
Progress Publishing Co. 

Blackjack Card Counting 
Blackjack Card Counting
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The purpose of Card Counting
Necessary condition for Card Counting
How Card Counting works
Running and “True” Counts
Increase in Expectation
A History of Card Counting
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The Purpose of Card Counting
Blackjack Basic Strategy is the best way to play if the only information a player has is his two initial cards and a dealer’s upcard. The Strategy does not require any information about the cards in a deck or a shoe. A player follows the decisions of the Basic Strategy without making any deviations from them under any circumstances. Basic Strategy was a giant step forward in improving player’s expectations and reducing the house advantage against a player. The best the previous strategies could do was to guarantee a loss to a player at the rate from 3.5% to 17% on the total amount wagered. Basic Strategy significantly cut down a house percentage and improved player’s expectations to almost a 0% for a singledeck game and about 0.55% for the multiple decks. A player’s disadvantage became small but a player was still a guaranteed longterm loser. In result, the obvious question about Basic Strategy was how to improve the Strategy in order to give a player a positive average return and make him a longrun winner. That became the purpose of the Blackjack method known as “Card Counting”.
Necessary Condition for Card Counting
There is one condition of the game, which makes Card Counting possible. Under that condition all played out cards do not return back to the deck or shoe, but, instead, go to the discard tray where they stay until the shuffling point is reached. If that condition is not present, the cards go back into play after every round. In result, a player always faces a freshly shuffled deck or shoe. In that case Card Counting is completely useless as a practical method of Blackjack play. The existence of that condition depends entirely on the casino. In many casinos, which use continuous shuffling machines, that condition is not present. In many other casinos there are tables where the dealers shuffle or shuffling machines are not used for continuous shuffles – in these casinos the players can try Card Counting.
If the cards are removed from the play after every round, the composition of the deck or shoe changes in terms of the relative quantities of the cards with small numerical values (2, 3, 4 etc) and the highvalue cards (nines, tens and aces). That change can affect player’s odds positively or negatively for the next few rounds. If the previous rounds of play removed a lot of small cards, the deck or shoe become favorable for a player. A player has better odds of winning due to the relative excess of the high cards still in play. High cards help a player by giving him more blackjacks, which are paid at 3 to 2 when the dealer’s are paid at even money only. They give a player better double downs. They also bust a dealer more often – a dealer must draw to his stiffs of 12 to 16 while a player has an option of standing with those hands. On the other hand, if the big cards are out of play, a player looks at a disadvantageous shoe or deck full of small cards. Small cards hurt a player by helping a dealer to bust less and make a pat hand more often when he has stiffs. A player gets good hands less often including blackjacks. His double downs don’t turn into strong pat hands because of the drawn small cards. The table below shows the changes in Basic Strategy player’s expectations caused by the removal of individual cards. The numbers were worked out by Julian H. Brown (Thorp “The Mathematics of Gambling”). They are for a singledeck game under the assumption of 0.02% Basic Strategy’s expectation. The table shows that the removal of high cards (9, 10 and Aces) decreases the player’s expectations. The removal of small cards has an opposite effect.
Cards  2  3  4  5  6  7  8  9  10  A 
Effect in %  +0.38  +0.44  +0.58  +0.74  +0.44  +0.29  0  0.18  0.51  0.6 
Thus, if the necessary condition is in place, the previous rounds of play in Blackjack affect the probable outcomes of the future rounds. The mathematicians call that situation the case of “Dependent Trials”. The outcomes of the current trial and future trials depend on the results of the previous ones. The case when the previous results do not affect in any way the current and future results is the case of “Independent Trials”. For ex, Roulette is a game of independent trials. If a player bets on the Red the odds are 1.11 to 1 against him guaranteeing him a disadvantage of 5.26%. If he manages to hit Red 1000 times in a row, he’ll be facing exactly the same odds and disadvantage when he’ll bet on the Red for the 1001^{st} time. The other example of “Independent Trials” is the game of Keno. If a player buys a straight 6spot ticket and wants to match 4 numbers, the odds are 34 to 1 against him. If he hits 4 numbers 5555 times in a row, he’ll be facing the same odds against him hitting 4 numbers on the 5556^{th} time.
