Card counters need to understand frequency distributions and how to use them to determine win rate and fluctuations.
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Card Counting and Frequency Distributions

 
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How to Use Frequency Distributions to Determine Your Card Counting Win Rate and Fluctuations (An Excerpt from Beat the 6-Deck Game)

By Arnold Snyder
© 1983, 2005 Arnold Snyder

[Thanks to Sam Case for lots of help with the original 1987 edition of this report. And thanks to Radar for editorial assistance on the new edition.]

TABLE OF CONTENTS
Foreword — Important
What is a "Frequency Distribution?"
Reading the Charts
50% Penetration
65% Penetration
75% Penetration
85% Penetration

Technical Appendix
Analyzing Betting Strategies Not in This Report
Standard Deviation
Playing Multiple Hands
Accuracy
Frequently Asked Questions

FOREWORD — IMPORTANT
This is a guidebook for serious blackjack players who are attempting to get the edge over the casinos. Its purpose is to help you work out your win rate, fluctuations, and optimal blackjack betting strategy in the six-deck game. I’m making some assumptions about the users of this book.

1) I assume you are familiar with the game of casino blackjack as it is played in most legitimate casinos. By this I mean that you understand the rules of the game sufficiently to play comfortably, correctly employing the available rule options. This is not a primer.

2) I assume you know basic strategy for the game you are playing. Basic strategy decisions for the various hands have been known for some fifty years now. Virtually all books on card counting—and there are dozens of legitimate card counting systems in publication—provide essentially the same basic strategy decisions.

3) I assume you are using a legitimate card counting system to obtain an advantage at the game. Such a system would provide you with plus and minus values to apply to the various cards, with instructions on how to vary your bet and alter your playing decisions according to your count. If you are not a card counter, the information in this book will be of little use to you. If you are a card counter, the information herein will help you maximize your profits in the games available to you.

4) I assume that you have an understanding of the most common rule options offered in U.S. casinos. There are, in fact, dozens of rule options used in various casinos, and most books on card counting explain the most common variations available.

5) I assume that the blackjack games you play in are traditional blackjack games. The information provided in this guide does not apply to "Spanish 21," Superfun 21," single-deck blackjack where "BJ Pays 6-to-5," Internet blackjack variations like "Pontoon" or "Blackjack Switch," etc.

In fact, the information presented here will be of no use to those who play in web casinos, even in traditional blackjack games, because web casinos reshuffle the cards between every round of play, and the information in these pages is all based on deck "penetration." This book deals with games in which at least 50% of the cards in play are being dealt out between reshuffles.

In other words, I assume that the user of this guide is a knowledgeable card counter. I will not spend much ink within these pages explaining the basics. I will recommend my own book, Blackbelt in Blackjack , for any blackjack player who wants to employ a card counting system or use other professional methods of winning at blackjack, from the beginner level to the advanced professional player.

The purpose of this book, Beat the 6-Deck Game, is to take a good card counter and turn him into a 6-deck expert. If you play in games with various numbers of decks, then I advise you to invest in the other reports in this series, which cover 1-deck, 2-deck, 4-deck, 6-deck and 8-deck games. The best way to comprehend the mathematics employed in this series is to read with a pocket calculator handy. Don’t be afraid to scribble in the margins with a pencil. All of the charts and numbers may appear forbidding at first glance, but the math is easy if you follow along.

You will not have to perform any of the mathematics within these pages while you are playing at the tables. The purpose of this book is to give you a clear understanding of how the various conditions you will find in the casino will mathematically affect your potential for winning—and how you must alter your attack on a game in order to beat it. This is not done while playing.

In fact, I have made every attempt to thoroughly analyze all of the most common attacks—and many uncommon attacks—on the various games, so that you will not have to do any math whatsoever if you prefer not to. I have tried to explain the charts clearly enough that you will be able to understand them at a glance. Please read the text carefully so that you can use the charts easily and accurately.

Arnold Snyder

What is a "Frequency Distribution?"


Definition: As used in this book, a frequency distribution is a table of numbers that tells us how frequently any player (or house) advantage occurs in a casino blackjack game.

In blackjack, we can analyze a card counting system and figure out our win/loss rate by using a frequency distribution. We know that the count is continually going up and down, and that sometimes the house has the edge, sometimes the player.

But to know exactly how much I expect to win or lose in a specific game, using some specific betting spread based on my count, I need more details. What I need to know is how often the different house/player advantages occur. That is, what are the frequencies of player advantages of 1%, 2%, 3%, etc.? And what are the frequencies of house advantages of 1%, 2%, 3%, etc.?

A player advantage of 2% over the house may occur 3 or more times per hundred hands, or only 3 times (or fewer!) per thousand hands, depending on the number of decks in play and how deeply into the deck(s) the game is dealt between reshuffles. The differences in advantage frequencies between a game with deep penetration and one with poor penetration make a HUGE difference to a card counter’s potential win rate.

In fact, many blackjack games are not beatable with any practical card counting strategy. And in many beatable games, your edge over the house is so small that the inevitable bankroll fluctuations will spell doom for the player with limited funds.

