On the Math Behind OPP
by Kim Lee
[From Blackjack Forum Vol. XXV #1, Winter 2005/06]
© 2005 Blackjack Forum Online
[Kim Lee has been contributing to gambling publications, including Blackjack Forum, for many years. His review/explanation of the OPP appeared in the same 2005 online issue of BJFO as Zilzer’s article, explaining the math behind it and discussing some of his thoughts on making the OPP more powerful. – A.S.]
In the latest Blackjack Forum (XXV #1, Winter 2005/06), Carlos Zilzer introduces the “OPP” counting system (an acronym for one per person). This count seems almost too simple to work and too good to be true. In brief it counts low cards 2-6 as +1 and subtracts 1 for every hand, including the dealer’s. Then it bets large when this running count reaches a sufficient level.
Simulations agree the OPP count is profitable and earns a significant fraction of the returns associated with conventional counting systems. This article analyzes the math behind OPP to show why it works. It explains why some modifications don’t work, while others improve the profitability.
All counting systems (including shuffle tracking and sequencing) are based on predicting the cards to be dealt. Usually you count the cards seen to give information about remaining cards. Most systems count some combination of high cards minus low cards. But you can count anything correlated with the cards such as blackjacks or busted hands. One author even recommended counting ashtrays on the dubious theory this was correlated with cards!
The Theory Behind the OPP
The OPP system is based on the observation that there are on average 2.6 cards per hand, or equivalently .38 hands per card. This was the basis for Jake Smallwood’s KWIK count, the Speed Count, and my own Comp Count. OPP counts low cards as +1 and then subtracts 1 for every hand. This is similar to counting low cards as +.62 and counting all other cards as -.38. Indeed, this would be a marginally profitable system if you could actually add +.62 and subtract -.38 quickly in your head.
But counting hands instead of cards has an intriguing feature. High cards have a greater effect of completing hands than low cards. For example, you would only get two Tens per hand, but you might get three Sevens or more low cards. There are fewer high cards per hand than low cards per hand. Equivalently there are more hands per high card. So high cards have a bigger impact than low cards on the negative portion of the OPP count.
We can approximate the OPP count by a conventional card counting system. MathProf kindly ran some correlations of to measure the average effects of different cards on the OPP count. Here are the average effects of different cards on the OPP running count in a double deck game:
| A | -0.40 |
| 2 | 0.85 |
| 3 | 0.79 |
| 4 | 0.74 |
| 5 | 0.70 |
| 6 | 0.65 |
| 7 | -0.39 |
| 8 | -0.44 |
| 9 | -0.46 |
| Ten | -0.51 |
This makes a lot of sense. The low cards have a positive effect, but less than +1 because they also contribute to completing hands (which the OPP count subtracts). The effects of other cards are negative, and the larger cards have larger negative effects. It is easier to see these effects if I double them and round off.
| A | -0.8 |
| 2 | 1.7 |
| 3 | 1.6 |
| 4 | 1.5 |
| 5 | 1.4 |
| 6 | 1.3 |
| 7 | -0.8 |
| 8 | -0.9 |
| 9 | -0.9 |
| X | -1.0 |
This looks like a pretty reasonable counting system except for the negative effect of the Seven and Eight.
These effects of removal help explain why OPP works better than Jake Smallwood’s KWIK Count. The KWIK Count works opposite OPP; it counts Aces and Tens as -1 and adds +1 per hand. The KWIK Count would show largest positive effects for the Nines and Eights, and the smallest positive effects for low cards. This is not highly correlated with the players’ advantage because the small cards should count more Eights and Nines.
The OPP count is slightly unbalanced, it tends to rise at roughly +1 per deck. This makes it effective as a running count system. You can use any system in running count mode, including High-Low. While the running count is not perfect, it is highly correlated with the truecount. Unbalanced running count systems are better in this regard because they recommend betting big at high counts that typically occur late in the shoe. Since high running counts occur at deep penetration, unbalanced running counts are highly correlated with unbalanced truecounts.
Improving the OPP
So how can we improve the OPP Count? We could count Sevens as 1. This would fix the negative effect of the Seven on the OPP Count. But it would also further unbalance the system by +4 per deck.
Note that the OPP count is only about half as volatile as High-Low or KO. So adding an imbalance of +4 would make the OPP+7 count twice as unbalanced as KO.
This is not particularly a problem in handheld games. In fact MathProf’s recent simulations show the unbalanced OPP+7 Count outperforms the original OPP count in double deck games. However, the OPP+7 Count overlooks opportunities early in a shoe. Therefore MathProf’s simulations show that the OPP+7 Count is only superior to the original OPP Count in shoe games if one uses a very large spread.
So which is better, the original OPP Count or OPP+7? Simulations show the original count is better for low spreads in shoe games and OPP+7 is better for large spreads or handheld games.
But we need to consider the users of these counts. They are probably recreational players who want to have fun and earn comps. They probably don’t have the bankrolls for large spreads nor the discipline for backcounting. Ideally they would use the OPP+7 Count in good handheld games. But if they only have access to shoe games with a limited play-all spread then they may be better off with the original OPP Count. ♠