Zenfighter: UNLLI Revisited

[Zenfighter]

										 

										 

			                        UNIT         NORMAL	      LINEAR	     LOSS	 INTEGRAL			 

										 

										 

 z	  0.00	          0.01	          0.02	          0.03	          0.04	          0.05	          0.06	          0.07	          0.08	          0.09 

										 

0.00	0.398942	0.393962	0.389022	0.384122	0.379261	0.374441	0.369660	0.364919	0.360218	0.355557 

0.10	0.350935	0.346353	0.341811	0.337309	0.332846	0.328422	0.324038	0.319693	0.315388	0.311122 

0.20	0.306895	0.302707	0.298558	0.294448	0.290377	0.286345	0.282351	0.278396	0.274479	0.270601 

0.30	0.266761	0.262959	0.259196	0.255470	0.251782	0.248131	0.244518	0.240943	0.237404	0.233903 

0.40	0.230439	0.227011	0.223621	0.220267	0.216949	0.213667	0.210422	0.207212	0.204038	0.200900 

0.50	0.197797	0.194729	0.191696	0.188698	0.185735	0.182806	0.179912	0.177051	0.174225	0.171432 

0.60	0.168673	0.165947	0.163254	0.160594	0.157967	0.155372	0.152810	0.150280	0.147781	0.145315 

0.70	0.142879	0.140475	0.138102	0.135760	0.133448	0.131167	0.128916	0.126694	0.124503	0.122340 

0.80	0.120207	0.118103	0.116028	0.113981	0.111962	0.109972	0.108009	0.106074	0.104166	0.102285 

0.90	0.100431	0.098604	0.096803	0.095028	0.093279	0.091556	0.089858	0.088185	0.086537	0.084914 

1.00	0.083316	0.081741	0.080190	0.078664	0.077160	0.075680	0.074223	0.072789	0.071377	0.069987 

1.10	0.068620	0.067274	0.065949	0.064646	0.063365	0.062104	0.060863	0.059643	0.058443	0.057263 

1.20	0.056102	0.054961	0.053840	0.052737	0.051653	0.050587	0.049539	0.048510	0.047499	0.046505 

1.30	0.045528	0.044568	0.043626	0.042700	0.041791	0.040898	0.040020	0.039159	0.038313	0.037483 

1.40	0.036668	0.035868	0.035083	0.034312	0.033555	0.032813	0.032085	0.031370	0.030669	0.029981 

1.50	0.029307	0.028645	0.027996	0.027360	0.026736	0.026124	0.025525	0.024937	0.024360	0.023796 

1.60	0.023242	0.022700	0.022168	0.021647	0.021137	0.020637	0.020147	0.019668	0.019198	0.018738 

1.70	0.018288	0.017847	0.017415	0.016993	0.016579	0.016174	0.015778	0.015390	0.015010	0.014639 

1.80	0.014276	0.013920	0.013573	0.013233	0.012900	0.012575	0.012257	0.011946	0.011642	0.011345 

1.90	0.011055	0.010771	0.010493	0.010222	0.009957	0.009698	0.009445	0.009198	0.008957	0.008721 

2.00	0.008491	0.008266	0.008046	0.007832	0.076229	0.007419	0.007219	0.007025	0.006835	0.006649 

2.10	0.006468	0.006292	0.006120	0.005952	0.005788	0.005628	0.005472	0.005321	0.005172	0.005028 

2.20	0.004887	0.004750	0.004616	0.004486	0.004359	0.004235	0.004114	0.003996	0.003882	0.003770 

2.30	0.003662	0.003556	0.003453	0.003352	0.003255	0.003160	0.003067	0.002977	0.002889	0.002804 

2.40	0.002720	0.002640	0.002561	0.002484	0.002410	0.002337	0.002267	0.002198	0.002132	0.002067 

2.50	0.002004	0.001943	0.001883	0.001825	0.001769	0.001715	0.001662	0.001610	0.001560	0.001511 

2.60	0.001464	0.001418	0.001373	0.001330	0.001288	0.001247	0.001207	0.001169	0.001131	0.001095 

2.70	0.001060	0.001026	0.000993	0.000961	0.000929	0.000899	0.000870	0.000841	0.000814	0.000787 

2.80	0.000761	0.000736	0.000711	0.000688	0.000665	0.000643	0.000621	0.000600	0.000580	0.000560 

2.90	0.000541	0.000523	0.000505	0.000488	0.000471	0.000455	0.000440	0.000425	0.000410	0.000396 

3.00	0.000382	0.000369	0.000356	0.000343	0.000331	0.000320	0.000308	0.000298	0.000287	0.000277 

										 

[Zenfighter]

Professor Griffin taught us how to extract practical applications from his famous UNLLI chart (page 87, TOB 4th Edition). Basically it can be used to approximate a player?s gain from perfect strategy decisions at a given depth level. But we?re mere mortals and not computers, which means that it would seem to be more practical to approximate those gains for a player using a given point count. For this given purpose, you will obviously need to know what the correlation coefficient is for the selected play of your choice. More math, damm! But this is not rocket-science. It can be done. For the true fanatic, the procedure can be learned from careful reading of the book.

Enter his first ?magical? equation, also known as b

b = 51* √ [(ss*(N-n)/13*(N-1)*n], where

ss = Sum Ei ^ 2 {for i = 1 to 10} and any given (i) = appropriate effect of removal,
N = number of cards in the full pack, and
n = number of cards remaining.

