
|       -48 |  0.00000000042684590 | -0.90693430656934315 |
|       -52 |  0.04034992234489480 | -1.00000000000000000 |


    Note that there is no TC floored equal to zero or equal to minus one. The index is clearly +1 since from +1 the expected value is positive.

    Code:
    +----------+-------+-----+-----+------------+------------+----------------------+----------------------+
|   Play   | Decks |  CR | IRC |  TC Index  |  RC Index  | Total EV >= TC Index | Total EV >= RC Index |
+----------+-------+-----+-----+------------+------------+----------------------+----------------------+
|    Ins   |   1   |   0 |  -4 |         1  |      1     |  0.08558354668387291 |  0.08558183737933568 |
+----------+-------+-----+-----+------------+------------+----------------------+----------------------+


    The decimal index is +1.1

    Sincerely,
    Cac
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  7. 10-03-2022, 12:26 PM #7
    k_c
    k_c is offline
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    Oct 2012
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    Quote Originally Posted by Cacarulo View Post
    Read again the methodology that I sent you earlier. The answer is there.
    Once you assemble the arrays you will see it more clearly. This is what I get from the arrays:

    Code:
    |       104 |  0.00871477100434657 |  2.00000000000000000 |
|        94 |  0.00000000000157696 |  1.81282495667244370 |
|        93 |  0.00000000002582721 |  1.79999999999999982 |
|        92 |  0.00000000291990962 |  1.77090909090909099 |
|        91 |  0.00000001888279004 |  1.75000000000000000 |
|        89 |  0.00000011539482804 |  1.72727272727272685 |
|        88 |  0.00000061443999346 |  1.70000000000000018 |
|        86 |  0.00000290322938725 |  1.66666666666666630 |
|        85 |  0.00000000001263757 |  1.64705882352941169 |
|        84 |  0.00001232995767432 |  1.62500000000000000 |
|        83 |  0.00000000204895869 |  1.60000000000000075 |
|        81 |  0.00004748601158011 |  1.57142857142857140 |
|        80 |  0.00000010700247691 |  1.53846153846153832 |
|        79 |  0.00000000000264831 |  1.52631578947368407 |
|        78 |  0.00016704038144613 |  1.50000000000000044 |
|        76 |  0.00000000104259939 |  1.47058823529411731 |
|        75 |  0.00000280244582382 |  1.45454545454545459 |
|        74 |  0.00000001001713137 |  1.43750000000000000 |
|        72 |  0.00054354543772864 |  1.40000000000000013 |
|        71 |  0.00000000033898419 |  1.36842105263157876 |
|        70 |  0.00000043364161695 |  1.35714285714285721 |
|        69 |  0.00004387543124798 |  1.33333333333333348 |
|        68 |  0.00000215788328434 |  1.30769230769230749 |
|        67 |  0.00000003892371046 |  1.29411764705882293 |
|        66 |  0.00000000005561459 |  1.28571428571428559 |
|        65 |  0.00169413476984208 |  1.25000000000000000 |
|        62 |  0.00000142216800880 |  1.20011666964613362 |
|        61 |  0.00003468047098998 |  1.18181818181818166 |
|        60 |  0.00000012776135551 |  1.16666666666666696 |
|        59 |  0.00044653704872974 |  1.14285714285714279 |
|        58 |  0.00000078431276579 |  1.11764705882352944 |
|        57 |  0.00011494953741622 |  1.10000000000000009 |
|        56 |  0.00002466219327307 |  1.07692334951209756 |
|        55 |  0.00000387542778391 |  1.06250000000000000 |
|        54 |  0.00000038131483604 |  1.05237716471420817 |
|        52 |  0.00557287748507175 |  1.00000000000000000 |
|        49 |  0.00000098985500915 |  0.95024721217439700 |
|        48 |  0.00006574386419129 |  0.93037593984962408 |
|        47 |  0.00024697563712821 |  0.90909115571936860 |
|        46 |  0.00000495213522480 |  0.89473684210526305 |
|        45 |  0.00091596628308714 |  0.87500000000000000 |
|        44 |  0.00017428659431586 |  0.84629621318248338 |
|        43 |  0.00002064258472653 |  0.83333333333333326 |
|        42 |  0.00000084422952156 |  0.82608695652173880 |
|        41 |  0.00355000323823723 |  0.79999999999999993 |
|        40 |  0.00000500312850994 |  0.77276324321463064 |
|        39 |  0.00053207026460195 |  0.75197575923828264 |
|        38 |  0.00004212583313604 |  0.73676600341560983 |
|        37 |  0.00235642369969060 |  0.71428571428571419 |
|        36 |  0.00001024331819497 |  0.69565217391304324 |
|        35 |  0.00021420777749855 |  0.68736137655442942 |
|        34 |  0.00160784843995766 |  0.66666534073775563 |
|        33 |  0.00119028157534114 |  0.63725719265324676 |
|        32 |  0.00079238019541892 |  0.61562675968216229 |