These Beat the Deck guides will not teach you how to count cards. But they will teach you how to choose games that can be beaten. If I know how often the player advantages and disadvantages occur in a game, I can figure out how much of a betting spread I need to beat the game. I can also figure out how much I should bet at each specific count in order to get the highest win rate I can obtain. And I can use these frequencies to figure out a betting strategy for minimizing my bankroll fluctuations.

There are two ways to draw up a frequency distribution. You can run a computer simulation of your system and then just look at the data the computer spits out. Or, you can use a mathematical formula for deriving the precise data you seek. There are arguments in favor of both methods. In this book, and in all of my Beat the Deck reports, I have used a combination of these two methods.

My goal in these reports is to provide data that can be used by all card counters, not just card counters using some specific count system. The advantages that occur in a casino blackjack game are about the same for all valid card counting systems. Some of the "advanced" systems can squeak out a bit more of an edge over the house, and some of the simpler systems are slightly weaker, but the actual differences are relatively small.

The information presented in these Beat the Deck reports is for an "average" card counting system. They will be highly accurate for a player using the Hi-Lo count from my book, Blackbelt in Blackjack. The more advanced Zen Count, also from Blackbelt in Blackjack, will be slightly stronger. The Red Seven Count (same book) will be slightly weaker; but the charts will be pretty accurate for all of these systems, as well as most other popular counting systems, especially in helping you to choose games and estimate the betting strategies needed to maximize both your profits and your chance of survival.

Technically, we cannot answer the question, "How frequently will a card counter have a 2% advantage over the house?" unless we first specify five conditions:

    1. What are the rules of the game?
    2. What counting system is the player using?
    3. How many players are at the table?
    4. How many decks are in play?
    5. How many cards is the dealer dealing out prior to reshuffling?

For the sake of simplifying our analysis, we’re going to comment briefly on the first three conditions, so that we can concentrate on conditions #4 and #5, which are most important.

#1: The Blackjack Rules

There are dozens of common rule variations in U.S. casinos. Each rule has some positive or negative effect on the basic strategy player’s expectation as well as the card counter’s potential win rate.

One factor that simplifies our analysis is that most casinos in the U.S. have settled on a set of rules that gives the house approximately a ½% advantage over the player. In other words, the player starts at a disadvantage of -0.5%. In fact, 90% of the traditional blackjack games in U.S. casinos have a house advantage between -0.4% and -0.6% off the top of the deck.

[Editor's note: This has been changing in recent years, since the introduction of substandard payouts on naturals and the spread of such games. If you're playing a game where blackjack get paid 6:5 or even money, you're playing a game with a much higher house edge.]

Likewise, most rule variations have only a minor effect for a card counter who is helped (or hurt) by the rules’ limitations on his play. So, for the sake of simplicity, all of the frequency distribution charts in this book assume that the house has a ½% advantage over the player off the top of the deck. This is your standard Las Vegas 6-deck game, where the dealer stands on soft 17 and the player may double down on any two cards.

If the dealer his on soft 17, but the player may double down after splits, the starting advantage is about the same. With late surrender allowed and resplitting of aces, with the dealer stands on soft 17, the house edge is only about 0.3% off the top. If you find yourself playing in a 6-deck game where the rules are more or less favorable than -0.5% off the top, you may use these frequency distributions as they appear here, but remember to adjust your final expectation up or down accordingly.

It would be more accurate to develop a separate frequency distribution chart for each set of rules—but you would not find the final result to be far from the simplified method I am advising. Because 6-deck games with starting advantages very different from –½% are rare, we will not bother analyzing them in more detail.

#2: The Card Counting System

Most valid card counting systems perform within one- to two-tenths of a percent of each other in computer simulations. So, instead of presenting the distribution to show how frequently each positive and negative "true" count will occur, I’m providing the data on how frequently each positive and negative advantage will occur.

It is, of course, necessary for you to know how your count relates to your advantage. If you use any of the counting systems from my book, Blackbelt in Blackjack, which adjust the running count according to the True Edge method, then your true edge count (minus the house advantage off the top) is your advantage. If you are using Stanford Wong’s Hi-Lo Count from his book, Professional Blackjack, then each true count (or count-per-deck) is equal to a 0.5% change in your advantage.

With a higher level count such as Wong’s Halves Count, Revere’s Point Count, or Uston’s APC, each count-per-deck is equal to (approximately) a 0.3% change in your advantage. If you are using a card counting system from a book that does not clearly explain the value of each point increase, then I would advise you to seek out a more advanced text. The system you are using may be valid, but if the author fails to provide you with a method for determining when you get an advantage, and how much of an advantage you get in percent as the count rises, then the book is just too elementary for a serious player.

With an unbalanced "running count" system, such as the simple Red Seven Count or Knock-Out Count, the values of each running count are constantly changing as the deck is depleted. In analyzing the Red 7 system with frequency distributions, it is important to note that your "pivot" will always reflect a 1% positive change from your starting advantage.

With simplified systems like these, you can use the charts in these Beat the Deck guides to compare the potential profitability of the games available to you, and to get a handle on how deep the penetration you will need, the betting spread required, etc. But to get really usable data on fluctuations, bankroll requirements, optimal betting strategies, etc., you will either need to use the True Edge method of running count adjustment, or switch to a balanced card counting system like the Hi-Lo Lite.

