Algebraic Indices for Unbalanced Count Systems and Fractional Insurance Indexes
©Copyright ETFan December 12, 2010  

Did you know the hi-lo index for insurance can be calculated on the back of a (large) envelope? And the right answer for one deck is 17/12, the exact answer for infinite decks is 10/3, and the correct answer for any other number of decks can be found with linear interpolation? 

This article attempts to explain and extend the ideas behind Arnold Snyder’s 1980/81/82 pamphlet: Algebraic Approximation of Optimum Blackjack Strategies. The extended system can handle unbalanced true counts, such as TKO or C. Membrino’s true-counted Red 7, as well as balanced counts, such as hi-lo. The formulas developed will be more complicated than Arnold’s, since we have easy access to computers nowadays, freeing us to abandon simplicity in favor of utterly preposterous precision. The system can be used for any strategy decisions where Effects Of Removal (EoRs) are available. In addition, a formula will be given that produces insurance indexes in the form of simple fractions.  

Even though a lot of number crunching is involved, the ideas behind the system are fairly simple. If you accept the concepts of proportional deflection and linear transformation of a count as espoused by Peter Griffin in Theory of Blackjack (ToB), then you must grant that the results are exact. Unfortunately, this will not stop debate on the subject, since decks of cards aren’t precisely normal in their distribution, making proportional deflection an approximate theory. Therefore, indices generated by simulators may be slightly more accurate. On the other hand, simulators sometimes generate different indexes at different penetrations, and who needs that… ;?  

At any rate, from the standpoint of optimizing expectation (as opposed to SCORE) these algebraic indexes are almost certainly “close enough” by any standard. The system cannot provide surrender indexes, because of the way EoR tables have traditionally been layed out.  

I’ll try to describe the method in such a way that anyone versed in elementary probability theory and Peter Griffin’s work can understand. Anyone unfamiliar with ToB will assuredly not fully comprehend a word I’ve written. In particular, readers should feel comfortable with the material in Appendix A of Chapter 7 and the reasoning in Appendix C of Chapter 5. You must also know how to use the “Effect of Removal” (EoR) tables in Chapter 6.  

To illustrate, I’ll develop a hi-lo insurance index for 1 deck, and a true-counted Red 7 insurance index for 6 decks. 

Exact Insurance EoRs  

Very precise insurance indexes are possible because we can derive exact insurance EoRs. An EoR is the amount you need to add to the full deck EV to get the EV for the 51 cards after a particular card is removed. I.e. EV(full deck) + EoR = EV(51 cards with specified removal), or rearranging: EoR = EV(51 cards with specified removal) – EV(full deck). [Note: EV stands for “Expected Value,” also known as the expectation, in this case for the insurance bet only.]  

Let’s find the insurance EoR for a 5. Assume a 1 unit bet …  

There are 16 Ts in a full deck and 36 other cards. If the hole card is a T, you win 2 units, if the hole card is one of the others, you lose 1 unit. EV(full deck) = 16/52 * 2 + 36/52 * (-1) = -1/13  

If you remove a 5, there are now 51 cards remaining, still with 16 Ts, but now with 35 others. EV(51 cards with a 5 removed) = 16/51 * 2 + 35/51 * (-1) = -1/17  

So EoR(5) = (-1/17) – (-1/13) = 4/221 ~= 1.81% as listed in ToB, Chapter 6. 4/221 is the exact EoR, 1.81% is the decimal approximation.  

This same EoR = 4/221 can be used for any number of decks, and any number of cards removed, by using the conversion outlined on the fourth page of Chapter 6, ToB, of 51/(cards remaining). For example, if you remove four 5s from a 6 deck shoe, the total EoR will be: 4*(4/221) * 51/308 = 12/1001  

(We’re already at the point where a CAS — Computer Algebra System — program or calculator would come in handy. Examples of CAS are Mathematica, Maxima, XCas, and calculators like the HP 50g or TI-89. You could also do the fractions by hand or throw caution to the wind and take my numbers on faith.)  

Since we know the EV at the top of a 6d shoe is -1/13, we know now that the EV after four 5s are removed is: -1/13 + 12/1001 = -5/77, which you can verify by exact calculation. You should calculate the expectation directly, based on four non-tens removed from a 6d shoe, to convince yourself that -5/77 is the exact answer (ignore the ace upcard, just this once). If you can follow the argument thus far, you have a very good model of how EoRs work for any strategy — not just for insurance. On the other hand, if you can’t calculate the -5/77 EV, you probably should get off now. This article may exceed your level of preparation.  

Using reasoning similar to the above, we can show that the insurance EoR for any card other than a T is also 4/221, and that the EoR for a T is -9/221. The EoR for all 52 cards sum to zero.  

You can multiply all the insurance EoRs by 221 (a “linear transformation”) to get the perfect, balanced, insurance count of 4 4 4 4 4 4 4 4 4 -9 (we follow Griffin’s convention of listing tags in numerical order from ace through ten). This is Thorp’s old ten count “parameterized as a point count,” according to ToB Chapter 4. These tags have correlation 1 with the EoR, so if you know how to calculate a correlation coefficient, you can quickly correlate any set of tags with 4 4 4 4 4 4 4 4 4 -9 to get the an Insurance Correlation (IC) slightly more accurate than what you get using the three digit EoRs listed in ToB. 

A Word about Linear Tranformations of Count Systems  

A linear transformation is the act of adding a constant to every tag in a count, and/or multiplying every tag by a constant. A little thought should convince you that linear transformations will not degrade the information accessible to an arithmetically adroit counter. For example, suppose all the hi-lo tags were doubled. You of course will adjust your indices and your betting ramp, but at the end of the day, you’re no better nor worse off than when you started. Similarly, you could subtract 1 from all the tags, and unbalance the system, but at any given penetration, it’s trivial to convert back to hi-lo.  

In general a set of tags, tagi for i = 1, 2 … 52, defines a count system S. If, for constants m and b, each tagi(S) x m + b = tagi(S’), then system S’ has linear equivalence with S. The correlation between S and S’ will be 1, so the correlation between either count and any third set of numbers (such as EoRs) will be identical.  

For example, if S = hi-lo, m = 2 and b = 0.5, we have the count:S’ = -1.5 2.5 2.5 2.5 2.5 2.5 0.5 0.5 0.5 -1.5which is nothing more than hi-lo in disguise. This new count has an insurance correlation of 76%, same as the insurance correlation for hi-lo. Betting and playing correlations are also identical.  

Or if you let m = 3/13 and b = 1/13 and apply those to Thorp’s ten count, S = 4 4 4 4 4 4 4 4 4 -9, you generate S’ = 1 1 1 1 1 1 1 1 1 -2 — the well known Noir count.  Both counts have an insurance correlation of 100%.  

Calculating the One Deck Insurance Index for Hi-lo  

Now to find the one deck insurance index for hi-lo, we will perform six steps:

1) Remove the upcard (ace for insurance) as well as any player cards from the shoe.

2) Balance the count with a linear transformation.

3) [A common sense check] Find the effect on each individual card when you increase the TC by 1, using the principle of proportional deflection.

4) Write an equation that finds EV (or EV delta, for strategies other than insurance) based on 3), the EORs, and the starting EV delta.

5) Solve for TC when the EV delta = 0.

6) Convert this TC back to the original count to get your index.  

As follows: 

1) Remove an ace from the deck. Now there are 51 cards in the “shoe” and the count is no longer balanced. 

