Visit almost any casino in the United States and youll notice that the most popular table game is blackjack. According to recent statistics, we [the average American adult] now wager some $500 yearly on the game. And how do we fare? Not so well, it turns out. An average casino blackjack table earns a tidy $200,000 per year for its operators.
Casinos beating gamblers? Nothing earthshaking about that. What is surprising, though, is that it doesnt have to be this waya skillful blackjack player can beat the casinos at their own game. In fact, with proper play, almost all of the nations ten thousand blackjack tables can, quite legally, be beaten. This is accomplished by a technique called card counting.
Whats the idea behind card counting? Simply stated, a player uses information about cards already played to determine the favorability of the remaining pack. Not too difficult, yet the vast majority of blackjack players do not count cards. Why not? Because this simple idea has been somewhat difficult to implement. Until now, the most popular counting systems have been variations of the original point-count systems developed 35 years ago. And, though effective, these systems fall a bit short in the user-friendly department. Not helping matters are the casinos themselves, who are all too happy to educate you about the difficulty of card counting. There is the mistaken idea, often put forth by the casinos, that card counting requires a mind like Star Treks Mr. Spock, with a memory tantamount to that of a super-computer.
While card counting has never required superhuman skills, it remains relatively inaccessible to most of us. Theres simply too much to think about, and under highly distracting circumstances. But a new, very simple card-counting technique, developed by Ken Fuchs and myself, dubbed the Knock-Out system, aims to change all that.
What makes blackjack special?
You may be wondering what makes blackjack different from other gambling games. Why is it that a skilled player can beat blackjack, but has no mathematical hope of ever beating a game like craps? It boils down to the mathematical concept of independent and dependent trials.
In craps, every roll of the dice is independent of past and future rolls. What was rolled just now, or during the last hour, is irrelevant to what will be rolled in the future.
In blackjack, hands subsequent to the initial deal are dependent events. This means that past events can, and do, influence what happens in the future. In other words, the cards already played affect the composition of the remaining deck, which, in turn, affects your future chances of winning. Card counting identifies those times when the deck composition favors the player, allowing him to bet more when he has the advantage (see Counting Gumballs, below).
A little background
Card counting has become synonymous with the name of Edward Thorp. In 1962, Dr. Thorp published the first card-counting treatise on blackjack, the New York Times best-selling Beat the Dealer. In it were the words that would send the casino industry reeling: In the modern casino game of blackjack, the player can gain a consistent advantage over the house.
For the better part of six decades prior to Thorps groundbreaking work, people played blackjack based on hunches and whims, which resulted in fat casino profits (though a few intuitive players employed card-counting concepts in a seat-of-the-pants approach). But now, a prominent mathematician had demonstrated that a skillful player could gain the upper hand. Indeed, in the original edition of Beat the Dealer, Thorp presented several methods for beating the game, including the first point-count system in which each card was assigned an integer tag (a point value reflecting the cards relative strength).
Unfortunately, some in the casino industry misrepresented these ideas. John Scarne wrote of card counting as chicanery. To some extent, this type of sentiment still exists in the industry, where, in extreme cases, card counters are looked upon more as freeloaders than as skilled artisans. And most casinos implemented rules changes in an attempt to counteract the impact of Thorps findings.
Everyone had a different take on what card counting represented. In his book Blackjack for Blood, Bryce Carlson writes: Their paranoia out of control, the Las Vegas casinos snapped! The effect on the average player was disastrous. A suave PR man, however, saw it differently in a Newsweek article dated April 13, 1964. In his view, casino operators had merely eliminated the fringe benefits of the gamenamely, the right to double most bets and to split hands of two aces. Edward Thorps nonchalant reply (in the same Newsweek article) to all the hoopla: Instead of five hours, now Ill have to play seven to make the same money.
In the past 35 years, many experts have refined and expanded on Thorps original work. The list reads like a Whos Who of blackjack, made up of luminaries from the academic, playing and casino-operating communities. I could not do all of these individuals justice in just a few words. Rather, I will focus on a few whose insights have particular relevance in the advancement of our knowledge of this game, with respect to the Knock-Out system.
Peter Griffins landmark book, The Theory of Blackjack, delves into the game in unprecedented detail. Many enterprising students have applied Griffins ideas to develop novel techniques to beat blackjack.
Arnold Snyder has long been a proponent of simplification, and his book, Blackbelt in Blackjack, was the first to espouse a so-called unbalanced system for blackjack (more on this later).
Don Schlesingers work in determining the Illustrious 18 reinforced the notion that card counting could also be used to adjust playing strategy, and showed that most of the possible gain due to playing-strategy variation accrued from only a few situations.