#3: Other Players at the Blackjack Table

In six-deck games a frequency distribution will look different if one player is at the table than if more than one player is playing. Technically, you would need seven different frequency distributions to estimate your potential win rate in games where the number of players varies from one to seven.

The problem with drawing up "accurate" frequency distributions to cover all possible situations is that there are hundreds of possible situations and these situations are always in flux. You might have five players at the table, with 75% (4½ decks) dealt. The very next shoe you might have three players at the table, with closer to 80% (4¾ decks) penetration, etc. The next shoe, with six players at the table, a new dealer comes on who deals more slowly, cuts the penetration to about 72%, delivering fewer hands per hour. Imagine the possibilities, with anywhere from one to seven players at the table, varying penetration between shoes, and the idiosyncrasies of various dealers.

All of the distributions presented in this guidebook assume you are alone at the table. If you are not alone at the table, the distribution will still be fairly accurate, assuming you use all of the information available to you when making your betting and playing decisions. It will be slightly more advantageous for you to sit at third base, or as close to third base as possible, as this will allow you to play your hands with more information based on the hit cards of players who must play before you. As a rule, in all shoe games, when playing with other players at the table, always seek out situations in which you can see as many cards as possible prior to playing your hand.

#4: The Number of Decks in Play

A frequency distribution for a six-deck game differs so widely from a frequency distribution for a four-deck game or an eight-deck game that it is necessary to draw up separate distributions based on the number of decks in play. Do not attempt to use the distributions in this guidebook to approximate your advantage in any game other than a 6-deck game. Separate guides are available from the publisher for one-deck, two-deck, four-deck, and eight-deck games.

#5: Card Counting and Deck Penetration

Once you know how many decks are in play and the house advantage off the top afforded by the rules, the prime concern of the card counter is the deck penetration—i.e., how many cards are being dealt out prior to reshuffling. In a 6-deck game, if a dealer will not deal another round if more than three decks have been used, use the 50% penetration chart to analyze this game.

A six-deck game with less than three decks dealt out is generally a waste of time for card counters, so I have not bothered to analyze such games. Technically, if such a game had exceptionally good rules—such as double on any two cards, double after splits, dealer stands on soft 17, resplit aces and surrender—and/or if you could use a very large betting spread, say 1-to-25 units or more, then a 6-deck game with poor penetration might show slim profits for a card counter.

However, if you look at the pitiful returns on the typical 6-deck games with 50% dealt out, you can imagine what a waste of time it would be to attack a 6-deck game with lesser penetration. Most 6-deck games should be analyzed with the 65% (4 decks) and 75% (4½ decks) dealt charts. An 85% (5 decks) dealt game is way better than an average profit opportunity. Use the 85% chart only for dealers who deal down into the last deck prior to reshuffling. Remember, 85% of six decks is 265 cards. An 85%-penetration dealer would be one who would only shuffle when he had 47 or fewer cards undealt.

How often do you find a shoe dealer who would deal another round with less than a deck remaining in the shoe? Such dealers are extremely rare, but they do exist. Low stakes players are more likely to run into such situations. If you do find yourself in such a situation, use the 85% chart to analyze your possibilities.

Reading the Frequency Distribution Charts

All of the frequency distribution charts are set up identically to the chart on the facing page.

The top line reads: 6 Decks, 75% Dealt, Bet Strategy = 1:2:4:8:16. This tells us that the chart applies to a 6-deck game in which 75% of the cards (4½ decks) are being dealt out between shuffles. "Bet Strategy = 1:2:4:8:16" would apply to a player who is spreading his bets from 1 to 2 to 4 to 8 to 16 units. This might be from $1 to $16, or $5 to $80, $10 to $160, etc.

6 Decks 75% Dealt Bet Strategy = 1:2:4:8:16
Adv.HandsSys 1Sys 2Sys 3Sys 4Sys 5Sys 6Sys 7Sys 8
-4.5%0.011111111
-4.0%0.011111111
-3.5%1.011111111
-3.0%2.011111111
-2.5%3.011111111
-2.0%4.011111111
-1.5%8.011111111
-1.0%13.011111111
-0.5%34.011111111
+0.0%13.021111111
+0.5%8.542111111
+1.0%4.584 2 11111
+1.5%3.5168 4 2 1111
+2.0%2.01616 8 4 2 111
+2.5%2.01616 16 8 4 2 11
+3.0%1.01616 16 16 8 4 2 1
+3.5%0.51616 16 16 16 8 4 2
+4.0%0.01616 16 16 16 16 8 4
+4.5%0.01616 16 16 16 16 16 8
+5.0%0.01616 16 16 16 16 16 16
+6.0%0.01616 16 16 16 16 16 16
+7.0%0.01616 16 16 16 16 16 16
+8.0%0.01616 16 16 16 16 16 16
 