2) To balance the count (making it much simpler to apply Griffin’s formulas) we need to subtract 1/51 from all the tags, which is a linear transformation. So we have three aces that are tagged -52/51, four each of 2s, 3s, 4s, 5s and 6s tagged 50/51, four each of 7s, 8s and 9s tagged -1/51, and 16 Ts tagged -52/51. If you add the tags all up, they come to zero, and this new count is guaranteed to provide the counter with exactly the same information as a hi-lo counter at any given penetration.  

We could multiply all the tags by 51, to make them integers, and it might make life a little easier for a counter using the tags in the real world, but our calculations would actually become slightly more complicated — and we have no intention of using these tags in the real world.  

A subtle point concerning this new “rebalanced” count is how it reveals that the so-called “neutral” cards are now slightly correlated with the count. Because of the removal of the ace, a high count now portends a slightly higher density of 7s, 8s and 9s, and you can see that clearly now, because 7s, 8s, and 9s have small negative tags. 

3) For each of the 51 cards, a given True Count (TC) implies that, on average, it’s no longer one, single card. The principle of proportional deflection says it’s “deflected” from the value of 1, by an amount proportional to the TC and the individual (balanced) count tag. Under this model, cards can assume fractional values in order to represent the “average” situation. The formula for each card is: 1 – TC*Tag*51/y, where y = the sum of (tag squared) for all 51 cards. TC, here, is expressed on a per card basis, so a TC of +1/52 corresponds to a full deck TC of +1. There will still be 51 cards in the deck after the cards are deflected. In this case:y = 19*(-52/51)^2 + 20*(50/51)^2 + 12*(-1/51)^2 = 1988/51.  

If we happen to have a per-card TC of +1/52, the average number of aces would then be: 

3*[1 – (1/52)*(-52/51)*51/(1988/51)] 

And the average number of Ts would be:

16*[1 – (1/52)*(-52/51)*51/(1988/51)] 

And the total number of 2s, 3s, 4s, 5s and 6s, on average:

20*[1 – (1/52)*(50/51)*51/(1988/51)] 

And the total number of 7s, 8s and 9s, on average:

12*[1 – (1/52)*(-1/51)*51/(1988/51)]  

If you add these all up, you’ll find the total number of cards is still 51. It’s also fairly easy to prove that it adds up to 51 for any TC you choose to plug in. Finally, if you use any number other than 1988/51 for y, the sum of tags for the 51 cards would no longer equal -51/52 as it must, showing our deflection formula is accurate. (Please take your time to verify each of these statements. You don’t have to take my word for anything.) 

4) Now the nice thing about the 1d insurance case is we don’t have to convert EoRs. If we use our TC formula for the various cards, the number remaining at the end will be 51. So we’ll forego multiplying by 51/51 (but we must remember to convert when we tackle 2 or more decks). The number “removed” for each of the 51 cards, is just the expression after the “1 -“. E.g. for one given ace, remember the formula was:

1 – TC*(-52/51)*51/(1988/51) 

So the amount removed for that ace is:

TC*(-52/51)*51/(1988/51) 

which will be a negative removal (i.e. an addition) for a positive TC. So combining all this information with the EoRs, and remembering to add the full EoR for the ace removed off the top, we deduce that the total insurance EV, for any given TC is: 

EV = -1/13 + 4/221 +

3*TC*(-52/51)*51/(1988/51) *(4/221) +

20*TC*(50/51)*51/(1988/51) *(4/221) +

12*TC*(-1/51)*51/(1988/51) *(4/221) +

16*TC*(-52/51)*51/(1988/51) *(-9/221)  

This is similar to the example on the fourth page of Chapter 6, ToB, showing how to use the EoR tables, just a tad more tortuous for us mathe-masochists. 

5) To find the per-card index for our “rebalanced” count, we set this last expression equal to zero and solve for TC: 

TC/(1988/51)/221*[3*(-52)*4 + 20*50*4 + 12*(-1)*4 + 16*(-52)*(-9)] = 1/13 – 4/221

TC*51/1988/221*[10816] = 1/17

TC = 497/10608 

[As a check, solving: 

16*(1 – TC*(-52/51)*51/(1988/51)) * 2 = 51 – 16*(1 – TC*(-52/51)*51/(1988/51) 

for TC also returns 497/10608. This last expression just says there are twice as many non-tens as tens in the deck. So you can solve for TC without bothering with EoRs, just from the expression for Ts in 3). But I’m using EoRs to develop a more general system that can handle any strategy — not just insurance.] 

6) Now to get the hi-lo per-card TC from this “rebalanced” count at any penetration, all you need to do, believe it or not, is subtract 1/51. Then to convert to a full deck TC, you multiply this per-card TC by 52. So our surprisingly simple answer is: index = (497/10608 – 1/51) * 52 = 17/12  

There are other ways to look at it and think about it, but the blankety-blank mess above is about as simple as it gets for this problem. Luckily we live in the age of computers, and once you codify the six steps in a program or spreadsheet you can immediately harness the system to kick out an index for any strategy decision. 

Second Illustration — Six Deck Insurance Index for Red 7  

We follow the same six steps… 

1) Remove an ace from the shoe. Now there are 311 cards in the shoe and the count is (still) unbalanced. 

2) To balance the count we subtract 13/311 from all the tags, which is a linear transformation. So we have 23 aces that are tagged -324/311, 24 each of 2s, 3s, 4s, 5s and 6s tagged +298/311, 12 red 7s tagged +298/311 and 12 black 7s tagged -13/311, 24 each of 8s and 9s tagged -13/311, and 96 Ts tagged -324/311. If you add the tags all up, they come to zero, and this new count is guaranteed to provide the counter with exactly the same information as a Red 7 true counter at any given penetration. 

3) The sum of squares:y = 119 * (-324/311)^2 + 132 * (298/311)^2 + 60 * (-13/311)^2 = 77892/311 

The formula for each card is:

1 – TC*Tag*311/y 

The rest of this step is left as an exercise for the interested reader. 

4) The amount removed for one of the 23 remaining aces is:TC*(-324/311)*311/(77892/311)which will be a negative removal (i.e. an addition) for a positive TC. Combining this type of information with the EoRs, and remembering to adjust EoRs by a factor of 51/311 while remembering to add the full EoR for an ace off the top, the total EV for a given TC is: 

EV = -1/13 + 4/221*51/311 +

23*TC*(-324/311)*311/(77892/311) *(4/221*51/311) +

132*TC*(298/311)*311/(77892/311) *(4/221*51/311) +

60*TC*(-13/311)*311/(77892/311) *(4/221*51/311) +

96*TC*(-324/311)*311/(77892/311) *(-9/221*51/311) 

5) Set EV equal to zero and solve for TC (A CAS is highly recommended):

TC = 149293/2418336 

6) index = (149293/2418336 – 13/311) * 52 = 2015/1944 ~= 1.04 

[Note: This is the full deck true counted Red 7 insurance index assuming the normal Initial Running Count (IRC) for 6 decks of -12. People using another IRC will need to adjust the index accordingly.] 

Comparison with Arnold Snyder’s Algebraic Method  

In the case where only the dealer’s upcard is removed from one deck the 6 steps can be collapsed to this formula:

index = 52/51 * m * y / i + 52/51 * t

Where m is precisely as the Bishop defined it, namely:

-(Griffin’s 11th column) – EoR(upcard).

And y is the same as Snyder’s “p” except it is based on sum of squares of the modified tag values, instead of the original tags. And i is the inner product, also based on the modified tags instead of the originals.  

