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HOW TO BEAT SINGLE DECK BLACKJACK
Version 1.01
Copyright 1991, Michael Hall
----------------> Part 1: The Basics
Part 2: About the Strategy Charts
Part 3: The Strategy Charts (LONG)
Introduction
============
Here is the long-awaited article on Hi-Opt I, covering everything
from the rules of blackjack to basic strategy to card-counting to
multiparameter tables for experts. Novices should read this article.
People interested in "just the facts" about the strategy should
just read Part 2 and Part 3, in separate articles.
Help for the novice blackjack player
====================================
The basic idea of the game is to get a total less than 21 that is
higher than the dealer OR to not bust (go over 21) when the dealer
busts.
All single deck games are dealt face down. You receive the first
two cards face down, and any subsequent cards face up. Use one
hand to hold the first two cards. Don't touch the others.
Insurance is a side bet for up to half of your original bet. It can
only be placed at the start of a round when the dealer has an ace
showing. An player who is not counting cards should never take insurance.
Insurance pays 2-1 only if the dealer has blackjack.
Splitting can be done 3 times, to produce up to 4 hands. If you have
a pair and wish to split them, put your cards face up in front of your
bet, and push out a bet equal to your original. The dealer splits the cards
apart and deals a card to the first one, which you play normally,
and then the dealer deals a card to the second one, which again you
play normally.
Doubling can be done on any two cards. First put your cards face up
in front of your bet, and then push out a bet equal to your original,
and you will receive exactly one more card. (If it is a pair, you
may have to verbally tell the dealer whether you are doubling or
splitting.)
Standing versus hitting is the most common and important decision. To
hit you scratch your held cards on the surface of the table. Standing is
indicated by pushing your cards under your bet. When you split,
you instead use the protocol for face-up games - hitting is indicated
by tapping the table, and standing is indicated by a waving motion
parallel to the table.
Aces can be counted as either 1 or 11. A "soft" total means you have
an ace and can use it as 11 without going over 21; "hard" means you
aren't counting an ace as 11 in your total.
Basic Strategy
==============
You must first learn basic strategy, whether your goal is to become
a professional card counter or just to survive a weekend at
the blackjack tables.
With basic strategy, you eliminate the house edge on most Las Vegas
single deck games. With the Frontier's special rules, you can get
a +0.3% advantage with just basic strategy. Basic strategy is the
computer proven *best* way to play, unless you are counting cards.
The basic strategy for typical single deck blackjack is given in
another part of this article.
About Nevada
============
Single deck rules and conditions vary tremendously throughout Nevada.
But yes, Virginia, single deck blackjack is alive and well.
Vegas Strip rules are dealer stands on soft 17, double any first
two cards, resplit up to 3 times, no doubling after splitting,
and blackjack pays 3 to 2. Proper basic strategy on this game
makes it an even game. Many casinos on the Vegas strip use
these rules.
Vegas Downtown rules are like Vegas Strip rules, except that the
dealer hits soft 17. This swings the odds slightly in the house's
favor against the basic strategist. Many casinos in downtown Vegas
use these rules.
Northern Nevada rules are like Vegas Downtown rules, except that
doubling is restricted to totals of 10 and 11 only. This really hurts
the basic strategist.
Advantage
=========
Advantage is your winnings divided by your action. Your action is
the total amount of money you wager. To find out how much of an
advantage you have with basic strategy for a single deck game,
use the table below. Start with +0.02% for Vegas Strip rules,
and then add the adjustments for rule differences for the game
you wish to analyze.
% RULE
----- ---------------------------------
-0.78 Doubling on 11 only
-0.26 Doubling on 10 and 11 only
-0.21 No non-ace pair splitting
-0.19 Dealer hits soft 17
-0.16 No splitting of aces
-0.13 Doubling on 9, 10, 11 only
-0.11 No hole card (European)
-0.02 No resplitting of non-ace pairs
+0.03 Resplitting of aces
+0.11 Six card automatic winner
+0.13 Doubling after splitting (DAS)
+0.14 Drawing to split aces
+0.16? Suited BJ pays 2-1
+0.24 Doubling on 3 or more cards
Help for the aspiring card-counter
==================================
For single deck blackjack, I recommend a simple counting system card
called Hi-Opt I. Although it is one of the simplest systems to start
with, it can be extended to be more powerful than nearly any other
counting system - and you never have to forget what you have already
learned.