Hands Bet/100:100100100100100100100100
Average Bet: 3.05 2.29 1.82 1.46 1.23 1.09 1.03 1.01
Gain/Hand: 0.029 0.022 0.016 0.008 0.003 -0.001 -0.003 -0.004
Win Rate %: 0.95% 0.97% 0.86% 0.58% 0.22% -0.12% -0.30% -0.37%
 
Units/Hour*: 2.90 2.22 1.55 0.84 0.27 -0.13 -0.31 -0.38
S.D./Hour*: 58.7 46.9 37.9 28.0 19.9 13.6 11.6 11.1
 
Units/10 Hours*: 29.0 22.2 15.5 8.4 2.7 -1.3 -3.1 -3.8
S.D./10 Hrs*: 185.5 148.2 119.7 88.6 62.9 43.0 36.6 35.0
 
Units/100 Hours*: 290 222 155 84 27 -13 -31 -38
S.D./100 Hours*: 586.7 468.8 378.6 280.2 198.8 135.8 115.6 110.8
 
Units/1000 Hours*: 2900 2215 1553 843 270 -130 -310 -375
S.D./1000 Hours*: 1855.4 1482.3 1197.2 886.2 628.5 429.6 365.7 350.4
 
Units/10000 Hours*: 29000 22150 15525 8425 2700 -1300 -3100 -3750
S.D/10000 Hours*: 5867.2 4687.6 3785.8 2802.3 1987.6 1358.4 1156.3 1108.2
* Based on 100 hands seen/hour, betting the number of hands in Hands Bet/100.

The first column, labeled Adv., shows the various player advantages, positive or negative, in percent. The Adv. column lists the various advantages that will occur in the game from a 4.5% house advantage to an 8% player advantage. If the advantage has a minus sign (such as -3.5%), then it is a house advantage. If positive (3.5%), then it is a player advantage. Neither the house nor the player ever has an advantage greater than +/- 3.5% in this game. With greater penetration (85%), the player sometimes sees an advantage of up to 4.5%.

The second column, labeled Hands, shows how many hands per 100 will occur with the advantage shown in the first (Adv.) column. Example: In this 6-deck game with 75% (4½ decks) dealt, a player advantage of 1% will occur about 4.5 times per hundred hands. A house advantage of 1% will occur about 13 times per 100 hands. Note that the house advantage of 1% occurs three times as often as the player advantage of 1%. This is because we assume the house has a ½% edge (0.5%) off the top. In a sense, the house gets a running start on the player.

Following these two columns, there are eight columns labeled Sys1, Sys2, and so on through Sys8. These columns identify eight different betting systems in which the player is using a 1:2:4:8:16 betting spread.

Note that each of these columns has a list of numbers, all of which are 1, 2, 4, 8, or 16. These numbers are the player bet size in units.

In Sys1, for instance, we see that the player bets 1 unit whenever the Adv. column is negative (meaning the house has an advantage). The player’s bet rises to 2 units when the advantage is 0.0% (dead even), 4 units when the advantage is 0.5%, 8 units with a 1% advantage, and 16 units with any advantage greater than 1%. In all of the betting systems, the player is using a 1-to-16 spread. The only difference in the systems is the advantage at which the player raises his bets from 2 to 16 units.

It is very important that you understand how to read this top portion of the chart, which describes the card counter’s betting strategies in this game. All of the charts in this report, and in all of the Beat the Deck reports, use this same format. Most card counters, if spreading their bets from $5 to $40, would simply say, "I was using a 1-to-16 betting spread." This may be true, but it is very important to know what your advantage is when you are raising your bets. If you don’t provide this information, you cannot analyze your expectation in the game.

So before you go on to the explanation of the bottom portion of the chart, which provides the analyses of the different betting systems, look at the Sys1 through Sys8 betting strategies and be sure you understand how these betting systems differ from each other. Now, on to the bottom portion of the chart ...

The "Hands Bet/100" line shows how many hands out of every 100 hands played the player would expect to place bets. In all of the examples shown in our sample chart, the player bets on all 100 hands. If the player were table-hopping, however, and not betting on negative expectation hands, this would be indicated in the charts by placing zeros in the SYS columns for all of the (dis)advantages where the player placed no bet, and the Hands Bet/100 would be lower than 100.

If you flip forward a few pages, you will see numerous charts where the betting system (Sys) columns have zero (0) entries when there is a house advantage. You will note that in these instances, the Hands Bet/100 reflects the actual number of hands per hundred that the card counter placed bets on.

If your method of play is to not sit down, but to wander from table to table as you play, making your decisions to exit a game whenever your count indicates that the house edge has gone up to a point where you refuse to place even a 1-unit bet, then you will use the chart(s) where the Sys columns show the zero (0) entries. These charts are easily identified by the top line, as the heading will read: Bet Strategy = 0:1:2, or 0:1:2:4, etc.

Just beneath the Hands Bet/100 line, the "Average Bet" line indicates the player’s average bet per hand played in units for each betting system. For instance, in using Sys1, note that the player’s average bet is 3.05 units, while in using Sys8, the average bet is only 1.01 units. This is because the Sys8 player is waiting longer to raise his bets.