Hence, the famous Algebraic Approximation formula:

index = m * p / i + 52/51 * t

is closely related to the procedure above. After calculating a dozen different strategies using Griffin’s most recent EoR tables, I can state that indexes calculated with 1) through 6) above rarely differ from Snyder’s until you reach the third digit, which is insignificant to the player’s hourly win rate.  

For example, the one deck insurance index under the ’81 system for hi-lo is 1.4332 while the method above yeilds 1.4167 . It’s impossible to construct a set of cards with 51 cards or less having a TC that falls between those two numbers. Thus, the two indexes are in practice indistinguishable.  

The main value of this new procedure is in expanding the algebraic system to unbalanced true count systems. In addition you get to entertain your friends with insurance indexes in the form of exact fractions! 

Toward Perfect Insurance — A Challenge  

Griffin states that “insurance is linear,” and we know EoRs can provide exact expectations for insurance bets, and that perfect insurance decisions are possible with the Noir count, for example. So it might seem reasonable to expect our procedure to provide exact insurance indexes, or at least the best possible index for every situation. Unfortunately, for most counts this is only possible when you have different indexes at very deep penetrations.  

To see this, imagine one deck dealt down to the last two cards — one of which is the dealer’s hole card. A hi-lo count of zero portends the following probabilities for the two unseen cards: ace-low — 60/446, ten-low — 320/446, mid-mid — 66/446. So an insurance bet of one unit has a positive expectation of +17/223, and a simulator should produce an index of 0 in this extreme situation.  

However, I challenge anyone to find an insurance index that’s more accurate than the one generated by this algebraic system, for any popular count system throughout a realistic range of penetrations. 

Extension to Unbalanced Running Count Systems  

Several simulations lead me to this rule of thumb: the best index for a conventional unbalanced count, such as Red 7, is the running count corresponding to the true count index at the most profitable penetration that can actually occur in the game you’re facing — or just under that penetration. The most profitable penetration normally occurs right before the last hand is dealt from the shoe.  

For example, for Red 7, since the six deck true counted insurance index is 2015/1944, I would look at the RC where 4.5 decks had been dealt out and calculate: RCindex = 1.5 x 2015/1944 = 1.55, which dovetails nicely with Snyder’s recommended index of +2 for shoe games. Again, this is the index when you start with an initial running count of -12, which is necessary to make the final running count equal to zero once all the cards in the shoe are counted.  

Simulations might reveal some exceptions to my empirical rule of thumb. 

General Linear Insurance Index Formula:  

Applying our six steps to the general insurance case and simplifying (don’t try this at home) yields: 

index = 

52 (d s s – 48 d s t – 4 d y + 4 a a + 48 a t – 2 a s + y)
—————————————————————
48 (52 d t – d s + a – t) 

where d = number of decks, a = ace tag, t = tens tag, s = sum of all tags over one 52-card deck, and y = sum of squares of tags over one deck.  

For example, for Red 7, 6 deck case, d = 6, a = -1, t = -1, s = 2, and y = 42. So index = 

52*(6*2*2 – 48*6*2*(-1) – 4*6*42 + 4*(-1)*(-1) + 48*(-1)*(-1) – 2*(-1)*2 + 42)           
…..all divided by…..
48*(52*6*(-1) – 6*2 + (-1) – (-1))  

= 2015/1944 as above.  

It is assumed, for unbalanced counts, that an IRC of -s*d is used, so true count division can work normally.  

Changing the 52 in the numerator to 26 gives the half deck index; changing it to 13 gives a quarter deck index; and changing it to 1 returns a per-card index.  

Some interesting conclusions can be drawn from the General Insurance Index Formula. For example, you can use it to show that when a = t (as in hi-lo or Red 7), interpolation by reciprocal of the number decks (1/d) is possible. Interpolation can also work perfectly in some other cases, e.g. for any perfect insurance count, such as 0 0 0 0 0 0 0 0 0 1. a = 0, t = 1, s = 16, y = 16, and index = -52/3, for any d save d = 1/36 (which is unlikely).  

The GLII formula works for insurance, but for other strategies I revert to the six steps above. A formula would get pretty hairy.  

Here is an input string for the formula that you can copy and paste for use in some CAS programs:

52*(d*s*s – 48*d*s*t – 4*d*y + 4*a*a + 48*a*t – 2*a*s + y)/(48*(52*d*t – d*s + a – t)) 

Here is a formula for the infinite deck case: 

index =             

52 (s s – 48 s t – 4 y)   
————————–
48 (52 t – s)  

And an input string for infinite deck:

52*(s*s – 48*s*t – 4*y)/(48*(52*t – s))  

Here are some insurance indexes:

                                         Thorp  Noir
decks    hi-lo     Red 7         Zen    (4/-9) (1/-2)
  1      17/12    -65/108      923/309   52/3     0
  2      19/8     247/648     2587/621   52/3     0
  3      97/36    689/972     1417/311   52/3     0
  4     137/48    377/432     1183/249   52/3     0
  5      59/20   1573/1620    7579/1557  52/3     0
  6     217/72   2015/1944    3081/623   52/3     0
  7     257/84     13/12     10907/2181  52/3     0
  8      99/32   2899/2592   12571/2493  52/3     0
infinite 10/3     221/162       16/3     52/3     0

With the General Linear Insurance Index Formula, all the problems of the world can be solved! 
– ETF

Increasing the Power of the Easy OPP Count: The Advanced OPP Count

By Carlos Zilzer
[From Blackjack Forum Vol. XXVI #1, Winter 2007]
© 2007 Carlos Zilzer

[In 2007, Carlos Zilzer provided his “Advanced OPP,” that he revved up by incorporating T. Hopper’s “counter basic strategy.” If you’re tempted to play the OPP Count, use the advanced version, which is just as easy as the original version once you learn T. Hopper’s new basic strategy. – A.S.]

It’s been about a year since I first presented the OPP count to the public. The Easy OPP is the simplest card counting system available, and the easiest to learn. Since the publication of my article presenting the OPP, I have learned a lot, but the most rewarding thing has been the hundreds of letters and emails from grateful people who are now going to the casinos with a different view of the game.

In this article, I will provide information on how to improve the efficiency of the Easy OPP count without increasing the difficulty of use. The proposals and simulations in this article are oriented to six-deck shoe games. I will present the data for eight-deck games in a future article.

Card Counters’ Basic Strategy to Increase the OPP Count’s Power

One of the simplest ways to make the Easy OPP more powerful is to use a different basic strategy geared toward the card counter. A counter-oriented basic strategy increases winnings by making the strategy correct for when the counters’ biggest bets are placed. For example, standard basic strategy calls for a player to hit his 16 versus a dealer’s 10 of the dealer. In more advanced card counting systems, playing strategies call for players to stand on a 16 versus a dealer’s 10 once the count reaches a certain level.

A counters-oriented basic strategy will call for you to stand all the time on 16 versus a dealer’s 10, because the counter’s winnings at high counts will be larger than the losses at low counts for this play. Many other deviations from standard basic strategy have the same effect.

Card-counting analyst T. Hopper has developed a basic strategy that optimizes the winnings for card counters without changing strategy with the count. At the end of this article, you will find charts of T. Hopper’s counters-oriented basic strategy from his free e-book T-H Basic Blackjack. The charts for T. Hopper’s counters’ basic strategy are below.

A simulation of one billion rounds using standard 6 deck S17 rules shows an increase of return on investment (ROI or “score”) in the range of 15.2% to 16.7% (depending on the bet spread) for using T. Hopper’s counters’-oriented basic strategy rather than standard basic strategy. This represents an increase in winnings of greater than 0.2 units/100 rounds.

