SINGLE DECK

Level = 13 cards
Penetration = 25%

ACES COUNTED | PROB | EXACT PROB

----------0------------| .3038 | .30381752011

----------1------------| .4388 | .438847539016

----------2------------| .2135 | .213493397359

----------3------------| .0412 | .0412004801921

----------4------------| .0026 | .00264105642257

----------------------- Total | .99999999308

Level = 26 cards
Penetration = 50%

---------0------------| .0552 | .055220888297

---------1------------| .2497 | .249699879926

---------2------------| .3902 | .390156062385

---------3------------| .2497 | .249699879926

---------4------------| .0552 | .055220888297

----------------------- Total | .9999975988

Level = 34 cards
Penetrat ion = 65.38%

---------0-----------| .0113 | .0113029827214

---------1-----------| .1025 | .102480376674

---------2-----------| .3170 | .317048665336

---------3-----------| .3979 | .397864991792

---------4-----------| .1713 | .171302982578

----------------------- Total | .99999999908

DOUBLE DECK

Level = 26 cards
Penetration = 25%

---------0------------| .0910 | .0910291513375

---------1------------| .2667 | .266676950397

---------2------------| .3241 | .324086571663

---------3------------| .2131 | .213098019724

---------4------------| .0828 | .0827914603658

---------5------------| .0194 | .0194283960325

---------6------------| .0027 | .00268418629396

---------7------------| .0002 | .000199197498624

---------8------------| .0000 | .0000060653084517

------------------------Total | .99999999857

Level = 52 cards
Penetration = 50%

---------0------------| .0029 | .00292162135496

---------1----------- | .0270 | .0270087663036

---------2----------- | .1048 | .1048057562

---------3----------- | .2230 | .222990970638

---------4----------- | .2845 | .284545769824

---------5----------- | .2230 | .222990970638

---------6------------| .1048 | .1048057562

---------7------------| .0270 | .0270087663036

---------8-------------| .0029 | .00292162135496

------------------------- Total | .99999999878

Level = 69 cards
Penetration = 66.35%

---------0------------| .0001 | .0000913744431098

---------1------------| .0018 | .00180138187846

---------2------------| .0148 | .0147837547266

---------3------------| .0660 | .0660341044453

---------4------------| .1757 | .17573592312

---------5------------| .2856 | .285570875072

---------6------------| .2769 | .27691721219

---------7------------| .1466 | .146603229983

---------8------------| .0325 | .0324621437819

------------------------Total | .99999999961

If you have a look at chapter 7, appendix A, from the TOB, Prof. Griffin listed there the probability that a particular running count will occur (Hi Opt II) as a function of three different penetration levels. These probabilities are exactly, that is, they were extracted without using the Normal Approach, as been explained clearly by him on pages 92/93 (4th edition) .

I have skipped for this article the same tentation. That means, all the data here was extracted by pure combinametrics. The funny problem we have had here on the Theory Page, a few months ago, has convinced me that our host?s solution was the correct one, if you are interested in accurate figures, of course.

The following formula has been used to extract the probabilities

[ ( C (Ta,Na) * C (N-Ta, L-Na))/ C (N,L) ] = p

Where,

N = total number of cards
L = level been elected
Ta = number of total aces in play
Na = number of aces counted
p = probability

One example will suffice:
You?re playing a DD game and at the 26 card level , you realize horrorified :-), that 7 aces are already gone.

Therefore,

p = [( C (8,7) * C (96,19))/ C (104,26)]

Solving here, p= [ (8 * 5.61513222 * 10^19)/ (2.2551010150 * 10^24)]

what yields p = .000199197498624 or .0002 if you prefer.

To finish I?d say that the 75% penetration level was deliberately not included, because
it looks to yours truly here, more fictionary than for real . I?ll be mistaken, naturally.

So enjoy, or at least do me a favour, just smile :-)

Regards
Z