Beneath the Average Bet line, the "Gain/Hand" shows how many units the player can expect to win for every hand he plays with each betting system. (If this is a negative number, then it shows how many units the player can expect to lose per hand played over the long run.)

In our sample chart, all of the examples are positive. If you flip forward a few pages, however, you will see many examples where the Gain/Hand is negative, which means that the card counter would actually be losing money in the long run in that game with that particular betting system. This occurs most frequently when the counter’s betting spread is too small to beat a game, especially because the percentage of cards being dealt out between shuffles is insufficient.

Beneath the Gain/Hand line, the "Win Rate %" line shows the player’s long run expectation in percent over the house. (If this were a negative number, it would indicate the house’s advantage over the player.)

Note that with Sys2, the player’s win rate in this game, using a 1-to-16 betting spread, maximizes at 0.97%. Using Sys6, 7 or 8, the player’s win rate is negative. This is because the Sys6, 7 and 8 bettors are waiting too long to raise their bets.

Beneath the Win Rate % line, we find the hard data we’re most interested in: how much money (in units) we can expect to make, on average, for every hour of play, every ten hours of play, every hundred hours of play, etc., in this game. Note that the "Units/Hour" is always based on seeing 100 hands per hour. A crowded table will be slower than this, and if we’re playing heads up, or with only one other player at the table, we’ll probably play more than 100 hands per hour.

Note that if we employ Sys2 for our betting strategy, we can expect to earn 2.22 units per hour, or 22.2 units per 10 hours of play, etc. To get our win rate in dollars and cents, we simply multiply this number by our betting unit. For instance, if my 1-to-16 betting spread is obtained by spreading from $10 to $160, then my unit is $10, and my expected Sys3 win rate in dollars is:

2.22 x $10 = $22.20 per hour.

Note that under each Units/Hour line is a line that shows "S.D./Hour," "S.D./10 Hours," etc. S.D. stands for standard deviation. This is the normal fluctuation from your expected win, in units. We can use this measure to estimate how much we should be betting in any given game, with any given betting strategy, based on the actual size of our bankroll.

Standard deviation is a commonly used statistical measure. In our Units/10 Hours example for Sys2, for example, you can see that we have an expected win of 22.2 units, with a standard deviation of 148.2 units. What does this mean?

It means that two-thirds of the times that you play for ten hours (or more precisely 1000 hands) under these conditions, your outcome will be within one standard deviation of your expected win. And 95% (or 19 out of 20) of the times that you play 1000 hands under these conditions, you will be within two standard deviations of your expected win. You will be within three standard deviations of your expected win 99+% of the time. (For a comprehensive discussion on standard deviation, see Blackbelt in Blackjack.)

In our Sys2 example, this means that after 1000 hands, we would expect to be ahead by about 22.2 units. Two-thirds of the time we’ll actually be within one standard deviation (148.2 units) of this expectation. In other words, we’ll actually be somewhere between –126 units and +170.4 units two-thirds of the time.

Ninety-five percent of the time we’ll be within two standard deviations (148.2 x 2 = 296.4 units) of our expected win. In other words, we’ll be between –274.2 units and +318.6 units. And we are virtually assured of always being within three standard deviations (148.2 x 3 = 444.6 units) of our expected win—i.e., between –422.4 units and +466.8 units.

Professional players use the standard deviation to calculate the best betting unit for their particular bankroll. For example, it would be foolish to spread your bets from $5 to $80 in this game if your total bankroll was only $400. Since your expected win per 1000 hands is only $5 x 22.2 = $111.00, and your standard deviation on 1000 hands is 148.2 units, and 148.2 x $5 = $741, a small negative fluctuation within one standard deviation could easily wipe you out.

To spread your bets from $5 to $80 in this game, you would be relatively safe (meaning likely to survive) in ten hours of play if you entered the game with enough of a bankroll to withstand a fluctuation of two standard deviations. Since one standard deviation is $741, and our expected win is $111, a negative fluctuation of 2 S.D.s would set us back $111 - (2 x 741) = $1371. So, a bankroll of about $1400 would survive about 95% of the time (19 out of 20 times).

Using the charts in this report, you will begin to see patterns in the numbers. For instance, every betting spread has an ideal point at which to raise your bet in order to maximize your expected unit win. The more units you bet at positive advantages, the greater your expected win in units, but also, the greater the fluctuations (and the bigger the bankroll needed to survive).

If you lower your top bet substantially in order to cut the flux, however, you may not be able to get a significant edge over the house. The problem you face as a professional gambler is to balance the need to aggressively pursue return (expected win) on your investment dollars with the overriding need to protect your bankroll.

Using the charts in this book, you can quickly see what the return is with any betting strategy along with its associated risk.You can also use these charts to compare profit opportunities when multiple games are available to you. If one casino deals out only 65% of the cards between shuffles, while another deals out 75%, you will almost always find the game with deeper penetration to have greater profit potential.

This may not always be the case, however. If the casino with the deeper penetration is very paranoid about card counters and shuffles up any time a player raises his initial bet by a factor of 4, while the casino with lesser penetration has no problem when you spread from $5 to two hands of $40 ( a 1-to-16 spread), then this more aggressive betting strategy will often compensate for poorer penetration.