The mathematicians who wrote card counting books were stressing “uniqueness” of Blackjack as a “Dependent Trials” game. That uniqueness is not inherent to the game. It exists as long as casinos choose to let it to exist. If they will choose to use continuous shuffling machines, or tell the dealers to shuffle after the first hand against the players who win consistently, then Blackjack will turn into an “Independent Trials” game just like Roulette, Keno and many other casino games. Card Counting will become impossible to use in principle and the only best choice left for the players will be using Basic Strategy. However, since the millions of card counters in the last 30 years were the reliable source of casino Blackjack revenues, it is easy to speculate that in foreseeable future casinos will allow Blackjack to stay, at least on some tables, a “Dependent Trials” game.
Thus, as long as the casinos keep the played out cards in the discards tray instead of immediately putting them back into a shoe or a deck, then, at least in theory, the card counting method is possible to use in Blackjack play.
How Card Counting works
Card counting aims to improve Basic Strategy expectation through gathering information about the cards left in the deck or shoe. That information helps a card counter to determine when he faces a favorable deck or shoe and when he does not. The excess of tenvalued cards and aces still in play creates favorable conditions and give an edge to a player over the house. A player takes advantage of these conditions in two ways. Since he has an edge over the house he increases his bet size – usually in proportion to his advantage. Also, he makes correct deviations, on the basis of the acquired knowledge, from the Basic Strategy playing decisions to make them more precise. When the composition of the deck in terms of the remaining cards is hostile to a player – there is an excess of small cards still in play – he always bets minimum amount. Thus, when a card counter has an edge he wins more often and with big bets. When the house has an edge, a player loses more frequently but the bets are minimal.
Card Counting gathers information about the cards to be played by using a count. Card Counting creates a count through assigning positive and negative point values to the cards of the different ranks. Positive values are given to the small cards, because the removal of these cards from play is beneficial for a player. Negative point values are attached to high cards for the opposite reasons. After that card counting comes down to a simple arithmetic of adding and subtracting point values. It has nothing to do with memorizing specific cards.
Let’s use for illustration purposes a very popular and simple “High – Low” card counting strategy to see how card counting works in principle. It was devised by Harvey Dubner in 1963. That particular strategy assigns point values to the specific cards according to the table below:
Cards  2  3  4  5  6  7  8  9  10  A 
Point Values  +1  +1  +1  +1  +1  0  0  0  1  1 
As soon as the dealer begins to deal a new deck or shoe, a player starts his count from zero and then adds 1 every time when he sees a small (2, 3, 4, 5 and 6) card and deducts 1 when any 10valued card (10, J, Q and K) or an Ace shows up. He ignores 7, 8, and 9 because their assigned point values are equal to 0. If, for ex, the first 7 cards are 2, 10, K, 4, 8, A, J, then the count is +1 1 1 +1 +0 1 1 = 2. The value of the count at any moment is the sum of the assigned positive and negative point values of all the cards that have been played out until that moment.
The value of the count gives player information about the composition of the deck or shoe in terms of the proportion of high and small cards. If the count is highly positive, that obviously means that a lot of small cards have left the deck and the rest of the deck is very rich in high cards. High cards make a deck favorable for a player. A player bets big to take a full advantage of his existing at the moment edge over the house. The abundance of high cards also warrants deviations from Basic Strategy decisions. For ex, Basic Strategy recommends hitting hard 15 when a dealer’s upcard is a 10valued card. The excess of high cards in the deck makes the probability of busting hard 15 very high. If the count is + 4 or more the correct decision for a player in this case is to stand with his 15 instead of hitting. A card counter can make that deviation thanks to the information about the rest of the deck acquired with the help of his count. Basic Strategy, for ex, also always recommends against taking an insurance since the payoff on the insurance bet is 2 to 1 but the odds are at best 2.06 to 1 against it (in a single deck when a player does not have tens). Card counter can take insurance if the count is +2 or more (single deck assumption) because of a higher probability of a hole card being a tenvalued card. If, on the other hand, a count is negative, a player is at disadvantage since many 10s and Aces have already left the deck. A player bets small and also makes adjustments to Basic Strategy decisions on the basis of his current negative count. If, for ex, a player has a hard 12 and the dealer’s open card is 4, Basic Strategy recommends standing. However, if the count is 1 and less, the correct decision will be hitting instead of standing, because an excess of small cards still in play will help a player to improve his hand without busting. There are 100 or so deviations from Basic Strategy that can be made with the help of the “High – Low” count.
