Blackjack Card Counting Efficiency 

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Comparison of Card Counting Systems

Betting Efficiency and Correlation Coefficient

Insurance Efficiency and Correlation Coefficient

Playing Efficiency

Relative importance of betting, playing and insurance efficiencies

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Comparison of  the Card Counting Systems 

Peter Griffin (“The Theory of Blackjack”) developed analytical methods for comparing different counting systems. The comparison was done on the basis of efficiency of a system in performing its important functions. These functions include spotting favorable conditions of the deck or shoe, helping to make profitable insurance bets and making correct deviations from the Basic Strategy at the proper times. Efficiency shows how an actual result compares to the maximum possible under ideal circumstances. Efficiency is measured by the coefficient of efficiency equal the actual result divided by the theoretical maximum.

Betting Efficiency and Correlation Coefficient 

Betting efficiency of a counting system shows how good the system is in identifying favorable conditions of a deck or shoe. As soon as the favorability has been established, a player increases his bet to maximize the profit. Thus, betting efficiency is an important characteristic of the system. A card counting system assigns point values to different cards. The removal of every card has a positive or negative effect on the deck favorability. It makes sense to evaluate a betting efficiency through measuring the correlation between the point values of cards and the effects of their removal on the favorability of the deck. The correlation means the relationship that is not based on chance alone. Correlation is measured by the correlation coefficient, which measures the degree of correlation. Theory of Probability gives a basic formula for correlation coefficient, which is equal the covariance of two random variables divided by the product of their standard deviations. In our case two variables are a point value of the card (P) and the effect of the removal of this card from the deck (E).

QEP = Cov(E,P) ∕σ(E)σ(P)  (*)

Cov(E,P) is covariance of E and P; σ(E) and σ(P) are standard deviations of E and P;

Cov(E,P) = M(EP) – M(E)M(P); σ(E) =√D(E)  and σ(P) =√D(P)    ;

M is mathematical expectation and D is variance;

D(E) =  ∑(E-M(E))²∕ 13  and  D(P) =  ∑(P-M(P))²∕ 13 ;

M(E) = 0 because the sum of all removal effects equals 0 and M(P) = 0 for obvious reason for balanced systems. In result (*) simplifies into:

 QEP = ∑EP  ∕ (√∑ E²   ) (√∑ P²   )  (**)

Griffin used (**) to determine betting efficiency of different counting systems. The table below illustrates the calculation of betting correlation coefficient for High – Low (Hi – LO) counting system. 

Different counting systems have different betting correlations. Those systems that count an Ace as – 1 (like for ex. Hi-Lo system) have better betting efficiencies than the systems that count Ace as 0. The reason is that the removal of the Ace has a negative effect on deck favorability and counting Ace as – 1 makes a counting system more sensitive to the changes in the condition of the deck.

Insurance Efficiency and Correlation Coefficient

Insurance efficiency reflects the ability of the card counting system to help a blackjack player to make insurance bets, which have positive expectations. The same Griffin’s methodology that is used to evaluate betting efficiency can be used for insurance efficiency. Expectation of insurance bet is determined by the ratio of the ten to non-ten cards in the deck or shoe. The high ratio can lead to a profitable insurance bet.  Thus, the removal of a ten-valued card has a negative effect on expectation, while the removal of a non-ten card has an opposite positive effect.

If we’ll take into consideration a sequence of the cards and how many ten and non-ten cards a player might have, we’ll find that an insurance bet for a single deck has approximate  -5.845% expectation. If we’ll remove one non-ten card from the deck, an expectation will be approximately -4%. That means the effect of the removal of any non-ten card is -4 – (-5.845) = +1.845%. If we’ll remove one ten-valued card, an expectation on insurance bet will deteriorate to approximately -10%. The effect of the removal of a ten-valued card is -10 – (-5.845) = -4.15%. Griffin’s numbers (“The Theory of Blackjack”) are 1.81% for effect of removal of a non-ten card and -4.07 for a ten-valued card. The table below shows the calculation of insurance correlation coefficient for High – Low (Hi – Lo) system.  

Playing Efficiency

Playing efficiency describes how good the card counting system is in recommending deviations from the Basic Strategy. Playing efficiency can’t be very high for any card counting system for an obvious reason. Counting system assigns fixed point values to the cards, while some cards may behave as small or big cards in different situations. A fixed and never changing point value can’t represent precisely something that has a dynamic nature. Ace is an obvious example because of its dual value of 1 or 11. In result it acts as a big card when a player draws it to his starting hand of 10 or less. It turns into a small card when a player gets it trying to improve his stiff hand of 12 – 16. Even 7, 8 and 9 play to some extent a dual role of big cards when they bust high stiffs and of small cards when they help to improve low stiffs.

Griffin’s findings (“The Theory of Blackjack”) showed that max possible for playing efficiency is about 70.3% for a level-ten count, which is impossible to use in a real game due to ten different and very high numbers used as point values. The best playing efficiency of around 69% can be achieved with level-three and four systems (which are also not easy to use without mistakes). The systems that count an Ace as 0 have better playing efficiency than the systems assigning -1 point value to an Ace. For ex, High – Low system has a playing efficiency of 51% when Hi-Opt I that counts Ace as 0 enjoys higher 57% efficiency.

Multi-parameter systems, which also use side counts of specific cards in addition to the primary running count, can increase in theory playing efficiency beyond 70.3. However, these systems tremendously overload the memory and increase the mental work during the game to the point when it is impossible for a blackjack player to use them in casino.

Relative importance of betting, playing and insurance efficiencies of card counting systems

Relative importance of different efficiencies depends on the relative contributions to the overall card counting gain from bet increase, deviations from Basic Strategy and taking profitable insurance bets. The contribution from insurance efficiency is probably less than 10%. According to Griffin (“The Theory of Blackjack”), playing efficiency and betting efficiency are equally important if the bet spread is 1 – 4. With bigger betting spreads betting efficiency delivers bigger input than playing efficiency, but according to Thorp (“The Mathematics of Gambling”) the ratio will not exceed 60 to 40 in favor of betting efficiency.

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