John Gwynn, Jr. and Jeffrey Tsais paper Multiparameter Systems for Blackjack Strategy Variation was the first to study the effects of slight errors in what is known as the true count, and their minimal impact on a count systems performance.
To understand how card counting works, consider a carnival gumball machine filled with 20 gumballs10 blue and 10 white.
You are allowed to bet (whenever you want to) that the next ball to come out is blue. If a blue ball comes out, you win an amount equal to your wager. If a white ball comes out, you lose your wager. At the beginning, your chances are even. If you bet $1, you have a one-half chance of winning $1 and a one-half chance of losing $1. Your advantage (or, as mathematicians would say, your expectation) is zero.
Lets say, however, that the first gumball to come out is white. Since there are now 10 blue and only 9 white balls left, you have the advantage! It should be clear that any time there are more blue balls than white, you have the advantage on your blue bet. Any time there are more white balls remaining, you are at a disadvantage.
What is the game plan, then? Simple! Bet more when you have the advantage, and bet less, or not at all, when the house has the advantage. This way, when all is said and done, you should come out on top.
So lets develop a gumball count. A simple method to keep track of whether or not we have the advantage is to realize that removing white gumballs from the machine helps us. That is, every time a white ball is taken out of play, our expectation (for betting on blue) goes up slightly. The opposite is true when a blue gumball is removed.
One simple way to count the gumballs is to assign a tag of +1 to each white gumball and -1 to each blue gumball. Starting at zero, we can keep a running count, or total, of all balls seen. To illustrate, if the first ball to exit was white, our count would be +1. If the next ball was blue, our count would drop back to zero, and so on.
What does this do for us? Well, the count is a reliable gauge of where the advantage lies. If the running count is positive, we have the advantage. If it is negative, the house has the advantage. If the count is neutral, the game is even. You can play this game yourself and log the resultsyoull be able to verify the recipe.
But first, basic strategy
While gumballs work well to demonstrate the underlying concept of card counting, the game of blackjack is more complicated. In particular, it requires a study of the proper play of the hands before tackling advanced techniques. Thats where basic strategy comes in. Thorp wrote: One of the characteristics of basic strategy is that, by using it, you will be considerably luckier than the average player. So what is this so-called basic strategy?
Basic strategy is a fixed set of rules by which you should play your blackjack hand. People who employ basic strategy are commonly said to be playing by the book, because in any given situation (with no knowledge of deck composition), this is the optimal way to proceed. Maxims such as Always split Aces and Eights and Always stand with 17 or more are examples of basic strategy.
Most blackjack enthusiasts already know at least an abbreviated form of the strategy. All thats needed for these bettors to become primed for card counting is to learn it perfectly, for even when counting, you will be playing the hands according to basic strategy most of the time. (An accurate basic strategy is not hard to find in the better blackjack books on the market. One is included in Knock-Out Blackjack.)
Counting cards: The old way
Once youve mastered basic strategy, youll be ready to progress to card counting. Ever since Thorp let the card-counting genie out of the bottle, a select group of players have mastered the technique and earned a consistent profit playing blackjack. And select group means just thatthe problem with traditional balanced systems is that they require some fairly potent mathematical brainpower to implement. Allow me to demonstrate.
Like all card-counting systems, balanced counts assign a tag to each card (+1, -1, etc.). And, as in our gumball example, a running count (this time of cards) is kept and continually updated. Now imagine the following two-deck scenario. You have a running count in your head, say +3. Its time to bet and the dealer is waiting on you. But before you can bet, the system requires that you convert the running count to a standardized measure, which is called a true count. Heres how it goes. While remembering the running count, you need to look over at the discard rack and estimate the number of decks already played. Lets say you estimate 1-1/4. Now you think to yourself, 2 (decks) less 1-1/4 leaves 3/4 unplayed. Okay, now divide the running count (+3, remember?) by the number of decks unplayed, and round down toward zero to get the true count. Quick, whats the answer? (Its 4.) Finally, you size your bet according to the true count of +4, and make the bet. Its necessary to repeat this process before you make every big wager.
Now that the wager is made, you have to go back to the running count (Still got it?) and count the cards as theyre dealt. When its your turn to play (dealer waiting on you again), it may be necessary to go through the process of converting to the true count again to decide how to play your cards. This constant conversion between running count and true count is mentally taxing, prone to error, and leads to quick fatigue. It certainly detracts from the enjoyment of the game.
The mental gymnastics necessary to effect the true-count conversion are only part of the problem. Some systems are far more difficult due to demands such as those described below.
Multiple-level tags: Many systems count by more than one numeral, incorporating higher-level tags such as -3 to +3 inclusive. Theyre called multi-level counts. If you think counting up and down by 1 is tough, try counting up and down by 2, 3, or 4 at a time.