Insurance Bet for the Advanced OPP Card Counting System

Although the OPP does not count the 10-value cards, for counts equal to or greater than +11 in six-deck games (or +17 if starting the count from +6 as my original article suggests), the insurance bet is recommended. Taking insurance at these counts will increase your ROI (or score) 4% more.

The Penetration Effect on the Power of the OPP Count

One thing I have learned about the OPP from the feedback I’ve received from players is that, with the OPP, there is more risk to high bets early in a shoe.

Kim Lee’s article, “On the Math Behind the OPP”, helped me to understand many things about the differences between the OPP and other card counting systems. Even though the OPP is an unbalanced count, it is very different from an unbalanced count like the Red7.

For example, with the Red7 count, it is possible to make a true count conversion or true edge adjustment using fractional methods to estimate the true count or true edge at any running count in any part of the shoe. But with the OPP, this is a very difficult task because the OPP does not have a “pivot” that equates to the same edge at any level of penetration.

With the OPP, the counter’s edge will increase different amounts at the same count at different levels of penetration. A running count of 12 (starting the count at 6 as recommended in my first article) will represent a larger edge after 3 decks out of 6 have been played than the same running count of 12 if it happens at the beginning of the shoe.

Some time ago, I began suggesting to players to avoid any bet increase until the first deck was in the discard tray; it was easy to explain that a deck is approximately the width of the middle finger. After that I started to receive good reports from the same people telling me that they had noticed a significant increase in their winnings after applying that simple rule.

Now I will present a more comprehensive analysis and advice.

To develop advice for improving the performance of the OPP, I modified ET Fan’s PowerSim Card Counting Simulation Software to report sim results deck by deck. Then I ran simulations of 6-deck shoe games with a very deep penetration (the maximum possible to avoid shoe overflow with a 1 billion round simulation).

At the end of the simulation I got six charts indicating the running OPP count, the number of rounds played in that count, the edge for that count and the variance for that count per deck played. The simulation also returned a seventh chart with the overall results of the one billion rounds. All the simulations were run using T. Hopper’s counters’ basic strategy, and insurance at counts of 11 (17) and above.

The tables below are extracts of these simulation results, showing the part of the tables for running counts 0 to 11. The running count numbers assume an initial count of 0 (not 6).

Results for the First Deck
RC	Frequency	Win Rate	Variance
0	45099453	-0.0048985	1.385056
1	22533049	-0.0039873	1.382429
2	17437729	-0.0023652	1.378858
3	12403039	-0.0015394	1.373733
4	8301513	-0.0005589	1.371465
5	5292419	0.0026138	1.368379
6	3205931	0.0021953	1.364121
7	1847678	0.0033142	1.359199
8	1012187	0.0059672	1.356724
9	525366	0.0073825	1.354168
10	258580	0.0048979	1.351327
11	121608	0.0085356	1.346574

Results for the 2nd Deck
RC	Frequency	Win Rate	Variance
0	15544347	-0.0061163	1.388518
1	15711450	-0.0041119	1.384026
2	15128594	-0.0030499	1.380121
3	13898680	-0.0011779	1.375426
4	12198110	0.0003807	1.370685
5	10220651	0.0023369	1.367378
6	8189436	0.0038852	1.362689
7	6268648	0.0044798	1.358611
8	4596761	0.0059538	1.352943
9	3225899	0.0086714	1.351015
10	2166535	0.00917	1.346022
11	1395226	0.0113644	1.342142

Results for the 3rd Deck
RC	Frequency	Win Rate	Variance
0	13068780	-0.0079109	1.392763
1	13538813	-0.0056508	1.387848
2	13543243	-0.0033995	1.381609
3	13059870	-0.0017427	1.376070
4	12164731	0.0003312	1.370765
5	10936276	0.0029875	1.365025
6	9491670	0.0046452	1.359720
7	7953935	0.0066228	1.354781
8	6429897	0.0076003	1.349389
9	5017369	0.0103951	1.343530
10	3776907	0.0112456	1.339024
11	2745790	0.0125243	1.332236

Results for the 4th Deck
RC	Frequency	Win Rate	Variance
0	12302783	-0.0099998	1.400835
1	13067271	-0.0069178	1.394555
2	13400161	-0.0039155	1.385639
3	13286241	-0.0012346	1.378854
4	12721486	0.0010927	1.370608
5	11766974	0.0040554	1.363121
6	10512181	0.0067284	1.356291
7	9060393	0.0084723	1.348629
8	7545402	0.0124984	1.341781
9	6061063	0.0135348	1.333774
10	4700803	0.0157303	1.326587
11	3518665	0.0186512	1.320435

Results for the 5th Deck
RC	Frequency	Win Rate	Variance
0	12344303	-0.0164626	1.41875
1	13715205	-0.0112253	1.406535
2	14617344	-0.0065245	1.393547
3	14920325	-0.0019819	1.381162
4	14603698	0.0031037	1.36922
5	13679086	0.0070817	1.356935
6	12269032	0.0109532	1.345168
7	10531009	0.0148428	1.333483
8	8639179	0.0177164	1.322816
9	6785582	0.0210233	1.312948
10	5090888	0.0241871	1.301468
11	3656873	0.0268388	1.290593

Results for the 6th Deck
RC	Frequency	Win Rate	Variance
0	5964172	-0.0295393	1.455209
1	7198634	-0.019633	1.431447
2	8166379	-0.0105634	1.409773
3	8695839	-0.0020113	1.387818
4	8699057	0.0051575	1.365654
5	8156422	0.0122596	1.34639
6	7158520	0.0183577	1.326588
7	5888341	0.024383	1.307584
8	4525689	0.0287781	1.289022
9	3252617	0.0318885	1.272016
10	2180576	0.0360891	1.255976
11	1362531	0.0408519	1.240604

One thing I learned from the simulation results was that even in the first deck, there is an edge at counts of +5 and higher (or +11, if starting from 6). However, closer analysis of the simulation results shows that the edge is too small to justify a bet increase. This is typical behavior for any unbalanced count: There is an edge at the pivot, no matter the number of decks played. But what we really want to know is when that edge justifies a bet increase.

When to Increase Your Bet with the Advanced OPP Count

A look at the numbers indicates that the count at which a player obtains an edge equal or greater to 1% gets lower with the number of decks played. In the first deck, the running count (RC) must be 14 to get a win rate of 1%; in the second you get that edge at an RC of 11; and in the third deck you get it at an RC of 9. In the fourth deck you have a 1% edge at an RC of 8, while in the fifth and last deck the 1% edge comes at 6.

Another way of looking at this is to say that the deeper we are in the shoe, the higher a win rate any particular RC represents.

As modifications to the SCOCALC program to calculate the optimal bet ramp from the data by deck was a major work, I introduced the data into a spread sheet and used a recursive trial and error macro in Visual Basic to determine the optimal bet ramp and score.

The optimal bet ramp shown below rounds the optimal bet to the nearest whole number.

