On 19 Nov., 12:40, Lastwa...@googlemail.com wrote:

The predictability of Gambling
Author: Dieter Selzer-McKenzie
The Roulette-Wheels WurfWeiten Kesselgucken
It doesn't require an extensive mathematical back¬ground to look at
the 38 identically-sized spaces on an American roulette wheel (note
the 35-1 payoff on a single number) and conclude that the game is
unbeatable. With a 1138 chance of having a number come up on the next
spin and the 35-1 payoff, it is easy to calculate the often-quoted
expectancy of the player of -5.26. The odds for other wheels,
especially the Wheel of Fortune, appear even more against the player.
The unbeatability of the roulette wheel is based on the mechanical
perfection of the wheel--such a con¬clusion is based on the assumption
that the ball has an equal chance of landing in each pocket. This may
or may not be true, although Allan Wilson, in The Casino Gambler's
Guide, and others give fairly convincing evidence for the existence of
biased wheels--wheels sufficiently biased to overcome the house
advantage.
The very mechanical perfection of the wheel, however, would suggest
the applicability of the laws of physics to prediction of the next
number, whether the game is roulette or the Wheel of Fortune. Just as
the future position of a planet can be predicted quite accurately, so
can an understanding of the physical laws at work minimize the
uncertainty surrounding the resting place of the ball or the final
position of the wheel.
It is not possible, of course, to obtain an exact predic¬tion. But
this is not absolutely necessary to assure a profit. As Marvin Karlins
has pointed out in his book Psyching Out Vegas, "Simply being able to
predict which half of the wheel the ball will plunk into would give
the player such a whopping edge that he could go for the
chandeliers...and make it"
The following two chapters investigate the promise of this approach to
beating the wheel as well as discussing some of the difficulties that
might arise im¬plementing such a strategy in the casino environment
It was the spring. I was finishing my second year of graduate physics
at U.C.L.A. In the course of the next year I would make three
decisions that would shape my life for the next 28 years. I married
(my present wife, Vivian), I changed my field of study from physics to
mathematics, and I began to toy with the fantasy that I could shatter
the chains of poverty through a scientifically-based winning gambling
system.
I was living in Robison Hall, the student-owned cooperative. For $50 a
month and four hours work a week, we got our room and board. I had
lived in the co-ops for nearly six years of undergraduate and graduate
work, on a budget of about $100 a month. Part of this came from
scholarships and, in the early years, I got some help from home. But I
was basically self-supporting like most of the other 200 or so co-op
residents.
I attended classes and studied from 50 to 60 hours a week, generally
including Saturdays and Sundays. I had read about the psychology of
learning in order to be able to work longer and harder. I found that
"spaced learning" worked well: study for an
hour, then take a break of at least ten minutes (shower, meal, tea,
errands, etc.). One Sunday afternoon about 3 p.m., I came to the co-op
dining room for a tea break. The sun was streaming through the big
glass windows. (Robison, designed by Richard Neutra in the '30s, was
very radical for that time. It had so many big sheet glass windows
that it was often called "the glass house.") My head was bubbling with
physics equations, and several of my good friends were sitting around
chatting.
In our mutual poverty the conversation readily turned to fan¬tasies of
easy money. We began to speculate on whether there was a way to beat
the roulette wheel. In addition to me, the group included math majors
Mel Rosenfeld and Andy Bruckner (now professors of mathematics at U.C.
Santa Barbara), Tom Scott, and engineering major Rick Rushall. After
all these years it's hard to be sure of exactly who said what, but we
began the discussion by acknowledging that mathematical systems were
impossible. I'll demonstrate this in a future chapter.
Then we kicked around the idea of whether croupiers could control
where the ball will land well enough to significantly affect the odds.