The count value also helps a player to determine a correct bet size. If a count is negative the bet size is one unit. If the count is between +1 and +4 the recommended bet size to maximize player’s profit during the time when he has an edge is 2 units. If the count is +5 or more the bet size goes to 3 units and higher.
Running and "True" Counts
The count illustrated above is called a “running count”. It is a cumulative count based on addition or subtraction of assigned point values. The running count does not give precise information about the composition of the rest of the deck or shoe. The reason is that the same running count can have drastically different implications about the proportion of small and big cards still in play. If a running count is +8 in the beginning of an 8deck shoe that means an excess of roughly 1 card per every deck. The same +8 running count will indicate an excess of 4 cards per deck with only 2 decks left in the shoe. To avoid the distorting effect of the number of decks a running count is made more precise and meaningful by adjusting it to the number of decks still left in play. Adjusted running count is called a “True Count”. For ex, if a running count is 6 and three decks are still remaining, the True Count will be 6 divided by 3 or 2. If a running count is +6 and there are 4 decks in play, the True count is +6 divided by 4 or = +1.5. Most of the counting strategies use the True Count values to determine correct deviations from the Basic Strategy. There are also a few strategies that use a running count.
Increase in Expectation
Our example above of the “HighLow” card counting strategy illustrates how card counting method works in principle. It improves Basic Strategy performance by using information about the proportion of the big and small cards in the rest of the deck or shoe. On the basis of that information a card counter deviates at proper times from Basic Strategy recommendations to make better playing decisions. The same information also helps to vary the bet size and maximize the profit. In result, a card counter’s expectation becomes a positive number guaranteeing him a longterm win. A Basic Strategy player has a negative expectation that assures him of eventual loss over a long period of time. The “HighLow” strategy produces theoretical expectation of around +1% on a total money bet.
A History of Card Counting
The history of Card Counting officially
began in 1962 when Edward S. Thorp published his groundbreaking book “Beat the
Dealer”. The book included first winning strategies – 5count and tencount
strategies. Thorp, also, laid out the general procedures for developing and
evaluating the capability of all “oneparameter” counting systems. Thorp’s work
in the field of card counting became the foundation for the future theoretical
analysis of the game and helped to create numerous counting systems. The
contemporaries of Thorp who also worked on devising winning Blackjack methods
were Richard A. Epstein (The Theory of Gambling and stochastic Logic) and Alan
N. Wilson (The Casino Gambler’s Guide).
The next step forward in Blackjack research took place in 1963 at the Fall Joint Computer Conference in Las Vegas where Harvey Dubner made public his “HighLow” counting strategy. In the following years major theoretical discoveries and developments were made by Julian H. Braun and Peter Griffin. Braun used IBM computer to refine Thorp’s calculations and improve Dubner’s “HighLow” system. Peter Griffin created exact methodology for comparison of the effectiveness of various strategies.
Within the period of the few decades since 1962 dozens of counting systems were created. They were all the same in principle, and the difference between them did not go beyond technical details of the specific counts used and relative ease or difficulty in application during the game. In terms of their efficiencies, measured by playing and betting correlations, and their theoretical advantages promised to a player they were all comparable.
Many good books dedicated to a card counting approach to Blackjack containing the wealth of mathematical and practical information were written over the years by Braun, Griffin, Revere, Humble, Wong, Uston and Snyder.
Copyright Progress Publishing Co.
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