1:16 Optimal Bet Ramp
RC	1st deck	2nd deck	3rd deck	4th deck	5th deck	6th deck
<4	1	1	1	1	1	1
4	1	1	1	1	3	5
5	2	2	3	4	6	11
6	2	3	4	6	10	16
7	3	4	6	8	13	16
8	5	5	7	11	16	16
9	7	8	9	12	16	16
10	4	8	10	14	16	16
11	8	10	11	16	16	16
12	6	10	14	16	16	16
13	3	14	16	16	16	16
14	16	16	16	16	16	16
15	16	16	16	16	16	16
16	16	16	16	16	16	16
>16	16	16	16	16	16	16

• The score for this game (91.35% penetration) and this bet ramp is $28.36
• The same game but with standard bet ramp independent of the depth returns a score of $24.47
• A game with the same conditions but using standard basic strategy returns a score of $20.13

So, the counter’s basic strategy, with insurance and a deck-dependent bet ramp, provide an increase of 40% in score from the simplest version of the OPP.

Using the same spread sheet I tested my initial recommendation to my readers to avoid increasing the bet until after the first deck had been dealt (bet 1 unit during the first deck). The results were as follows:

• The score changed from $28.36 to $28.30
• The win rate changed from 3.414 units/100 rounds to 3.403 units/100 rounds
• The standard deviation was reduced from 64.12 to 63.9

As you can see, there is very little cost to this simpler betting method.

The next step was to find a simpler optimal bet ramp and an easy way to remember it, keeping in mind that the principal objective of the OPP count was that it should be exceptionally easy to learn and to implement. The following is an easy-to-remember table using multiples of 2 units that are shifted up with each deck played.

IRC= 6 IRC=0 1:16 Optimal Bet Ramp
RC	1st deck	2nd deck	3rd deck	4th deck	5th deck	6th deck
<4	1	1	1	1	1	1
4	1	1	1	2	4	6
5	1	1	2	4	6	8
6	1	2	4	6	8	10
7	2	4	6	8	10	12
8	4	6	8	10	12	14
9	6	8	10	12	14	16
10	8	10	12	14	16	16
11	10	12	14	16	16	16
12	12	14	16	16	16	16
13	14	16	16	16	16	16
14	16	16	16	16	16	16
15	16	16	16	16	16	16
16	16	16	16	16	16	16
>16	16	16	16	16	16	16

This simpler betting ramp returns a score (now is better to call it ROI because it is a real-life rather than “optimal” bet ramp) of $27, a win rate of 3 units/100 rounds and a standard deviation of 59.7.

As a final test, and in order to compare “apples with apples”, I performed a simulation of the same game, same penetration, same seed but using the Red 7 count with the counter basic strategy and only the insurance index. The SCORE of that game is $42.5631. So we can say that the OPP with the proposed bet ramp and strategy has 63% of the performance of Red7 in the same conditions, which is much better that the 47.9% performance of standard OPP.

There are other variations to OPP that return higher scores but they mean modification of the tag values of the cards. These more advanced options will be presented in my next article. ♠

T. Hopper’s Card Counters’ Basic Strategy

---

HITTING AND STANDING

Stand	2	3	4	5	6	7	8	9	X	A S17	A H17
17	S	S	S	S	S	S	S	S	S	S	S
16	S	S	S	S	S	H	H	H	S	H	H
15	S	S	S	S	S	H	H	H	H	H	H
14	S	S	S	S	S	H	H	H	H	H	H
13	S	S	S	S	S	H	H	H	H	H	H
12	H	S	S	S	S	H	H	H	H	H	H
A7	S	S	S	S	S	S	S	H	H	S	H

RULES FOR HARD HANDS
Always stand on 17 or higher. Always stand on 12-16 vs. 2-6 and hit 12-16 vs. 7-A except:Hit 12 vs. 2Stand on 16 vs. 10

RULES FOR SOFT HANDS
Always hit soft 17 or lower. Always stand on soft 18 or higher, except:Hit soft 18 vs. 9 and 10Hit soft 18 vs. Ace if the dealer hits soft 17

Hitting or standing is considered only after all other options (surrender, split, and/or double down) have been exhausted.

DOUBLING DOWN

Double	2	3	4	5	6	7	8	9	X	A
11	D	D	D	D	D	D	D	D	D	D
10	D	D	D	D	D	D	D	D
9	D	D	D	D	D
8				D	D

DOUBLING DOWN, SOFT TOTALS

Soft Totals	2	3	4	5	6	7	8	9	T	A
(A,9)
(A,8)				D	D
(A,7)	D	D	D	D	D
(A,6)	D	D	D	D	D
(A,5)			D	D	D
(A,4)			D	D	D
(A,3)			D	D	D
(A,2)			D	D	D

With 44, for a total of hard 8, when double after split is allowed, splitting is preferred over doubling down. All other hands clearly fall into one category or the other. Never double on hard 12 or more or hard 7 or less.

SURRENDER (LATE)

Surrender (Late)	2	3	4	5	6	7	8	9	T	A
16								Sur	Sur	Sur
15									Sur	Sur
14									Sur
88									Sur
77	See separate chart

SURRENDER (EARLY)

Surrender (Late)	2	3	4	5	6	7	8	9	T	A S17	A H17
Hard 17										Sur	Sur
16									Sur	Sur	Sur
15									Sur	Sur	Sur
14									Sur	Sur	Sur
13									Sur	Sur	Sur
12										Sur	Sur
8											Sur
4,5,6,7										Sur	Sur
88									Sur	Sur	Sur
77									Sur	Sur	Sur

When it is allowed, early surrender is the first choice the player needs to make, even before considering insurance when the dealer has an ace. Late surrender is considered before all other choices after the dealer checks for blackjack. There is no difference between early surrender and late surrender against a dealer 9 or less.

PAIR SPLITS
No DAS / DAS

Pairs	2	3	4	5	6	7	8	9	T	A S17	A H17
(A,A)	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y
(T,T)	N	N	N	N	N	N	N	N	N	N	N
(9,9)	Y	Y	Y	Y	Y	N	Y	Y	N	N	/Y
(8,8)	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y
(7,7)	Y	Y	Y	Y	Y	Y	/Y	N	N*	N	N
(6,6)	Y	Y	Y	Y	Y	N	N	N	N	N	N
(5,5)	N	N	N	N	N	N	N	N	N	N	N
(4,4)	N	N	/Y	/Y	/Y	N	N	N	N	N	N
(3,3)	/Y	/Y	Y	Y	Y	Y	N	N	N	N	N
(2,2)	/Y	Y	Y	Y	Y	Y	N	N	N	N	N

No Double After Split
Always split aces and eights. Never split tens, fives, and fours Split 99 vs. 2-9 except vs. 7 Split 77 vs. 2-7 Split 66 vs. 2-6 Split 33 vs. 4-7 Split 22 vs. 3-7

When Double After Split is Allowed
Split all of the pairs listed above, and also the following: 99 vs. Ace if the dealer hits soft 17 77 vs. 8 44 vs. 4-6 33 vs. 2 and 3 22 vs. 2

With the European No Hole Card Rule
Play as above except: Never double down or split versus an ace or ten

When surrender is not available, splitting pairs is always the first choice to consider.

Note that 44 is treated as any other hard 8 unless double after split is allowed.