I will show later that this is impossible under the usual conditions
of the game. (The incredible thing is that logical reasoning could
even be used to settle such a question.) It was a short brainstorming
step to wondering whether wheels were im¬perfect enough to change the
odds to favor the player. Those in the group who "knew" assured me
that the wheels are veritable jeweled watches of perfection, carefully
machined, balanced and maintained. This is false. Wheels are sometimes
imperfect enough so they can be beaten. I had no experience with
gambling, or with casinos, or with roulette wheels, so I accepted the
mechanical perfection of roulette wheels.
But mechanical perfection, for a physicist, means predictability. You
can't have it both ways, I argued. If these wheels are very im¬perfect
the odds will change enough so we can beat them. If they are perfect
enough we can predict (in principle) approximately where the ball will
land. Suddenly the orbiting roulette ball seem¬ed like the planets in
their stately and precise, predictable paths. In
my mind there was that intuitive "click" of discovery that I would
experience again and again. Unknowingly, I had just taken the first
step on a long journey in which I would discover winning systems such
as those for blackjack and for the options market, and I would
accumulate a wealth I never imagined.
One side argued that it is a long way from prediction in princi¬ple to
practical prediction. My group said that, over and over, the story of
science has been a rapid leap from a theoretical vision (E = MC2) to
an unexpected practical result (nuclear power plants). By now our
initial group of people agreed that the idea had merit and might well
work. The novel debate attracted listeners, some of them cynical. They
challenged us to prove the idea worked. The ten minute "study break"
had run into a couple of hours. We adjourned with the half definite
idea of "doing something."
In the following weeks the idea kept coming back to me: measure the
position and velocity of the roulette ball at a fixed time and (maybe)
you can then predict its future path, including when and where the
ball will spiral into the rotor. (The rotor is the spinning circular
central disc where the ball finally comes to rest in numbered
pockets.) Also measure the rotor's position and velocity at a
(possibly different) fixed time and you can predict the rotor's
rotation for any future time. But then you will know what section of
the rotor will be there when the ball arrives. So you know
(ap¬proximately) what number will come up!
You can see that the system requires that bets be placed after the
ball and rotor are set in motion and somehow timed. That means that
the casinos have a simple, perfect countermeasure: forbid bets after
the ball is launched. However, I have checked games throughout the
world, including Reno, Las Vegas, London, Venice, Monte Carlo, and
Nice. Only in a few cases were bets for¬bidden after the ball was
launched. A common practice instead was to call "no more bets" a
revolution or two before the ball dropped into the center.
The simple casino countermeasure meant that there were two problems:
(1) find out whether exact enough predictions could be
made to get a winning edge, first in theory and then in the casino
itself, and (2) camouflage the system so the casinos would be unaware
of its use. If we could solve the prediction problem, the camouflage
was easy. Have an observer standing by the wheel recording the numbers
that came up, as part of a "system." Many do this so it doesn't seem
out of place. But the observer also wears a concealed computer device
with timing switches. His real job is to time the ball and rotor.
(Much later we settled on toe-operated switches, leaving both hands
free and in the open.) The computer would make the prediction and
transmit it by radio to the bettor. The bettor, at the far end of the
layout, would appear to have no connection to the observer-timer. The
bettor would have a poor view of ball and rotor and would not pay much
attention to them. To further break any link between timer and bettor,
I would have several of each, with identical devices. They would each
come and go "at random."
The important bets have to be placed after the ball is launched. A
bettor who only bet then, and who consistently won, would soon become
suspect. To avoid that, I planned to have the bettor also make bets
before the ball was launched. These would be limited so their negative
expectation didn't cancel all the positive expectation of the other
bets. I became a radio amateur (W6VVM) when I was 13 (back in 1945
when there weren't easy novice-class tests), so I thought I could
build the radio link and other electronic gadgetry.
This left me with the prediction problem to solve. More than a year
passed without much time for roulette: I got my Master's degree in
Physics and wrote the first part of my Ph.D. thesis on nuclear shell
structure (Mayer-Jensen theory). The mathematical problems that I ran
into led me in the fall to take graduate math courses. I needed so
many that I got my Ph.D. in math instead! And early . I got married. I
had been working as a tutor and one of my "students" was T.T.