Many players are amazed at how severe normal fluctuations can be, even when they have the edge in their favor. It is not uncommon for players who lose heavily in a casino to worry that the game may have been dishonest. Or they may question the validity of the card counting system they are using, or even their own abilities in employing the system.

If you suffer inordinate losses on a trip, you can use the data in these charts to see if your losses fall within the realm of what a mathematician would consider "normal."

The Technical Appendix, which follows the charts, explains how you can insert any betting system you want into the frequency distribution charts, and estimate your average bet, gain per hand, win rate in %, win rate in units, and standard deviation. If you are competent with a spreadsheet type program like Excel, you could use it to do all of the math for you. But it is not difficult to do it with a simple pocket calculator.

For players who would prefer not to do any math whatsoever, I have analyzed in the charts that follow most of the practical approaches a player might take to attempt to beat the various games.

Also contained in the Technical Appendix, for those who have a greater interest in the mathematics of blackjack, are explanations of how to figure out the standard deviation for any number of hands, using any bet spread. The charts show standard deviation data for 100 hands, 1,000 hands, 10,000 hands, etc., with specifically defined betting systems, but you may want data for 3,200 hands, or 620 hands, or some number of hands not defined in the charts. There is also a simple explanation of how to estimate the effects of playing multiple simultaneous hands.

Now, on to the charts . . .

Note: There are 44 pages of frequency distribution charts in the print version of this book. The charts contain Arnold Snyder's comments on each betting approach. These charts are not part of this online excerpt.

See the 2005 print version for the full collection of charts. Also, note that the frequency distribution charts for the 1-, 2-, 4-, and 8-deck games are very different from each other and from the 6-deck charts, and each has its own Beat the Deck book.

Technical Appendix:
How to Analyze Card Counting Betting Strategies Not Included in this Book
(Such as Betting Strategies for Big Player Call-In Blackjack Teams)

Let’s analyze one system completely so that you understand the math. We’ll take a 6-deck Las Vegas game with 75% penetration. You may double down on any two cards, including after splits, the dealer stands soft 17.

You leave the table when your advantage falls below –1%, otherwise play $2 off the top and on all other negative and zero expectation hands, $7 if you have a ½% advantage, and $12 if you have a 1% advantage or higher.

Using a sheet of lined paper, label the first column "Hands," and simply fill in the numbers from the Hands column from any of the "75% Dealt" charts in this report. (Flip back a few pages to look at any of these 75% charts, and you’ll see that the numbers I’ve filled in here match the numbers in the second column of those charts.)

Label the second column "Bet," and we’ll fill in these numbers later. Leave it blank for now. Label the third column "Total," and leave this blank also.

Label the fourth column "Adv.," and here we enter the numbers from any of the 75% charts that are in the Adv. column. Note that this is the first column in those charts. The only difference here is that we should enter the numbers here in decimal format, rather than as percentages. In our 75% charts, we show the advantage as -4.5%, while here we enter -0.045.

If you are unsure how to translate percentages into decimals, the method should be clear to you if you study the numbers I’ve already filled in briefly. Basically, 1% = .01, 1.5% = .015, 2% = .02, etc.

Label the fifth and last column "Gain," and leave this blank for now also. Finally, place multiplication signs between the Hands and Bet columns, and also between the Total and Adv. columns. And place equal signs between the Bet and Total columns, and also between the Adv. and Gain columns. (See the example below.) Here’s how the first five lines of your template should look:

Hands   Bet  Total Adv.  Gain
0.0x____=____x -.045 = ____
0.0x____=____x -.04 = ____
1.0x____=____x -.035 = ____
2.0x____=____x -.03 = ____
3.0x____=____x -.025 = ____

In other words, two of your columns are filled in using numbers from the charts in this book that show the player/house advantage, and the number of hands that occur at each advantage. We are now ready to fill in our betting system, then perform a couple of simple multiplication operations.

If you are familiar with computer spreadsheets, you can see that it would be very convenient to set up a template according to the above guidelines for analyzing betting systems and letting the computer do all the math. But all you really need is a pocket calculator and a sheet of paper. The math is simple once you have the chart set up.

In the second column, labeled "Bet," you enter your betting system. Since, in this example, we are leaving the table if our Advantage falls below 1%, we enter a zero (0) in this column wherever the Adv. reads negative from -.045 to -.015. We then enter a 2 unit bet for all Advantages from -1% (-.01) to zero, a 7 unit bet where our Advantage is ½% (+.005), and a 12 unit bet for all Advantages of 1% (+.01) or higher. Look at the numbers in the Bet column and see that you understand this process.