*77 VS. 10 AND ACE

Player 77 Hit/Stand
Decks	10	A S17	A H17
1	Stand	Hit	Hit
2	Hit	Hit	Hit
4+	Hit	Hit	Hit

Player 77 Late Surrender
Decks	10	A S17	A H17
1	Sur	Sur	Sur
2	Sur		Sur
4+	Sur

In single deck, when the player has 77, two of the four cards that could give him a 21 are no longer available. Even in double deck, the removal of two 7s out of the original eight is important. For this reason, 77 vs. 10 and 77 vs. Ace are the only two plays in the T-H Counters’ Basic Strategy where the number of decks must be considered in playing the hand.
[Editor’s Note: I’d like to thank T. Hopper for permitting his T-H Basic Strategy for Card Counters to be included in this article. —Arnold Snyder]

The OPP Card Counting System: A New, Easier Approach to Counting Cards

By Carlos Zilzer
[From Blackjack Forum Vol. XXV #1, Winter 2005/06]
© 2005 Blackjack Forum

Introduction by Arnold Snyder

[Many blackjack players find even the simplest of the traditional card counting systems too cumbersome to use comfortably in casinos. I think the Red Seven Count is about as easy a counting system as has ever been developed up to now, but it’s not the answer for everyone. Many players simply don’t play in casinos often enough to maintain their counting skills, and do not have the time to practice sufficiently for their occasional casino trips.

Carlos Zilzer has developed a unique method of counting cards that is even simpler than the Red Seven Count. In fact, he tells me he has taught a few of his friends to use his “OPP Count” and they mastered it in a matter of hours. When I first read his method, I didn’t doubt that an amateur could learn it very quickly, but it wasn’t clear to me whether it would have much value in comparison with more traditional counting systems where we always weigh the high cards versus the low cards as cards are dealt.

Because there was no commercially available blackjack simulation software on the market that could handle the new method Carlos had devised for counting cards, he wrote his own program and begun testing his system on a simulator of his own device. But when he discovered ET Fan’s PowerSim software, he realized that it was much faster and more versatile than his own program, so he began running more extensive simulation tests with PowerSim. When he discovered that PowerSim was producing results comparable to his own software program, he submitted an article to Blackjack Forum describing his system and the simulation results.

The OPP Count is such a radical simplification of traditional card counting methods that I wanted to see more data before publishing the system. I asked Carlos to run more extensive tests not only of his system as described in the initial article that he submitted, but of numerous variations of the system, along with simulations of the Hi-Lo Count in the same games for comparison with his method. I also requested that he set counters on his simulations so that I could see that cards were being dealt accurately and randomly.

Carlos spent weeks running the simulation tests I requested and sending me spreadsheets with printouts of his results. He then compiled some of the most important data from these tests into a new Blackjack Forum article describing his system, its development, and the results of his simulations. I am proud to publish Carlos Zilzer’s OPP Count as one of the first truly new methods of counting cards to be developed in many years—a method that is surprisingly strong for its incredible simplicity of use, and the ease with which most players could learn it. — A. S.]

The Easy OPP Count System

Part 1: The Basic Idea Behind Card Counting

The whole idea behind card counting is to determine how rich or poor the shoe is at any time with cards favorable to the player or the dealer. A shoe rich in high cards (tens and aces) is favorable to the player, while a shoe rich in low cards (2s through 6s) is favorable to the house.

All modern card counting systems that I am aware of assign a negative value to the high cards and a positive value to low cards. The main difference between the systems is the value assigned to the different cards.

As an example, the popular Hi-Lo system assigns the value -1 to the tens and aces and the value +1 to the low cards, 2s through 6s. As the cards come out of the shoe, players add the values of those cards to obtain what is called a running count, which can be a positive or negative number depending on the proportion of low to high cards dealt from the shoe. With balanced counts, before making a betting or playing decision, the player divides the running count by the number of decks that have not yet been dealt to obtain what is called the true count.

Counting this way is not easy, as the player needs to pay attention to the cards while they are being dealt and then must make mental calculations. There are easier counting systems that eliminate the calculation of the true count by assigning positive values to more cards than those with negative values. For example, the Red Seven Count developed by Arnold Snyder has the same card values as the balanced Hi-Lo Count, but assigns a value of +1 to the red sevens, creating an imbalance of +2 per deck.

The goal of a good card-counting system is to be as simple to use as possible without losing the power to accurately determine when the player has the advantage. The OPP count does exactly that: It is a very simple method to learn and use that will give you results comparable to the Hi Lo Count. It is a running count system requiring no math at the tables, and it’s even easier to use than the Red Seven.

The main difference between this new counting system and all of the traditional ones is that it uses new factors to determine the composition of the remaining decks. It is the easiest way to count cards with a high degree of betting accuracy.

Part 2: The Mathematical Coincidence

In a deck of 52 cards there are 20 high cards (tens, faces and aces). There are also 20 low cards (2s through 6s). Since 52 / 20 = 2.6, if you shuffle a deck and start dealing cards you will see one high card every two or three cards–or every 2.6 cards on average–and you will also see one low card with the same frequency.

It has also been determined with computer simulations that each player or dealer hand will receive an average number of cards very close to 2.6 cards in blackjack games with the European no-hole-card rule. (The actual number determined after the simulation of billions of hands using different simulators is 2.63.) With standard US rules, in which the dealer hand receives a hole card, the average number of cards per hand, including the dealer’s hand, is closer to 2.7.

Now we have something very interesting: Since the average blackjack hand contains 2.6 to 2.7 cards, and the proportion of low cards in a deck (and high cards in a deck) is exactly 1 out of every 2.6 cards, then it follows that the average player or dealer hand can be expected to receive one high card and one low card.

With the help of computer simulation (see my results below), it has been shown that any player or dealer hand will actually receive an average of about 1.02 low cards and 1.02 high cards per hand (with US rules where the dealer takes a hole card). In the table below, you will find the results of a 100 million hand simulation using PowerSim for a 6-deck game.

Results of the simulations counting high cards per hand
Total dealt cards:	548,128,487
Total dealt hands:	202,156,193
Total high cards:	205,973,883
Average total cards/hand:	2.7114
Average high cards/hand:	1.0189

Results of the simulations counting low cards per hand
Total dealt cards:	548,141,978
Total dealt hands:	202,156,772
Total low cards:	205,384,220
Average total: cards/hand:	2.7115
Average low cards/hand:	1.0160

This data is the basis of the OPP counting system. OPP stands for One Per Person. As I will explain, the number of low or high cards per hand will give a very good indication of the composition of the remaining decks.

Part 3: Developing and Testing the Easy OPP Card Counting System

I first tested counting the number of high cards that were dealt per round and comparing this number with the total number of hands dealt per round.

Imagine a blackjack table with three players and the dealer dealing the first round of the shoe. These are the first round hands:

Player 1	Ace, 10 — Blackjack
Player 2	10, 2, 7 — 19
Player 3	8, 6, 10 — (bust)
Dealer	10, 3, 6 — 19

As you can see there were 4 hands played and 5 high cards were dealt (an ace and 10 to Player 1, a ten to Player 2, a ten to Player 3, and a ten to the Dealer).

The OPP count was obtained by subtracting the total number of high cards dealt from number of hands played, in this case, 4 – 5 = -1, for an OPP running count of -1.

The next round the hands dealt were:

Player 1	10, 5, 5 — 20
Player 2	Ace, 2, 5 — 18
Player 3	8, 6, 7— 21
Dealer	9, 5, 3 — 17

At this time there were again 4 hands played, but only 2 high cards were dealt. Thus, the count for this round will be 4 – 2 = 2. The new running count is now -1 + 2 = +1 (The -1 is the running count from the previous round.)

This process continues until the end of the shoe, adding the running count of each round to the cumulative running count of the prior rounds.

None of the commercially available blackjack simulation programs could be adapted to test this new counting system, so it was necessary to develop a simulator for this job specifically. The first simulations I ran, using my own simulation program, were performed in the beginning of 2005 using a program specially written for the tests. It was slow but returned the data I was looking for. With the new powerful and fast open source PowerSim simulator, available on this Web site, simulations that used to take me about two hours are now performed in eight minutes.

