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Can Baccarat be Beaten

By

Card Counting

George Joseph
Las Vegas

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CAN "BACCARAT" BE
BEATEN BY CARD COUNTING
Both Baccarat and the game of 21 (Blackjack) are dealt from decks of
cards which are not shuffled after every hand. This fact creates dependent
gambling trials. The known cards played in previous hands give an indication of
the relative make-up of the cards remaining to be played. Casino games such as
Craps, Roulette and the Big Six are games of independent trials. The last number
the ball landed on in a Roulette Wheel does not affect any other spin of the
wheel....no one spin of the ball is dependent on any other spin. The same holds
true in a crap game. The odds of rolling an eleven are 2 chances out of the 36
possible two number combinations that can be made with a pair of dice. The
eleven can be made with a 6 on one cube and a 5 on the other or vice versa. Any
other numbers rolled before an eleven shows are inconsequential to the odds of
throwing an eleven. The odds are 2/36th or 1/18th....18 to 1 against....This is an
independent event.
No matter how many elevens are rolled in succession the odds for the next
eleven are still 18 to 1 against. Although independent events for games like
Blackjack can be computed, they are just theoretical mathematics. Take for
example the chances of being dealt a Blackjack in a single deck game. The
mathematics for this event looks like this.....

(2 X 4 X 16 ) = 128 = 0.0482654 = 0.0483

(52 X 51) 2652

The math is very straight ahead.....The "2" implies that an Ace or Ten could
be had on either the first or second card.....The "4" is the number of Aces in a
single deck.....The "16" is the number of Ten value cards....."52" is the full deck
of cards and "51" is the number of cards after the first card dealt. The probability
of being dealt a Blackjack is 0.0483 or 4.83%. This means that on average a
Blackjack will occur once every 21 hands. That’s the theoretical mathematics of
an independent trial or event. Suppose though in the real world that in the first go
round of play in a single deck game all four aces are dealt.....with no Blackjacks.

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The odds now are "zero" of a Blackjack being dealt until after the next shuffle-up.
Conversely, let’s suppose that twenty six cards are dealt from a single deck and
no aces or tens are dealt. The odds of a Blackjack occurring are now dramatically
increased. The math looks like this.....

(2 X 4 X 16) = 0.1969231 = 19.69%

(26 X 25)

The odds of being dealt a Blackjack have increased in this situation to

approximately 19.69% or about 1 chance in 5. The dependent events (dealing
26 cards without an ace or ten showing) have effected the independent question
of the probability of a Blackjack being dealt. For your information, a hidden
card count computer used by Blackjack players make these types of calculations on
an ongoing basis during play.
There are several so called betting "systems" which purport to offer an
advantage in casino gambling games. Given that most games are a series of
independent trials I prefer to call these systems....betting methods. The simplest
of all betting methods is the Martingale. The method calls for you to double up
when you lose. A $5.00 loss means a $10.00 bet on the next wager. A loss of
that $10.00 bet means a $20.00 wager on the next bet and so on. The problem
with any type of Martingale method is the customer, attempting to win the amount
of the first wager in the progression, runs into the table limit.
Extending, the above Martingale progression would look like this;

# 1 2 3 4 5 6 7 8 9 10 11
$ 5 10 20 40 80 160 320 640 1280 2560 5120

On a game with a $5,000.00 limit the customer could not make the 11th bet in
the progression. Keep in mind that the player has already lost the sum total of
the first ten bets...($5,115.00)....he would need to risk an additional $5,120.00
wager in order to win $5.00. On a $5.00 to $500.00 game, you couldn’t make the
eight wager, $640.00. (You can expect to lose 7 times in a row approximately
once every 87 hand sets…approximately once every 7 hours)

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A cancellation betting method is a little more intriguing. A series of numbers is

used such as the following;

10 20 40 60 80

The first bet is the sum of the two outside un-cancelled numbers.............
in this case......10 + 80 = 90. If the bet should lose, the two outside numbers are
canceled and their total is added to the end of the list.