Here's an interesting table, drawn from "Theory of Blackjack" and elsewhere:
PLAYING BETTING COUNT VALUES
SYSTEM EFFICIENCY CORRELATION A 2 3 4 5 6 7 8 9 10
Hi-Opt I .615 .88 0 0 +1 +1 +1 +1 0 0 0 -1
Hi-Opt I + A .635 .96 "
Hi-Opt I + A,7 .736 .97 "
Hi-Opt I + A,7,8 .811 .97 "
Hi-Opt I + A,7,8,9 .870 .97 "
Hi-Opt I + A,7,8,9,2 .891 .98 "
High-Low .51 .97 -1 +1 +1 +1 +1 +1 0 0 0 -1
High-Low + A .61? .97 "
Hi-Opt II .67 .91 0 +1 +1 +2 +2 +1 +1 0 0 -2
Zen .63 .97 -1 +1 +1 +2 +2 +2 +1 0 0 -2
Uston APC + A .69 .91 0 +1 +2 +2 +3 +2 +2 +1 -1 -3
(Side counted cards are listed after the "+" signs.)
Playing efficiency is a measure of how good a counting system is
a strategy, while betting correlation is a measure of how good it
is at betting. As you can see, Hi-Opt I + A,7 is better in both
respects than of the listed competitor systems, and Hi-Opt I + A is as
nearly as good as or better than most systems. Even straight Hi-Opt I
with no side count of aces is respectable.
Here is how you do the Hi-Opt I count. Initialize "running count" to
zero at start. Add one for each 3, 4, 5, or 6 you see and subtract
one for each 10 you see. Divide running count by estimated number
of unseen decks to get true count used in the strategy adjustment
table. Note that for single deck, this division becomes a multiplication.
For example, if there is 1/4 deck remaining and the running count is
+2, then the true count is +8.
You should only take insurance if the true count is +2 or above.
Don't be swayed by what cards you have (i.e., don't fall into the
insure-your-blackjack trap); it's a side bet, so only the count
matters.
The Kelly Criterion is a betting heuristic that minimizes your chance
of going broke while maximizing your long-run profits. To bet
consistently with the Kelly Criterion, you should divide your bankroll
into 300-400 units and normally bet 1-4 units on each hand. Your
optimal bet on a hand is a percentage of your *current* bankroll
equal to about .5R/D + B, where R is the running count, D is the number
of remaining decks (so R/D is the true count), and B is the basic
strategy expectation. Note that on a weekend trip you might not
need access to your whole bankroll, but you may need 100-150 units.
It is wise to limit your top bet to about 1/100 of your total bankroll,
but you can get away with slightly more for very high true counts.
Of course, if you understand the risks and are willing to accept
higher risks by using fewer units, more power to you.
Playing two hands at once is a often a good idea. You can afford to bet
around 30% more on two hands (combined) than on one hand, and
so you will make more money per hour without increasing your variance.
To play two hands, each bet must be twice the table minimum at
most casinos. Play the first hand before looking at the second,
except when the dealer has an ace showing and you are permitted
to look at both hands to decide if you want insurance.
After you have mastered basic strategy and betting according to the
count, you can bet more accurately if you keep a side count of aces.
For each excess seen ace, temporarily subtract 1 from the running count
- if there are excess remaining aces, add 1 to the running count.
Then compute the true count for betting. For example, suppose you
have seen the first 1/4 deck, but have not seen any aces. This
means there is 1 excess remaining ace, since you normally expect
to see one ace for every quarter deck (since there are four aces
in a deck). So, you would adjust the running count by +1 for betting
purposes.
Strategy adjustments are another improvement. Here you deviate
intelligently from basic strategy, according to the true count.
The strategy adjustment table is given and explained in another part of
this article. The strategy adjustment table is refinement; you get
most of the benefit of counting from bet size variation, and you
should do fine if you avoid strategy adjustments at first. The
strategy adjustments will just buy you an extra 0.4% or so.
The multiparameter strategy tables for side counts are a refinement on a
refinement, and should only be attempted by counters with considerable
experience with basic Hi-Opt I. They'll buy you an extra 0.1% or so.
Here is what a card counter looks at to rate a blackjack game, in rough
order of their importance:
1. NUMBER OF DECKS. The fewer the better.
2. PENETRATION. The % of cards dealt before the shuffle is very important.
The number of spots being played can impact the penetration.
3. RULES. A near zero or positive basic strategy expectation helps.
4. MINIMUMS/MAXIMUMS. The game must be affordable and profitable, and
the player's betting should not be too much bigger than the minimum,
lest he attracts unwanted attention.
5. CONTENT DEALERS/PIT CRITTERS. It helps if the employees are happy
in general and happy about your business too.