|        31 |  0.00053494269277241 |  0.60003142672513032 |
|        30 |  0.00058621117922467 |  0.58461036843821323 |
|        29 |  0.00024998750897103 |  0.56785934865976717 |
|        28 |  0.00002061509815479 |  0.55440322652129603 |
|        26 |  0.02831763219996664 |  0.50000000000000000 |
|        23 |  0.00017953786589658 |  0.44632471376367122 |
|        22 |  0.00114804178525575 |  0.43265146754851591 |
|        21 |  0.00209148765373934 |  0.41546099791878849 |
|        20 |  0.00447136480366181 |  0.39093339124937432 |
|        19 |  0.00056184058125404 |  0.37492946689727213 |
|        18 |  0.00515434319090128 |  0.36020785934513228 |
|        17 |  0.00627889499237681 |  0.33385621431956813 |
|        16 |  0.00331551428946029 |  0.31459958099531204 |
|        15 |  0.00164849153076835 |  0.30304918220143839 |
|        14 |  0.01180284911857132 |  0.28377075011933750 |
|        13 |  0.00831900989971149 |  0.25415565796320805 |
|        12 |  0.00444586248330500 |  0.23660929562690006 |
|        11 |  0.00491463032112485 |  0.22331489610256541 |
|        10 |  0.02168642985412893 |  0.19999999999999990 |
|         9 |  0.00644723933782312 |  0.17778520825362779 |
|         8 |  0.01422232357584844 |  0.15916454008745787 |
|         7 |  0.00944368618281479 |  0.14108362235274344 |
|         6 |  0.01711522515242123 |  0.12463070776752874 |
|         5 |  0.01605857014446175 |  0.10583558658162687 |
|         4 |  0.02216349669607403 |  0.08640204440221574 |
|         3 |  0.02408297229015970 |  0.06586101585896612 |
|         2 |  0.02559654723463824 |  0.04650772469795619 |
|         1 |  0.02940911552579521 |  0.02693192982359423 |
|        -2 |  0.04680045940086009 | -0.02585514096402246 |
|        -3 |  0.05839132204279256 | -0.04592230359816294 |
|        -4 |  0.07090358504550298 | -0.06614055023135565 |
|        -5 |  0.03502125088674130 | -0.08663693722547899 |
|        -6 |  0.04592331313814468 | -0.10415894876732154 |
|        -7 |  0.03468944744545854 | -0.12489370469198741 |
|        -8 |  0.03636714909623130 | -0.14466592458066274 |
|        -9 |  0.02420005793126845 | -0.16451936561462860 |
|       -10 |  0.02486080551606965 | -0.18353544056261387 |
|       -11 |  0.02038495612294155 | -0.20354595308019077 |
|       -12 |  0.01532869795078525 | -0.22259503687233592 |
|       -13 |  0.03894686613946247 | -0.24954723803196077 |
|       -14 |  0.00080989335537749 | -0.26795849902205870 |
|       -15 |  0.01093248186377699 | -0.27882703196169500 |
|       -16 |  0.01406786328230392 | -0.30114621791639995 |
|       -17 |  0.00481905158117618 | -0.31896326089162186 |
|       -18 |  0.01126052264892294 | -0.33420239515465161 |
|       -19 |  0.00762527797492671 | -0.35497267983234110 |
|       -20 |  0.00537093486559662 | -0.37316482762760189 |
|       -21 |  0.01801042182419816 | -0.39999853302049365 |
|       -22 |  0.00129906745042194 | -0.42137186852973024 |
|       -23 |  0.00396274715883349 | -0.43423415635535390 |
|       -24 |  0.00524302171262204 | -0.45346298679295921 |
|       -25 |  0.00289832384058324 | -0.47175178783928612 |
|       -26 |  0.01044042271488793 | -0.49957257348688450 |
|       -27 |  0.00022173003543806 | -0.50606119544374928 |
|       -28 |  0.00357199862289537 | -0.53351642333846871 |
|       -29 |  0.00079307750523909 | -0.55079027561422045 |
|       -30 |  0.00644165281717133 | -0.57141013454565792 |
|       -31 |  0.00032944637150629 | -0.59028248638169123 |
|       -32 |  0.00130362128163072 | -0.60131673732001456 |
|       -33 |  0.00418993054904777 | -0.62500000000000011 |
|       -34 |  0.00062257914666534 | -0.64668475755092125 |
|       -35 |  0.00278975680418676 | -0.66666577754524736 |
|       -36 |  0.00023555528866450 | -0.68407953715517900 |
|       -37 |  0.00187454179945584 | -0.70000491647215080 |
|       -38 |  0.00134223589377089 | -0.72646282678620899 |
|       -39 |  0.00086745082199328 | -0.74965167535242383 |
|       -40 |  0.00056156501838238 | -0.76910060208855813 |
|       -41 |  0.00036102366584841 | -0.78567292568131819 |
|       -42 |  0.00022880167207248 | -0.79999069392982158 |
|       -43 |  0.00022828057142051 | -0.81666643961004726 |
|       -44 |  0.00008039959467130 | -0.83653844120138621 |
|       -45 |  0.00002999390339164 | -0.85424028268551233 |
|       -46 |  0.00000425403966710 | -0.87313860252004571 |
|       -47 |  0.00000015328441153 | -0.89098134070490687 |
|       -48 |  0.00000000042684590 | -0.90693430656934315 |
|       -52 |  0.04034992234489480 | -1.00000000000000000 |