The first step was to verify that the count was producing logical results. We would expect the count to present a normal distribution of positive and negative counts with the majority of counts around zero, and a reasonable spread of counts on both the positive and negative side.

The chart showed the PowerSim results for 100 million rounds of a player using basic strategy in a six-deck game. About 80% of the hands were played with running counts between -6 and +6 and approximately 14% of the hands were played at a zero running count. But the graph showed that the count was negative more times than positive. This is not what we would expect for a traditional balanced counting system, which would tend to produce a more symmetrical pattern of counts on either side of zero.

The next step, however, was to see how well the player’s advantage correlated to the count. In order to have an accurate counting system there must be a strong relation between the count and the player’s advantage. This was the case in all negative counts, but the results at high positive counts were not well-correlated to the player’s expectation.

The chart showed that the relation between the running count and the advantage was almost linear up to counts of +12, but that the relation became erratic at counts higher than +12. I tried several simulations with PowerSim and always got inconsistent results starting at counts of +12 and above.

I next wanted to see what the overall player advantage would be using this counting system with a bet spread. I ran a 100 million-hand simulation using the following bet ramp: 1 unit up to count +1 and ramping up 2, 4, and 16 units respectively at counts +2, +3 and +4. The following results were obtained using the PowerSim simulator:

Advantage with 1-16 spread	0.3271%
Average bet (units)	2.683

Although it was possible to get an edge using this method, these results were very disappointing and were not comparable to the results obtainable with a traditional card counting system. The system’s power represented less than 40% of the betting gain that could be obtained with the standard Hi Lo Count in the same game.

Before dropping the idea I decided to reverse the OPP counting method. Instead of subtracting the number of high cards dealt per round from the number of hands in play, I decided to test the system subtracting the number of hands in play from the number of low cards dealt per round.

Here’s how it modified the counting method:

Imagine a blackjack table with three players and the dealer dealing the first round of the shoe. These are the first round hands:

Player 1	Ace, 10 — Blackjack
Player 2	10, 2, 7 — 19
Player 3	8, 6, 10 — (bust)
Dealer	7, 7, 5 — 19

As you can see there were 4 hands played and only 3 low cards were dealt (a 2 to Player 2, a 6 to Player 3, and a 5 to the Dealer).

The OPP count is obtained by subtracting the total number of hands played from the number of low cards dealt, in this case, 3 – 4 = -1, and that is the OPP running count.

The next round the hands dealt were:

Player 1	10, 5, 5 — 20
Player 2	Ace, 2, 4, 10 — 17
Player 3	8, 6, 7 — 21
Dealer	10, 4, 3 — 17

At this time there were also 4 hands played but 7 low cards were dealt. The count for this round will be 7 – 4 = +3. The new running count is now -1 + 3 = +2 (The -1 is the running count from the previous round.)

This process continues until the end of the shoe, adding the running count of each round to the cumulative running count of the prior rounds.

So I reversed the OPP counting method in this way, comparing the number of low cards dealt to the number of hands in play. We again have a normal distribution of the counts with more than 80% of the counts occurring in the range from -6 to +6. But this time the count distribution is more positive than negative.

This graph showed that there was a much more linear relation between the running count and the player expectation. The unpredictable results obtained at high counts when using the OPP method to compare high cards to hands dealt do not occur when we compare low cards to hands dealt. With this OPP counting method, the player starts to have an edge over the house when the running count reaches +5.

I then ran a simulation applying a bet ramp of 1 unit up to a count of +5, and ramping up to 2, 4, and 16 units at counts of +6, +7, and +8 respectively. The following results compare the OPP with the Hi Lo Count with a similar 1-to-16 bet ramp. (Both systems were tested using basic strategy only.)

	OPP	Hi Lo
Player edge with bet ramp	0.6359%	0.7723%
Average bet (in units)	2.907	2.547

As shown in the table above, the new system has a betting power equivalent to 82+% of the standard Hi Lo system, but with far greater ease of use.

A new and very simple count system has been born: the OPP Count system. The player counts only the number of low cards dealt per round and compares this with the number of hands played in the round.

Mathematical Considerations

Many players may wonder why counting the high cards versus hands dealt was less efficient than counting the low cards versus hands dealt. That is because the OPP method does not really have us comparing low cards with high cards as with a traditional card counting system.

Instead, we are comparing low cards or high cards with something that has a frequency distribution of its own—that is, the number of cards per hand. We know that the overall average number of cards per hand is 2.7, but that the number of cards per hand will also vary with the count. It is logical that the number of cards per hand will be higher when the count is rising because more cards will be required to complete the hands when extra low cards are being dealt.

Consider what happens when counting high cards per hand if we have a hand with 3 or 4 low cards, then hit with a high card and bust. Because this hand contains only one high card, it is counted as “zero,” since there is no discrepancy from the normal number of high cards (one) that should occur per hand.

If we were counting the low cards, this hand will accurately reflect the fact that the count has gone up in the player’s favor, since more than one low card has been dealt to it. It is less probable to have a hand with 3 or 4 high cards in it, because such a hand would require either a soft or stiff hand to start with that we hit with multiple aces. Multiple-high-card hands would be rare, while multiple-low-card hands would occur more frequently, and produce a more accurate measure of advantage.

Analyses of Two-deck and Eight-deck Blackjack Games

In order to configure a system usable in most situations, I performed simulations with PowerSim to determine the effectiveness of the OPP count in games with two decks and eight decks, again comparing the betting gain with the gain from Hi-Lo.

Two Decks

Applying a bet ramp with a spread of 1 unit up to a count of +1 and ramping up 2, 4, 6, 8 and 10 units respectively at counts of +2, +3, +4, +5 and +6, the following results were obtained using the PowerSim simulator (again, with both systems playing basic strategy only):

Two deck performance	OPP	Hi Lo
Player edge with bet ramp	0.8218%	1.0073%
Average bet (in units)	2.097	1.875

The power of the OPP count is again about 82% of the standard Hi Lo with no index play.

Eight Decks

Applying a bet ramp with a spread of 1 unit up to count +7 and ramping up 2, 4 and 16 units respectively at counts +8, +9 and +10, the following results were obtained using the PowerSim simulator (again with both the OPP and Hi-Lo playing basic strategy only):

Eight deck performance	OPP	Hi Lo
Player edge with bet ramp	0.4416%	0.5772%
Average bet	2.691	2.249

The betting power of the OPP count is shown to be about 77% of the standard Hi Lo in the eight-deck game.

Testing Alternative Counting Strategies

I also tested alternatives for the OPP system, including:

• Count only tens
• Count nine and tens
• Consider 3 to 7 as low cards
• Consider 3, 4, 5 and 6 as low cards
• Consider 2, 3, 4, 5, 6 and 7 as low cards

Playing Indexes for the Easy OPP Card Counting System

The only playing index tested so far for the OPP count is the insurance index, which adds an additional 0.009% edge to all games tested (2, 6 and 8 decks)

The running count insurance index for a 6-deck game is +12. For a 2-deck game the index is +5, and for an 8-deck game the index is +14.

[Snyder comments: It is not intuitive that the insurance gain from a 2-deck game with 75% penetration would be the same as the insurance gain from an 8-deck game with 75% penetration, as this would not be the case with a traditional card counting system.