Side counts: As if multiple levels and true counts werent enough, some systems would have you keep an extra count of certain cards, usually Aces. Theyre called multi-parameter counts. Imagine keeping two separate running counts going in your head. Every time you see a Two through King, you add its tag to one running count, and every time you see an Ace, you add to a different running count. When its time to bet (or play), you need to compare the number of Aces played with the average number that should have appeared at this point in the deck, estimate the discrepancy, then add to (or subtract from) the running count, prior to calculating the conversion to well, you get the picture.
Strategy-variation indices: Some systems require that you reference complicated strategy-index matrices. These sometimes have upwards of 200 entries that need to be memorized in order to realize a small gain in performance.
If these bloated systems make you want to stop before you even step into the ring, take heart. Its time to lighten up.
The Knock-Out System Theory
The Knock-Out System was developed under the philosophy espoused by Albert Einstein: Everything should be as simple as possible, but not more so. The K-O is a single-level, single-parameter count, designed to be highly effective using a bare minimum of strategy changes (if any). Most importantly, K-Os unbalanced nature completely eliminates the necessity to convert to a true count. Well get to that in a moment.
First, we must be able to do the requisite counting, which means we need to recognize which cards, as they leave the game, help us or hurt us. It turns out that as the little cards (2 through 7) are removed, our expectation goes up (much like it did when the white gumballs were dispensed). On the other hand, as the big cards (10, J, Q, K, A) are played, our expectation goes down. The middle cards (8 and 9) are neutral, and have little effect either way. Much like we did in the gumball example, we assign the following card tags to reflect these effects.
2 through 7: +1 (24
Note that there are more + than tags. Because the sum of the card tags is not zero, the K-O system is referred to as unbalanced. It is this unbalanced aspect that sets the K-O count apart from its balanced predecessors. It is also the unbalanced nature of the count that eliminates the need to make the brutal true-count conversions. We keep only the running count for all betting and strategy decisions.
So how does it work? Well use an example from the single-deck game to demonstrate.
After a shuffle, start with an initial running count of zero. Then just add or subtract, as applicable, according to the cards that are played. Easy enough, but how do we know when we have the advantage? Simple. The count indicates that the deck favors us when we reach a point that we call the key count. Any time the count is at or above the key count before a hand, we generally have the advantage. The higher the count goes, the greater our advantage is likely to be. In a single-deck game the key count is +2 (see The K-O in Application).
Armed with this information (and basic strategy), we now bet a lot when we have the advantage (the count is at +2 or more), and only a little when the house has the advantage (the count is below +2). Believe it or not, mastery of this simple plan is sufficient to beat the casinos at their own (single-deck) game. With slight alterations, the K-O also identifies when the player has the advantage in multiple-deck games.
Its important to note that I have not recommended that you vary the play of the hands from basic strategy. In fact, simply betting higher with the advantage generally accounts for 70 to 90 percent of the value of a card-counting system. The remaining profit comes from varying strategy as dictated by the remaining deck composition.
Once you are comfortable counting and betting accordingly, you may wish to add some Knock-Out clout by learning a few strategic plays (based on the count) to further increase your advantage. Ill comment on the most valuable of these plays: insurance.
Recall that insurance is offered to players when the dealer has an Ace showing. It is really a side wager on whether or not the dealer has a ten in the hole. A basic strategy player would never take insurance. But we are card counters! We should take insurance if there is a suitable preponderance of tens remaining. K-O allows us to accurately identify these situations in a simple way (the insurance running count value happens to be +3), again without the need for converting to a true count.
Similarly, other strategic plays may be added at your leisure, each one relying only on the running count.
You may be concerned
that something this simple must give up a lot in performance.
So what can card counting be worth to you? A good card counter will generally realize an advantage of some 1 to 2 percent over the house. That means that for every $1,000 you bet (about an hour of play assuming a betting spread of $10 to $50), you can expect to win $10 to $20. Everything scales with bet size, so the degree to which you pummel the casinos is up to you.
Are these profits guaranteed? Not in the short run. Its important to realize that the card counters advantage manifests itself over the long haul. In the short run, the game is an inevitable series of ups and downs, and entails risk thats commensurate with your betting level. Plus, dont forget that you have to be able to add to and subtract from the running count accurately enough to ensure a proper read of the deck. However, if you play properly and long enough (in a fair game), you are virtually guaranteed to come out ahead in the end.
Remember, too, that whether youre winning or losing, the casino thinks it has the upper hand. So even if you only play well enough to break even, you will probably still beat them due to compsthe freebies you can receive as a welcome player.