    Note that there is no TC floored equal to zero or equal to minus one. The index is clearly +1 since from +1 the expected value is positive.

    Code:
    +----------+-------+-----+-----+------------+------------+----------------------+----------------------+
|   Play   | Decks |  CR | IRC |  TC Index  |  RC Index  | Total EV >= TC Index | Total EV >= RC Index |
+----------+-------+-----+-----+------------+------------+----------------------+----------------------+
|    Ins   |   1   |   0 |  -4 |         1  |      1     |  0.08558354668387291 |  0.08558183737933568 |
+----------+-------+-----+-----+------------+------------+----------------------+----------------------+


    The decimal index is +1.1

    Sincerely,
    Cac

    OK thank you. If I'm lucky I'll be able to get it.

    I still think best TC index(ins count) for +EV is greater than 0.00. I think you get >= 1.00?
    > 0.00 would work for any number of decks.

    k_c
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  8. 10-03-2022, 12:43 PM #8
    Cacarulo
    Cacarulo is offline
    Senior Member
    Join Date
    Feb 2022
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    721

    Did you find this post helpful? Yes | No
    OK thank you. If I'm lucky I'll be able to get it.

    I still think best TC index(ins count) for +EV is greater than 0.00. I think you get >= 1.00?
    > 0.00 would work for any number of decks.
    Remember that indices are generally evaluated by "greater than or equal" (>=). Also note that you could say TC >= -1 or TC >= 0 since these TCs do not exist.
    The correct thing in this case is TC >= +1.

    Sincerely,
    Cac
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  9. 10-05-2022, 06:47 AM #9
    Cacarulo
    Cacarulo is offline
    Senior Member
    Join Date
    Feb 2022
    Posts
    721

    Did you find this post helpful? Yes | No
    OK thank you. If I'm lucky I'll be able to get it.

    I still think best TC index(ins count) for +EV is greater than 0.00. I think you get >= 1.00?
    > 0.00 would work for any number of decks.

    k_c
    Sorry k_c, somehow a line didn't come out in my TC listing. The TC equal to zero does exist even though the expected value is zero. This is the line I accidentally leaked:

    Code:
    |         0 |  0.08558354668387287 |  0.00000000000000000 |


    Sorry again. Anyway the index is still +1 which is where the EV starts to be positive.

    Sincerely,
    Cac
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