To get the numbers in the Total column, we simply perform the multiplication of the numbers in the Hands column and the numbers in the Bet column. To get the numbers in the Gain column, we multiply the numbers in the Total column by the numbers in the Adv. column.Finally, we add up all the numbers in the Total column and the Gain column.This is how it looks when it’s all filled in:

Hands   Bet  Total Adv.  Gain
0.0 x0=0x -.045 = 0.0000
0.0 x0=0x -.04 = 0.0000
1.0 x0=0x -.035 = 0.0000
2.0 x0=0x -.03 = 0.0000
3.0 x0=0x -.025 = 0.0000
4.0 x0=0x -.02 = 0.0000
8.0 x0=0x -.015 = 0.0000
13.0 x2=26x -.01 = -.2600
34.0 x2=68x -.005 = -0.3400
13.0 x2=26x 0 = 0.0000
8.5 x7=59.5x +.005 = +0.2975
4.5 x12=54x +.01 = +0.5400
3.5 x12=42x +.015 = +0.6300
2.0 x12=24x +.02 = +0.4800
2.0 x12=24x +.025 = +0.6000
1.0 x12=12x +.03 = +0.3600
0.5 x12=6x +.035 = +0.2100
0.0 x12=0x +.04 = 0.0000
0.0 x12=0x +.045 = 0.0000
0.0 x12=0x +.05 = 0.0000
0.0 x12=0x +.06 = 0.0000
0.0 x12=0x +.07 = 0.0000
0.0 x12=0x +.08 = 0.0000
TOTALS:341.5 +2.5175

There is one more operation we need to perform before we have all of the data we need for a complete analysis. We need to know how many hands (per 100) the player is betting on. If we add up the numbers in the Hands column, we see that they sum to 100. The player using this betting system, however, is not betting when the advantage is against him by more than 1%. So, we total up only the numbers from this column where the bet is greater than zero (0). We find: 13 + 34 + 13 + 8.5 + 4.5 + 3.5 + 2 + 2 + 1 + .5 = 82 Hands playedNow, let’s do the math . . .

To calculate your Average Bet per hand, divide the Total Units Bet by the total number of hands played:

341.5 / 82 = $4.16

To calculate your Gain Per Hand in dollars, divide the Total Gain by the total number of hands played:

2.5175 / 82 = .031

To calculate your Win Rate in per cent, divide the Gain Per Hand by the Average Bet:

.031 / 4.16 = .0075 or 0.75%

To calculate your Expected Win in dollars for any number of hands played, multiply your Gain Per Hand times the number of hands played. For example, your expected win after 225 hands played:

.031 x 225 = $6.98

Note: Since we entered the actual dollar amounts of our bets in the Bet Per Hand column, i.e., $2 = 2, we do not have to convert units to dollars. If we were spreading from two $5 chips up to twelve $5 chips, then we would have to multiply our gain per hand in units by 5 to get the gain in dollars.

To summarize, if we played in this 6-deck game, spreading our bets from $2 to $7 to $12, and we played approximately 82 hands of the 100 hands we saw per hour, we would have a win rate of 0.75%, which would give us a long run expectation of about $2.54 per hour (which is simply the 82 hands played per 100 seen times our gain per hand of .031 (@3 cents per hand)).

If we were spreading from two $5 chips ($10) to 7 chips ($35) to 12 chips ($60), our percentage win rate would be identical (0.75%), but our expectation in dollars would be about $12.70 per hour.

Using the above methodology, you should be able to analyze virtually any count-based betting strategy using the frequency distribution data from the charts in this guide.

What if you estimate that the penetration in not quite 75%, but greater than 65%?

You can get a pretty close estimate of your expectation with 70% dealt if you look at the results for 65% and 75%, then calculate the midpoint.

For example, if you look in the charts for using Sys2 with a 1-to-16 spread, you’ll find that your estimated win rate is 0.59% with 65% dealt, and 0.97% with 75% dealt out. If you estimate that in fact about 70% of the cards are being dealt out between shuffles, the midpoint between these win rates is 0.78%.

After you find the midpoint, always round this number down. The actual win rate is between 0.65% and 0.70%.You can use this same method of interpolating results for different levels of penetration for approximating your average bet, your win rate in units per hour, etc.

Standard Deviation in Card Counting

To calculate your standard deviation for any number of hands, this is how you do it:1. First, add up the number of hands played at each of the different bet sizes.

Using the same $2 to $12 sample betting system described above, we placed $2 bets on

13.0 + 34 + 13.0 = 60 hands

We placed $7 bets on 8.5 hands.

And we placed $12 bets on

4.5 + 3.5 + 2 + 2 + 1 + 0.5 = 13.5 hands

2. Now, square each bet size, multiply the bets squared by their respective number of hands played then add up these products. It looks like this: 22 x 60 = 240 72 x 8.5 = 416.5

122 x 13.5 = 1944

2600.5

3. Now, take the square root of this number and multiply it by 1.1, like this: √ 2600.5 x 1.1 = 56.094. Finally, divide this number by the square root of the number of hands played (per 100).

56.09 / √ 82 = $6.19

This is your standard deviation, in dollars, per hand.To find your standard deviation, in dollars, for any number of hands, multiply your standard deviation per hand times the square root of the number of hands.Examples:The standard deviation on 100 hands = 6.19 x √ 100 = $61.90The standard deviation on 1000 hands = 6.19 x √ 1000 = $195.74The standard deviation on 10,000 hands = 6.19 x √ 10,000 = $619.00

Playing Multiple Hands While Card Counting

If you play multiple simultaneous hands, you may estimate your average bet, gain/hand and win rate, by multiplying the number in the Hands column by the number of simultaneous hands you are playing at that count. Do not use this method, however, to estimate your standard deviation.