And so far, legions of card counters and enthusiastic researchers have taken Griffin on faith, right? Well, almost, I?d say. What does it all really mean?

Essentially what Professor Griffin did, was to apply statistical tricks like using N-1 in the denominator of the equation with the ultimate purpose of getting better estimates for the standard deviation of any given population where a sample has been taken. By definition, the standard deviation is the square root of the average squared deviations from the mean. Ouch! Consider the following:

The variance (sd is the square root of this, remember) for finding the expected number of any given rank by drawing n cards without replacement from a pack, let?s say of size N, equals:

v = n * 1/13 * 12/13 * ((N-n) /(N-1)), further refinements by introducing the squares of n and the EoR?s ones and voil?.

What matters is that:

b = adjusted standard deviation at the selected card level.

Q: ?Mr. Griffin please, I do not understand much about your complicated formulas??
A: ?You don?t have to worry, neither do I!?

Shall we trust this humorous ?liar?? Surely, we shall not.

Calculate z

Here,
z = m/b where
m = full-deck favorability
b = see above.

That means to standardize the full-deck favorability. In plain English: to write it in a form that allows it to be read under the Normal Curve. You have to divide by the standard deviation to accomplish this.

The dreaded UNLLI

A table that has an existence of its own, independently of any blackjack considerations TOB

Griffin didn?t add a single comment about its intricacies. So I was forced to
search through all the books available at home, but alas, zero success.

MathProf?s formulae to the rescue:

UNLLI (z) = ∫ {t = z to t = infinity}(t-z) N (t) dt

In plain English this seems to be an integration of the partial derivative function of our famous Standard Normal Distribution. Now comes the expected mathematician?s admonition, that this integral can be solved quite ?easily.?

UNLLI (z) = part_(dz/dt) f (z) ? (1 ? cum f (z))

If I?m reading MathProf correctly this sounds like:

The UNLLI at any given standardized level equals the partial derivative of the Standard Normal Distribution minus the complement of the cumulative distribution function of the SND at this given level.

I told you guys, quite ?easy.? Let?s stop the nonsense. No card counter would need this knowledge ever! A lot ?easier? sounds to me something like this:

What about just summing all the possible values of a standard normal variable (the positives ones) about a number, without forgetting to multiply them by their probabilities of occurrence? Sounds like a plan, no?

That?s all we need. We have our new UNLLI tables; newly devised EoR?s (see Don?s Domain, Theory & Math page and/or wait for the new BJA3 soft cover edition); and finally we?ll be able to compute our EV at any given selected depth of the pack.

EV (n) = b * UNLLI (z),

where z = m/b don?t forget.

A practical example with our new tables.

What is the player?s gain from perfect insurance when there are 26 cards left in a single deck?

1) Remove the ace in order to improve the accuracy. That is, adjust m; also N = 51

m = 7.6923 ?1.8100 = 5.8823

2) Extract b:

b = 51 * sqr [(95.8211*25)/13*50*26] = 19.2012

3) Standardize the full-deck favorability. That is find z:

z = 5.8823/19.2012 = 0.3064

4) Look for this value in our new UNLLI table:

UNLLI (0.3064) = UNLLI (0.31) = 0.262959

5) Find the expected gain at the 26-card level:

E (n) = 0.262959 * 19.2012 = 5.0491

6) Finally adjusting for the probability of occurrence:

Dealer?s Ace = 1/13 Insurance bet = ? of the original, therefore

(1/13) * (1/2) = 1/26 so we get only

5.0491 * (1/26) = 0.1942% as the total computed gain.

You should note that Prof. Griffin gives a 0.20% as the ?precise? figure in The Table of Exact Gain from Perfect Insurance (in 1/100 of a %). You can double-check this, inside his book.

What?s new, Zen? Yes, I can hear you guys. But what the heck. Math, like music and love, can increase a man?s happiness. Tested!

Enjoy

Zenfighter

P.S. Obviously, a greater degree of precision can be achieved with the employment of mathematical interpolation and/or exact UNLLI expectations (computing UNLLI (0.3064) e.g.) I have refrained from this, among other reasons, because the reader is not supposed to have math and/or programming abilities beyond the standard ones. Just our new table, the coming EoR?s ones and a pocket calculator shall suffice.

[Don Schlesinger]

Thanks to Zenfighter, our newest Master, for this marvelous piece of research. Professor Griffin would be very proud of you! :-)

Don

[Don Schlesinger]

V,

Please archive this post and the next one by Zenfighter.

Thanks.

Don

[alienated]

I don't want to sound like a broken record when it comes to commenting on your contributions, but it really is very much appreciated. Thanks once again.

A note for others: the post by MathProf that Zenfighter mentions also shows a way to use the UNLLI in a spreadsheet, which I have found very useful.

[Zenfighter]

Alienated

Note that if you set delta to 0.01 the spreadsheet will end feeding us at the z = 0.60 level. Now, you?re guessing correctly; the whole thing has been redone completely, to avoid the temptations of any type of interpolations.

My pleasure hearing you liked it. Btw, note that even an inveterate disbeliever like yours truly, is beginning to show his nose into the ST/AP stuff. If there is any rationale whatsoever, for this suddenly change of mind, my bet is having committed the ?sin? to listen carefully to these two guys called alienated and McDowell. What a misfortune! :-)

Regards

Zenfighter

[Cacarulo]