(10) 20 40 60 (80) 90

The next bet is the sum of the two outside un-cancelled numbers.......in this
case.....20 + 90 = 110....and so on.

The ability to remember and correctly negotiate simple addition may make
the user of this method a "wonderful forth grade student"....but it in no way affects
the future outcome of an independent trail game like Roulette or Craps. These
types of betting methods give the inexperienced player a sense of security. They
are not just guessing, but rather have a method to determine their betting strategy.
The betting strategy, (however simple or complex) does not evolve from any
dependency on previous events or gambling trials. Adding two numbers or
doubling the previous bet does not affect the 5.26 percentage of advantage against
a player on a roulette table. In reality the player is simply placing a larger bet with
the same mathematical disadvantage.
In games of dependent trials such as twenty-one, the doubling of a losing bet
may, in fact, put the wager at greater risk because of the cards previously played.
(The betting method may call for you to double a wager at a time when the
remaining cards have a positive expectation for the house.) The amount of the
wager is not arrived at by any mathematical involvement in the gambling process.
A player using a betting method feels secure because they add structure to an
otherwise random pattern of attempting to "out guess" the gambling game or
"hunch" bet.

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The game of twenty-one offers a mathematical model by which all other

games of dependent trials can be evaluated. Twenty-one card counting very
simply runs a ratio of big cards (10's and aces) to little cards (2,3,4,5,6). When the
ratio is lopsided in favor of big cards, meaning an abundance of big cards
remaining in the shoe to be played, the player has the advantage. Mathematically
speaking, the player and dealer have the same chance of receiving the same first
two cards and thereafter achieving the final totals. The advantage to the player is
derived because of the 3 to 2 payoff for Blackjacks, (obviously more plentiful due
to the abundance of big cards.) Double Down wagers and Insurance bets are also
enhanced by higher card values. Also because the house hits last and must fall into
a window of between 17 and 21, the big cards have a tendency to take the count
over 21. When the ratio of big cards to little cards is lopsided in favor of little
cards, meaning an abundance of little cards remaining in the shoe to be played, the
house has the advantage. There will be far less Blackjacks due to the abundance of
little cards. The Double Down and Insurance wagers are now less profitable for the
players. Because the house hits last and must fall into a window of between 17 and
21, the little cards have a tendency to keep the count between 17 and 21. Very
simply, when the count is plus (big cards left) the player makes "plus" bets,
conversely, when the count is minus (little cards left) the player makes "minus"
bets. Any count between a plus and minus is considered a "flat" count and
indicates "flat" bets. The plus/minus count is a system which is dependant on the
gambling process. The card counters’ playing and betting strategies are
determined by a direct correlation with the cards previously seen in play. This is
an over simplification of the complex question of card counting. However, the
theory of card counting says that a player will lose more hands in any given play
than they win. Because the card counter recognizes the mathematically
advantageous makeup of the remaining cards, larger wagers are won more often.
Understand that the dependent nature of the game of twenty-one allows for a card
count system and basic strategy method of play to be effective in identifying the
relative strength of dependent trials that occur throughout the model. In the
previous example where 26 cards were played and no "Aces" were dealt, the
probability of a Blackjack being dealt increased dramatically. The net result
however does not impact both hands (the dealer and player) equally.