The importance of playing single deck blackjack with good penetration
cannot be overstressed. Sneak some extra effective penetration
by sitting towards third base if you are using strategy adjustments -
your adjustments will be more accurate since they will be made after
seeing more cards, raising your advantage significantly.
The maximum edge that you'll hear knowledgable card counters claim to attain
in practice is about 1.5%. Around 0.9%-1.1% is more realistic, winning an
average of about 1.5 units per 100 hands played. Figure on getting in
about 100 hands per hour, so that's 1.5 units per hour you'll make. For
typical blackjack games, a 1-4 betting spread is sufficient to beat
the game with a good profit margin. Even flat betting will produce
a profit, though probably not enough for the average greedy card
counter. If the game has poor penetration or poor rules and a better
game cannot be found, then a betting spread larger than 1-4 will probably
be necessary.
As far as risk goes, the variance on a hand of blackjack with a
1-4 spread is about 15.5. The expected value will form a normal
distribution, as shown on the next page. Study these distributions
to get a feel for the kind of negative negative swings you will
experience, purely as a result of variance.
The game simulated here is Vegas downtown rules, 3 rounds of 3 spots,
with the counter at the last spot using Hi-Opt I, strategy adjustments
-6 to +6, ace adjustment for betting, true count accurate to
a quarter deck, 1-4 betting spread.
20,000 hands with 300 unit bankroll
Player #3
Average winnings: 291.90 (variance 15.547404)
Average action: 31582.54 (variance 6.373246)
Average worse loss: -105.47 (variance 31339.356156)
Average advantage: 0.924258 +- 0.072904 % (variance 1.383540)
Win/100 rounds: 1.51
FINAL BANKROLL DISTRIBUTION CUM %
=============== =================================================== ==========
BUSTED |********************************************* 7
1 - 50 |* 7
51 - 100 |* 7
101 - 150 |******* 8
151 - 200 |************* 10
201 - 250 |************* 12 LOSS
251 - 300 |********************* 16________
301 - 350 |************************ 19 PROFIT
351 - 400 |************************* 23
401 - 450 |***************************************** 30
451 - 500 |************************************ 35
501 - 550 |************************************************** 43
551 - 600 |*********************************************** 50 --MEAN
601 - 650 |***************************************** 57
651 - 700 |****************************************** 63
701 - 750 |******************************************* 70
751 - 800 |**************************************** 77
801 - 850 |************************************* 82
851 - 900 |***************************** 87
901 - 950 |*********************** 90
951 - 1000 |************* 92
1001 - 1050 |*********** 94
1051 - 1100 |************ 96
1101 - 1150 |******** 97
1151 - 1200 |****** 98
1201+ |************ 100
Initial bankroll: 300.00
Mean final bankroll: 591.90
In a day or two, a player might get in 2,000 hands. With an 80 unit
bankroll, the results are shown below:
2,000 hands with 80 unit bankroll
Player #3
Average winnings: 29.71 (variance 15.577170)
Average action: 2860.31 (variance 6.363973)
Average worse loss: -44.59 (variance 3387.431404)
Average advantage: 1.038872 +- 0.317410 % (variance 26.225822)
Win/100 rounds: 1.70
FINAL BANKROLL DISTRIBUTION CUM %
=============== =================================================== ==========
BUSTED |************************************************** 24
1 - 13 |* 25
14 - 27 |*** 26
28 - 40 |**** 28
41 - 53 |****** 31
54 - 67 |******* 35 LOSS
68 - 80 |********** 40_______
81 - 93 |********* 44 PROFIT
94 - 107 |*********** 50 --MEAN
108 - 120 |*********** 55
121 - 133 |******* 59
134 - 147 |*********** 64
148 - 160 |*********** 70
161 - 173 |********* 74
174 - 187 |********* 78
188 - 200 |********** 83
201 - 213 |***** 86
214 - 227 |******* 89
228 - 240 |******* 92
241 - 253 |***** 95
254 - 267 |**** 97
268 - 280 |** 98
281 - 293 |** 99
294 - 307 |* 99
308 - 320 |* 100
321+ |* 100
Initial bankroll: 80.00
Mean final bankroll: 109.72
Note: different scale than previous graph.
Team Play
=========
If you have studied the above graphs, you may be getting depressed,
given that the size of your bankroll may imply you will make below
minimum wage. Don't give up hopes of making money with blackjack
yet. The way to do it is to form a team.