If, for instance, you play two hands of four units each at a specified true count, you would underestimate the standard deviation if you simply added the extra hands into your calculations. This is due to the fact that simultaneous hands will have more of a tendency to have the same result, since they are both played vs. the same dealer hand.

If you instead estimated your standard deviation as if two 4-unit hands were a single 8-unit hand, you would overestimate the standard deviation. The actual standard deviation would fall somewhere between these two results.

A simple way to estimate your standard deviation on two simultaneous hands is to simply estimate the standard deviation on one hand that is 75% of the total amount bet on both hands. In the example above estimate your standard deviation on two 4-unit hands as if you were playing one 6-unit hand.

For three simultaneous hands, estimate your standard deviation as if you were playing one hand that is 60% of the total amount bet. In other words, with three simultaneous hands of $10 each, take 60% of the $30 total bet and compute your standard deviation as if playing one hand of $18. Again, this is a simplification, but it will give you a good ballpark estimate.

Accuracy

All of the frequency distributions, estimates of win rates, standard deviations, etc., in this book are approximations of what human players might expect in casino play. If the charts herein estimate your win rate at $20 per hour, then your actual win rate is probably between $15 and $25 per hour.

Don’t assume pinpoint precision. Even if we were to run a billion-hand computer simulation to obtain a highly precise estimate for a specific counting system, it would not necessarily provide a better estimate of your expectations in a real-world casino.

Casino dealers vary their levels of penetration. Different numbers of players at the table may affect the shuffle point, change the number of hands dealt per hour, etc. And even the best card counters make errors in "rounding off" their count adjustments and they apply different amounts of betting and playing strategy "camouflage" as needed in the casinos where they play.

Use the data in these charts to compare the profit opportunities in the games available to you, determine the betting spread you need to get a sufficient edge over the house, and estimate your bankroll requirements.

Finally . . .

The top blackjack pros are not all mathematicians, but they all do understand the basic math and logic of the game. If you study the concepts and the charts presented in this book, you will get a very good feel for the profitability of any 6-deck game you find.

Look at the kinds of betting strategies you’ll need to beat the 6-deck games with 50%, 65%, 75% and 85% penetration. If the penetration is poor, look for any possibilities of beating the game by leaving the table at negative advantages. Consider the possibilities of getting a bigger spread by playing two or more hands at favorable counts. If there’s no practical way to get a healthy edge on the house, then keep your money in your pocket.

Frequently Asked Questions

Q: Why do you provide so much information on games and betting strategies that don’t win? Everyone knows that bad penetration and small betting spreads don’t work for card counters.

A: There are many card counters, perhaps even a majority of them, who do not have professional aspirations, but play at moderate to high stakes in order to acquire casino comps. These players are primarily interested in knowing how to reduce the house edge to a break-even point.

Also, there are many books on the market that poorly explain the betting spreads needed to beat various games. Many amateur card counters believe that they can beat most games with a 1-to-4 spread. My goal in providing these Beat the Deck reports to players is to show what works, how well it works, and what doesn’t work, in any game with any level of penetration.

Q: Is there a quick way to use these charts to judge a game’s profit opportunities "at a glance?"

A: I scan the Win Rate % line first, looking for an advantage of about 1% or better. If I find it, I then look at the Hands Bet/100 in that column. The bigger this number, the better. If the Hands Bet/100 is less than 30, you’ll probably spend too much time standing around watching games, waiting to place a bet. The % advantage might be good, but your hourly unit win may be too small to be worth your time.

Q: Do you have any guidelines for judging when the standard deviation is tolerable?

A: The primary guideline is your own personal bankroll. Casino blackjack is a fairly high-risk investment for a card counter. The "long run" often takes a long time coming. A player who is trying to get an edge over the house in the neighborhood of 1% should look for a standard deviation where the expected win after 100 hours (10,000 hands) is at least half of one standard deviation, and after 1000 hours (100,000 hands), the expected win is twice as much as one standard deviation.

In other words, when looking through the charts for a good betting strategy, I look at the unit wins for 100 hours and 1000 hours, compared to their S.D.s. If my unit win expectation is 100 units after 100 hours, then I don’t want the S.D. to be much more than 200 units. If my unit win expectation after 1000 hours is 1000 units, then I don’t want the S.D. to be much more than 500 units. The greater the number of decks, and the worse the penetration, the more difficult it is for a game to meet these criteria.

Q: Why is the Win Rate % higher for one blackjack betting system, when the Units/Hour is higher for another betting system?

A: In a case like this, the system winning more units per hour is betting more units per hour. This may be due to either betting on more hands/100 seen, or increasing to bigger bets at smaller advantages. Whenever you see this, you can find the answer in the chart data by looking at how many hands were bet by each system, and/or at what advantage did each system raise its bets. ♠


For more information on how to win at blackjack and optimal betting strategies for card counters, see Arnold Snyder's Blackbelt in Blackjack. Arnold Snyder's Beat the X-Deck Game reports are available at Cardoza Books.

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