The OPP Count, however, does not correlate well with the insurance effects of removal since ten-valued cards are not actually counted. I suspect that OPP running count indices for some of the more important strategy decisions, however, especially standing on 15 or 16 v. Ten, will be found by Carlos and others through continued simulation tests, as the counting method will probably correlate very well with these decisions.]

THE OPP COUNT

Basic Strategy

Before learning any count system it is a must to master basic strategy . The use of basic strategy will provide the player with the minimum advantage for the house. The counting system will turn that small house advantage into a small advantage for the player. Not using basic strategy may give the house such a large edge that no counting system will be able to turn the advantage to the player’s side.

OPP Rules

Value of the Hands:

• A hand with no low cards (2 – 6) has a value of -1
• A hand with one low card has a value of 0
• A hand with 2 low cards has a value of +1
• A hand with 3 low cards has a value of +2 and so on
• Split hands are considered new hands

This value is independent of the total number of cards in the hand. For example:

Ace, Ace, 2, 2, 10, 3 has an OPP value of +2

2, 3, 5, 3, 6 has an OPP value of + 4

7, 7, 7 has an OPP value of -1

Ace, 10 has an OPP value of -1

6, 7, 8 has an OPP value of 0

9, 8 has an OPP value of -1

10, 6, Ace has an OPP value of 0

Counting the Cards

It is very easy to mentally count up or down in positive numbers. Considering that 80% of the hands will be in the range of -6 to +6, I recommend starting the running count for the OPP system at +6 at the beginning of a shoe. In this way, the count will very seldom reach negative numbers.

Wait for the dealer to deal the first two cards to each player, as well as the two dealer cards (upcard and hole card, if dealt). You can speak, make jokes, and drink your soft drink. You don’t have to count anything while the cards are being dealt. If someone gets a blackjack and the dealer pays him and places his cards in the discard try, subtract one from your running count. Then check the hand of each player as they are making their playing decisions.

	Beginning of the shoe:	Running count +6
Player 4	Ace, 10 (blackjack)	Running count +5 dealer took the cards to the discard try
Player 1	Ace, Ace, 2, 2, 10, 3	Running count +7
Player 2	2, 3, 5, 3, 6	Running count +11
Player 3	7, 7, 7	Running count +10
Player 5	6, 7, 8	Running count +10
Player 6	9, 8	Running count +9
Dealer	10, 6, Ace	Running count +9

If any player happens to split hands, they should be considered new hands and the value of the original hand should be ignored.

The best way to practice the system is to play blackjack at home. Use a deck and start dealing cards to three imaginary players and to yourself as the dealer. Practice the count as indicated above while you play each hand. In this way you will see and practice counting all possible hands that can appear in real life games. You will learn to count spilt hands and to count very fast while the dealer is collecting the cards after a dealer blackjack.

[Snyder comments: My method of counting with the OPP system is a bit different. Before the hands are dealt, I immediately subtract the number of hands in play. So, if I use Carlos’s suggestion to start counting at +6, with three players at the table each playing one hand, I make my count +2 before any cards are dealt (subtracting one for each player hand and the dealer’s hand).

Then, as each player plays his hand, I simply add the total number of low cards in it. When the next round starts, again assuming three player hands are being dealt, I’ll immediately deduct 4 from the current running count, and repeat as above. One of the best features of the OPP Count is that it makes back-counting and table-hopping a breeze. You can approach a table and quickly scan for the total number of low cards on the layout, then subtract the number of hands that were in play after you get your low-card count.]

You will see that the OPP is very easy to learn and to master. When you feel proficient counting the cards at home, you can practice in an Internet casino playing for fun money at a table with multiple players. Be aware that the count won’t mean anything in an Internet casino where the online software shuffles after every round. You will only use these games as a practice method.

After you have practiced enough at home, go to a casino and start counting while other people play. When you feel comfortable sit at a table and start playing using the OPP count.

You will see that you will get used to the system very fast, and that it really does not require 100% concentration except when each player is making his playing decisions. The only moment you will need to count quickly is when the dealer has a blackjack, as he will collect the cards of all the players’ hands very quickly.

Betting to Win

Once you have learned this very simple counting method the only thing you need to do is wait for the right moment to increase your bet.

If you have started at a running count of +6, wait until the count reaches +12 for a six-deck game, +14 for an eight-deck game, or +8 for a two-deck game. At these counts the advantage has shifted to the players. The higher the count value, the higher the player edge, and the bigger your bet should be. We call this point where the advantage turns to the player’s side the “pivot.”

Be very patient. Only about 18% of your hands will be played at counts over the pivot in shoe games, and the player advantage will not occur in all shoes. It is possible that you won’t have an advantage for several shoes, or even all night. Also remember that there is no warranty that you will win most of your hands when the count reaches the pivot. There is always “variance” in gambling results (you may call it luck). Remember that there will be many times when you will lose high bet hands.

The following table shows the recommended bet ramps for games with two, six, and eight decks, to get the best advantage for the player at the minimum risk. The first three columns are the running counts where you raise your bet, according to the number of decks in the game. The fourth column is the recommended bet in “betting units.” And the fifth and sixth columns show the bet in dollars for $5 or $10 units.

For example, if you are sitting at a $5 minimum six-deck table, you may set your bet unit to $5, and if the count reaches +12 you bet 2 times your betting unit, or $10. If the count reaches +14, you will bet 16 times your betting unit, or $80.

Running count8 deck	Running count 6 deck	Running count 2 deck	Units to bet	$5 table	$10 table
Less than +14	Less than +12	Less than +8	1	$5	$10
+14	+12	+8	2	$10	$20
+15	+13	+9	4	$20	$40
+16	+14	+10	16	$80	$160
+17 or more	+15 or more	+11 or more	16	$80	$160

If the count reaches values of zero or less during two consecutive rounds in the shoe it is recommended to stop playing until the beginning of the next shoe. This will save you a lot of money.

[Snyder comments: I will point out here that Carlos is playing primarily in South American casinos where card counters are uncommon and camouflage is not a big necessity. Players in the US or elsewhere must always bear in mind the heat factor. His two-deck betting strategy would likely get you booted out fairly quickly in many US casinos.

For playing shoe games in the US, a back-counting strategy would likely be more advisable (and more profitable) than a 1-16 spread. To back-count is to watch the cards being dealt on a table without playing, ready to jump in and play when the count goes up and you will be starting play with an advantage.

Table-hopping is an extension of back-counting that pros use in big casinos, which tend to have multiple pits with many blackjack tables. Table-hoppers leave the game when the count goes negative and start play at another table. They refuse to play when the casino has a big edge.]

The Insurance Bet

Basic strategy recommends not taking insurance, but card counting sometimes indicates an insurance advantage for the card counter.

With the OPP Count, if we start our count at +6 at the beginning of the shoe, the insurance index is +18 for the six-deck game, +20 for the eight-deck game, and +11 for the two-deck game. This means that if you are playing in a six-deck game and the running count reaches +18 or more, and the dealer has an Ace up, you should place the insurance bet, because you will win the bet more than 35 % of the time. Simulations with PowerSim have confirmed these index numbers.

Final Thoughts on the Easy OPP Card Counting System

OPP is a very simple system to learn and to use. Practice at home, go to a casino and stand near a table and practice the count while others play. I know people who have mastered the system in less than a weekend and now they are part of the select group of players who are making money playing blackjack. ♠

[Note: Although hundreds of independent researchers and mathematicians have proven that card counting is a winning strategy over the long run, gambling is always risky. Anything can happen in the short run. No one can ever guarantee that you will win even when you are playing with an advantage. Never gamble with money you can’t afford to lose.]