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The dealer and player each have the same mathematical chance of achieving
the Blackjack…(without cheating, no system can determine which hand will
receive which cards.) When however, the relative make up of the remaining cards
is "Ace Rich", the player has the advantage in terms of money. If the dealer
receives the Blackjack the house takes "Even" money from the player. Under the
same conditions when the player receives the Blackjack they are paid 3 to 2......
called "Time and a Half". When the dealer receives two aces he cannot split his
hand and take twice the player’s money. The player however being dealt two aces
can split the pair and double the money with two strong cards. The same analogy
holds true for doubling down on soft hands. The probability of the independent
event is the same for both sides but the relative make up and strength of the
remaining cards ("Ace Rich" configuration) allows the wager to be exploited by
the player.
The question is, given that baccarat is a game of dependent trials, shuffled and
dealt similarly with eight decks, does a card count or betting system exist which
identifies the mathematical advantage at any time for either the player or banker
sides and allows that advantage to be exploited.
As you know, there is no time in the game of Baccarat where-in the
customer or house ever has the option of asking for or refusing a card. The rules of
the game call for the cards to (in effect) play themselves. This is known as a
strategically static game…as opposed to Blackjack which is a strategically
dynamic game for the customers. The minimum number of cards that will be
dealt in any one Baccarat hand is four (4)....the maximum number of cards that can
be dealt is six (6). It would be a monumental task to attempt to analyze all
possible six (6) card subsets for an eight (8) deck Baccarat game.
The total number of possible six card subsets is as follows;
(416) ! = 6,942,219,827,088
(6)
Further, each subset would have to be looked at in all possible combinations;
that looks like this...........
(6) x (4) x 2 = 180 x (6,942,219,827,088)
(2) x (2)

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The above calculations of course take into account the four different suits
and the card values 10, jack, queen & king....all of which are irrelevant to the game
of Baccarat. The simplest of all six (6) card subsets to consider is one in which all
six cards are the same value. Suppose that the six remaining cards to be dealt in a
Baccarat hand were 2 - 2 - 2 - 2 - 2 - 2. You would of course know that the
resulting hand would be a "Tie"....6 - 6.
It is however, an academic question to analyze six card subsets because this
assumes there are times when a Baccarat shoe will be dealt down to six cards.
This is an unlikely (almost impossible) occurrence except through dealer error. If
however a six card subset was encountered, (without knowing the exact order of
the cards) then the following chart represents the advantage gained for each side;

SIX CARD BACCARAT SUBSETS

WAGER CHANCE IT IS AVERAGE EXPECTATION PER

FAVORABLE EXPECTATION HAND PLAYED (%)
"PLAYER" .150967 3.20 .4831

"BANK" .270441 3.26 .8818

"TIE" .339027 72.83 24.6909

Understand the above chart makes the very broad assumption that a dealer
will mistakenly or intentionally deal past the cut card. There is still no guarantee
that a six card subset will be produced.....given that the "Pad" created in a Baccarat
shoe is fourteen cards, the average ending subset is between eight and fourteen
cards.

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For your information I am including the relative expectations for various

numbers of card subsets. They are as follows;

PLAYER & BANKER BETS

(COMBINED)

NUMBER CORRELATION OPPORTUNITY

OF CARDS (%)

10 .64 .24(.07)

13 .74 .12(.04)

16 .78 .09(.02)

26 .89 .03(.004)

As you can see, a relatively small increase in the number of cards in a

given subset dramatically reduces the percentage of opportunity to engage any
type of mathematical advantage however derived......(intuitively or via the use
of a hidden computer.) Also understand that from a purest standpoint the
above numbers vary slightly from Player to Banker. The variance is so minute
as to be insignificant to the overall question so the set of numbers above can be
considered valid for both sides.
The number of cards in a given subset is only one issue to be considered
in the Baccarat/Card Counting question. Far more significant and practical are
the card values played and what (if any) indications they give about the relative
make up of cards left to be played. As you know, in a 21 game card counters
use a weighted numeric index to assign value to each card relevant to its potential
strength or weakness as it regards the dealer’s possible hand totals.

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A simple weighted scale is used to maintain a ratio of big cards to little cards.
Basic, mid-level and advanced 21 card count scales are represented in the table
below;