Several players can pool their bankrolls and bet as each had the
whole bankroll. If there are N players contributing equally,
then they will each make N times as much together than they would
individually. Or they can make slightly less and reduce the
variance significantly.
There are also schemes of cooperative play at the same table,
but it's usually best to have the players play at separate tables.
Barring and Countermeasures
===========================
Card counting is not illegal. However, the casino can kick you
out for whatever reason they choose. If they read you the Trespass
Act, *then* it will be illegal for you to return to the casino
and they can have you arrested. In extreme cases, casinos have
been known to break the bones of card counters, but if you are
playing for low stakes at a reputable casino, you shouldn't have
any such physical problems.
There are many things that a casino can do besides kick you out
to make it not worth your while to stay. They can shuffle the
deck any time the cards favor you, which will cost you 1-2% in
advantage, making the game unbeatable. Even if they are not
card counting, the dealer can simply shuffle any time a large
bet is placed. Or they can simply stop dealing deep into the
deck altogether.
When the pit critters (properly known as floor managers or sometimes
a pit boss) are hanging around your table, eyeing you, looking
through the discards or obviously card counting during play,
telling the dealer to shuffle early, restricting your betting
spread, etc. then this is called "heat".
Since heat results in poor playing conditions and may preceed
a barring, you should try to avoid it. Don't look like you
are counting cards! Become good enough at card counting that
you can simultaneously carry on a conversation. Talk to the
other players, the dealer, and the pit critters. Mix up your
betting pattern. Don't always bet the same at the top of
the deck. Limit your betting spread, as spreads wider
than 1-4 are usually not necessary to have a good advantage
and wider spreads will usually not be tolerated by the pit
critters. You can make an okay profit with even a 1-2 spread,
though 1-4 should be your goal. Use a lot of different colors of
chips, representing different dollar amounts, so that a pit critter
cannot at a glance figure out whether you raised or lowered your
bet from the previous hand. Try switching back between 1 and
2 hands to range your bet that way, unless this causes heat
itself.
Many casinos like for you to "color up" when you leave. This
means to exchange your chips for ones of higher denominations,
making it easier for you to carry, and trivial for the pit critters
to count. It's a good idea to leave a table with no chips, if possible.
You can accomplish this without losing by *discretely* squirreling chips
into yourclothes during play. Even if you can't get rid of all your chips,
your coloring up will be less embarassing. If you run out of chips
on the table before you're ready to go, pull out more cash. Make
sure you do color up if you leave with only a few chips on the
table - you want them to know that you took a "loss".
If you have been squirreling chips, then be discrete about cashing
them in. Only cash in initially what you had showing on the table.
Come back later to cash in the rest, or try to get rid of the chips
at another casino.
Comps
=====
You can generate extra low variance income via "comps" - complimentary
rooms, food, and other hotel services. Four hours of betting $25
minimums is enough to get a free room at most casinos in Vegas.
Lower levels of betting will get you a free meel. Even higher levels
of betting will prompt the casino to comp *everything* - room,
food, and even your plane tickets.
To be eligible for comps, just ask a floor manager to "rate" your
play. He will record your buy-in, betting level, and color-out
(chips you take with you.) After you have played for a while,
ask a floor manager or the pit boss for a meal or whatever.
This is a great way to reduce expenses and hence essentially get
some guarenteed income. Unfortunately, it's also a great way to
get nailed by the casinos, since you'll be in their computers,
and it will take just one pit critter comment "card counter"
appended to your record to ruin your play there until they
forget your face and you use a new name.
Further reading for the aspiring card counter
=============================================
Although this article gives you enough information to make money off
the casinos, I recommend purchasing at least two of these three books:
"Blackbelt in Blackjack" by Snyder
"Fundamentals of Blackjack" by Chambliss and Rogenski
"The World's Greatest Blackjack Book" by Humble & Cooper
These are available from the Gambler's Book Club in Las Vegas,
(800) 634-6243. You have a decent shot at finding the last book
in your local book store.
"Blackbelt in Blackjack" has a couple of good counting systems,
Red 7's and Zen. In addition, Snyder gives many good suggestions
for achieving an advantage in single deck blackjack without looking
like a typical counter.
"Fundamentals of Blackjack" is a new book with lots of useful tables.
The counting system (C&R count) is not recommended, however.
"The World's Best Blackjack Book" focuses on the Hi-Opt I
counting system that I advocate. It has lots of general information
that any card counter should know, though the authors of this book are
a little too paranoid about getting cheated.
Good luck and on to the tables! The strategy tables, that is. See
the next two articles.
HOW TO BEAT SINGLE DECK BLACKJACK
Version 1.01
Copyright 1991, Michael Hall
Part 1: The Basics
----------------> Part 2: About the Strategy Charts
Part 3: The Strategy Charts (LONG)
Description
===========
This article describes basic strategy and Hi-Opt I strategy tables
for single deck blackjack, Vegas Strip rules. The strategy information
was compiled from a variety of sources, including "The World's
Greatest Blackjack Book", Steve Markowitz's analytic strategy adjustment
program, and my empirical strategy adjustment program.
Caveats
=======
I do not guarantee that these tables are correct. If you find any
mistakes, or have any suggestions, please let me know, and I will
repost if necessary.
How to read the Basic Strategy table
====================================
Cross index your hand with the dealer's face-up card. If there is
something other than "...", it means "yes, do the corresponding decision"
- conversely, a "..." means "no, *don't* do the corresponding decision."
Read from the bottom up. First see if you should split ("spl"), then double
("dbl"), then stand ("sta"). If nothing applies, then hit.
For example, suppose you have two 8's, and the dealer has a 10
showing. You first check splitting - the table shows that you always
split 8's, since it has an "spl" for every dealer up card. As
another example, if you Ace-7 vs. Ace, you first check to see if you double.
There is a "..." there, so you don't double, and so you check to see if
you should "soft hit" it. Ace-7 is soft 18, which when cross-refenced
with the dealer's ace has a "sta", indicating you should stand on versus ace.
How to read the Hi-Opt I Strategy table
=======================================
Cross index as with the basic strategy table. Follow the basic
strategy, except possibly when there is a number there. If there
is a number and the true count is greater than or equal to it,
it means yes, do the corresponding action. Exception: when there
is an asterisk (*) after the number it means just the opposite -
do the decision only when the true count that number or below.
For example, using the previous example, you would deviate from basic
strategy and hit soft 18 vs. Ace if the true count were below
zero, since there is a 0 in that box. For counts of zero and above,
you would stand. Note that the strategy with a count of zero is
not always the same as basic strategy - use the strategy adjustments
in preference to basic strategy. As another example, suppose
you have 8-8 vs. dealer 10 (no double after split). The number there
is "+5*", meaning that for true counts of 5 or less, you split, but
for true counts of +6 or more, you would not split. (Exercise for
the reader: what play would you do in that case of a true count of +6?)
Most of the numbers are rounded off to -20,-15,-10,-5,-1,0,+1,+5,+10,+15,
or +20. In the September 1991 issue of Blackjack Forum, Arnold Snyder
showed that such a simplication resulted in no measurable loss in
performance. However, I have included a finer granularity of numbers
for hard standing, because my empirical results produced statistically
significant settings of these adjustments to their exact values.
Whether this will make a statistically difference in the overall results
of a few million hands, I rather doubt, but since it definitely makes
a difference in the results for these particular hands, I can't bear
to throw away that gain. If you wish, you may round the rest
of the numbers off the multiples of 5 (or +1/-1) to make things easier.
Learn the adjustments gradually, starting with 0, then adding +1, then
-1, then +5, and so on. The most important adjustments are the ones for
hard standing and doubling. You can and probably should completely ignore
the ones for splitting, except for the hand 10-10.
How to read the side count strategy tables
==========================================
The side count tables are for experts only. They are memorized in
addition to the normal Hi-Opt I strategy adjustment table. The best
order to learn the side counts is given in the first part of the
article in the table that compares the various card counting systems.
First learn to side count aces, then add 7's and other side counts as you
becomes a grandmaster of blackjack.
As far as I know, this article is the only place you can easily
multiparameter Hi-Opt I in its full glory. No book I've seen gives
Hi-Opt I strategy adjustments outside of -6 to +6, and no book
gives multiparameter adjustments for anything other than aces.
(Currently just aces and 7's are here, but I'll add the other cards
in later editions of this article.)
You find the side count strategy adjustment number the same way you find
the Hi-Opt I strategy adjustment number, by cross-indexing on the
appropriate table.
For strategy, you adjust the running count by the number of excess seen
cards for the side count times the side count strategy adjustment number.
As an example of how to use the side count adjustment number, suppose
you are side-counting 7's and are faced with 14 vs. 10 and a running
count of -1 at the 1/2 deck level. The Hi-Opt I strategy adjustment
number for standing hard 14 vs. 10 is +15, and the seven adjustment
number for this play is +5. Suppose you have seen all four 7's
remaining at the 1/2 deck level, meaning that there are two excess
7's removed from the deck. For strategy, you then must adjust the