CARD COUNT VALUES

2 3 4 5 6 7 8 9 10 ACE

Basic 1 1 1 1 1 0 0 0 -1 -1

Mid Level 2 3 3 4 3 2 0 -1 -3 -4

Advanced 5 6 8 11 6 4 0 -3 -7 -9

As each card is played in a 21 game the numeric value (for which ever
system) is added or subtracted, starting from zero, to gain a simple "Running
Count". The count continues from hand to hand and the relative strength of the
remaining shoe is determined. The "Plus" or "Minus" value arrived at also
indicates the amount of the next wager for a card counter. The simple running
count is converted to the "True Count" by dividing the number of cards left to be
played into the simple running count. This conversion from running count to true
count takes into account the number of cards left to be played. (A running count of
+6 is much stronger with only two decks left to be played versus the same running
count of +6 with five decks remaining. Therefore the conversion process from
simple running count to true deck count.) For your information every 0.5 point
of true count indicates one unit of wager.....a true count of +3 equals a 6 unit bet.
A card counter needs a True Count of +3 or greater in a six (6) deck game to
increase his wager. (Note: For every +1 point of True Count the player gains
0.5% advantage.)
Similar numerically weighted indexes have been developed for the game
of Baccarat. The most advanced work being done by Peter Griffin and Ed Throp.
The following table shows the "Ultimate Point Count" for Baccarat.
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Card Value Player Bet Banker Bet Tie Bet

Ace -1.86 1.82 5.37
2 -2.25 2.28 -9.93
3 -2.79 2.69 -8.88
4 -4.96 4.80 -12.13
5 3.49 -3.43 -10.97
6 4.69 -4.70 -48.12
7 3.39 -3.44 -45.29
8 2.21 -2.08 27.15
9 1.04 -.96 17.68
10,J,Q,K -.74 .78 21.28
________________________________________________________
Full Shoe % -1.23508 -1.05791 -14.3596
The Baccarat count is used in much the same manner as a 21 card count.
The significant difference is that both the Bank and Player sides can be evaluated
but with different numeric values. Let’s suppose the first hand of a Baccarat shoe
was a 3 & 4 for the Player’s side and a 9 & Jack for the Bank. The simple running
count for the Bank side expectation is as follows;

3 4 9 Jack
2.69 + 4.80 - .96 + .78 = +7.31

Although +7.31 seems to be a significant positive advantage for the next

Bank side wager, the simple running count needs to be adjusted for the number
of cards left to be played and the Full Shoe % expectation in the table above.
The true count expectation is as follows;

-1.05791 + (2.69 + 4.80 -.96 +.78) / (412) = -1.04016% (Bank Side)

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Similarly, the Player’s Side expectation can be estimated;

-1.23508 + (-2.79 - 4.96 + 1.04 - .74) / (412) = -1.25316% (Player Side)

The expectation for the Tie bet in this example would be;

-14.3596 + (-8.88 -12.13 + 17.68 + 21.28) / (412) = -14.3160% (Tie Bet)

The number 412 used in the above examples is the number of cards left to
be played after the four cards used in this first hand. (Eight decks of cards = 416
cards....minus the four cards played = 412.) This number is slightly misleading
in light that at the beginning of a Baccarat game the first card of the shoe is turned
face up and whatever its’ value, a similar number of cards are "Burned".....
discarded without their value being seen. In all of the above examples even though
the Baccarat "Running Count" was high, the "True Count" still showed a negative
expectation for all three bets.
In order to put the question to rest, similar evaluations can be made for the
most advantageous, (yet nearly impossible) condition. The cards whose removal
yield the strongest positive expectation for the Player side wager are the fives,
sixes and sevens. (In an eight deck shoe there are 32 each of the fives, sixes and
sevens.....96 cards total.) Assuming that the first 96 cards dealt were all of the
fives, sixes and sevens......depleting the eight decks of all 5's, 6's and 7's.....the
expectation for the next Player Side wager would be as follows;

-1.23508 + 32(3.49 + 4.69 + 3.39) / (320) = -.078%

Obviously the conditions above could never occur in actual play. The
calculations show however the futility of counting down even the most
advantageous conditions and expecting positive results.
Assuming the opposite side of the argument, which is using the count to
determine the least negative expectation rather than waiting for a positive count
to occur yields similarly futile results.

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Taking this posture would improve the negative expectation on the Bank
Side wagers by an average of 0.09%. This reduces the negative expectation on
the Bank Side from -1.06% to -0.97%.
Understand this proposition implies that you place the same average wager
on every hand, make all the necessary calculations and chose the side with the least
negative expectation.....and still you will lose at a rate almost equal to "Pure
Chance" minus approximately 1.0%.
(From an academic standpoint, a hidden body worn computer could easily be
utilized to perform the necessary calculations. The use of any electronic device
to predict the future outcome of a gaming event is of course illegal. From a
practical standpoint however because of the speed at which a Baccarat game is
dealt...(very slowly)...the same calculations could be made by hand on the
scorecard supplied to each customer as they are seated. Many Baccarat shuffle-up
scams have been accomplished by recording cards with nothing more than the
scorecard and pencil the house supplied.)
All available data shows that the ultimate point card count for Baccarat is
worthless under virtually all conditions to predict a positive expectation for the
Tie Bet.
From the absolute purest point of view, the numerically weighted Baccarat
point count can indeed identify those times when a positive expectation can be
gained. Below is a representation of the Baccarat Card Count program I
developed for use in analyzing Baccarat wins or losses in a casino;

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******************************************************************
<<<< B A C C A R A T C A R D C O U N T >>>>
Developed by George D. Joseph
Enter Cards…0,1,2,3,4,5,6,7,8,9…As You See Them Played
Press (S) For Reshuffle…..Press (E) To End
******************************************************************
RUNNING COUNT
Player Banker Tie Shoe Card Dealt Cards Played
-7.45000 7.31000 17.95000 412 0 4
******************************************************************
TRUE COUNT
Player Banker Tie Shoe Card Dealt Cards Played
-1.23516 -1.04017 -14.311603 412 0 4
******************************************************************

(Next…70 cards played….with an abundance of 10’s & 1,2,3 played)

******************************************************************
<<<< B A C C A R A T C A R D C O U N T >>>>
Developed by George D. Joseph
Enter Cards…0,1,2,3,4,5,6,7,8,9…As You See Them Played
Press (S) For Reshuffle…..Press (E) To End
******************************************************************
RUNNING COUNT
Player Banker Tie Shoe Card Dealt Cards Played
-108.36000 107.27000 514.64030 346 3 70
******************************************************************
TRUE COUNT
Player Banker Tie Shoe Card Dealt Cards Played
-1.54826 -0.74788 -12.87220 346 3 70
******************************************************************

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As you can see, the Running Count does in fact show a Bank Side
advantage, however the True Count still shows a negative expectation for both
sides. (I suppose you could look at the proposition as, “I know when I have less
than the theoretical disadvantage.”) (Next see an anomaly, Bank Side positive;)

(228 cards played….with an abundance of 10’s & 1,2,3 played)

******************************************************************
<<<< B A C C A R A T C A R D C O U N T >>>>
Developed by George D. Joseph
Enter Cards…0,1,2,3,4,5,6,7,8,9…As You See Them Played
Press (S) For Reshuffle…..Press (E) To End
******************************************************************
RUNNING COUNT
Player Banker Tie Shoe Card Dealt Cards Played
-204.37010 204.10010 1307.452000 188 4 228
******************************************************************
TRUE COUNT
Player Banker Tie Shoe Card Dealt Cards Played
-2.32216 +0.02773 -7.40507 188 4 228
******************************************************************
However a major assumption must be conceded for this to be the case.
The casino would have to allow a customer to sit out for long periods of time
without wagering at all. The customer would utilize the point count and wager
only at those rare times when a positive expectation is calculated for either the
Player or Bank side. The average expectation gained under these unrealistic
conditions would be approximately 0.07%. Assuming a one hour time frame to
deal a Baccarat shoe and also assuming a $1,000.00 bet the net profit from this
proposition equals 70 cents per hour. Extensive simulations further suggest that
theoretically the positive expectations under these conditions may only arise three
times out of every eight shoes. A player would then sit at the Baccarat table and
place three wagers in eight hours with a net expected return of $2.10 for the three
$1,000.00 wagers.
Far more dangerous are the many attempts at cheating at Baccarat and the
various forms of advantage play which will be addressed in future papers.
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