Thornton. He was an independently wealthy, knowledge-loving bachelor
of about 45, who had degrees in English and chemistry. Now he was
getting a degree in mathematics, just for the pleasure of it. He was
an excellent student who didn't need a tutor but had hired me simply
to learn faster and more efficiently. We shared bits and pieces of our
hopes, dreams, and enthusiasms.
After I had mentioned the roulette project, I was surprised and
touched by his gift of a half-sized wheel. It was black plastic
(bakelite?) made in France. I learned later that it cost the enor¬mous
sum of $25. Though I had thought about the roulette system off and on,
the gift of this wheel (sometime in 1958, 1 recall) got me to work
more seriously on it. My first idea was to use a home movie camera to
film the orbiting ball. I then plotted the amount the ball had
traveled versus the number of the frame of the film. I expected that
the pictures were taken at a uniform rate of 24 (?) frames per second
so I could plot (angular) distance traveled versus time as in Figure
4-1. Instead of a smooth graph like the solid line in Figure 41, my
first film showed a peculiar wavy structure, like the dashed line.
After thinking about this, I guessed that this was because the camera
did not run at uniform speed. By taking a movie of a stop¬watch that
timed in hundredths of a second, I found that the camera did vary in
speed. Photo stores confirmed this. The distortion of the curve in
Figure 4-1 is analogous to the way a musical tone is distorted by a
phono turntable whose speed varies slightly.
My next move was to take a movie of the rotating ball and the
stopwatch. This gave me an accurate time for each frame. But there was
still some "ripple" to the curves. (I later learned that even a slight
tilt would cause this.) Worse, I found that the curves were not
consistent from spin to spin. The situation was something like Figure
4-2. This meant the ball behaved differently from spin to spin. This
meant that the distance it traveled varied even with the same initial
velocity. This doomed predictability on my wheel.
I found with further experiments that my half-sized wheel was really
very irregular. The track was curved like a tube and the ball "rattled
around" erratically, up and down, as it orbited. The slick bakelite
surface was moulded, not machined. The ball also skidded and bounced.
And there was a horizontal junction which added irregularities to the
track.
But full-sized wheels were not like that. In December 1958, I made my
first visit to the casinos. I observed several regulation wheels and
found that the ball moved smoothly in its track. Also the track was a
pair of flat-beveled, carefully-machined surfaces, not a tube. When I
saw how good the casino wheels were, I was more convinced than ever
that prediction was possible. But I needed a full-sized wheel and some
good laboratory equipment to continue. How could I pay for it? I got
my Ph.D. in June of 1958 and was teaching at U.C.L.A. Though my wife
was finally able to stop working, we had no savings and I barely
supported us. I couldn't ask her to go back to work to buy me a
roulette wheel and to finance my pipe dream.
But I persisted. I simulated the study of the problem of whether the
roulette ball would, for the same starting velocity, travel about the
same distance along the track. I set up a little vee-shaped inclined
trough. I would start a marble from a fixed height (a mark on the
trough) and measure how far across the floor it roll¬ed. I was
encouraged but not surprised to fmd that the distance the marble went
could be predicted closely from the starting height.
One memorable evening when my in-laws were due for dinner, I ran
overtime on a marble experiment. They came into the kit¬chen wondering
why I hadn't come to greet them at the door. They found me rolling
marbles down a little wooden trough and across the floor. All over the
floor were little distance markers and pieces of tape.
In early 1959 Vivian and I spent time with Mel and Judy Rosenfeld,
working on a radio link for the casino test of my yet to be completed
roulette system. We took model airplane radio con¬trol equipment and
altered it somewhat. We succeeded in getting a workable but somewhat
inconvenient radio link. Then around March or April , I pushed the
roulette project aside. Twelve man years of blackjack calculations
arrived, courtesy of Baldwin, Cantey, Maisel and McDermott. I had
convinced myself (as described in Beat the Dealer) that I could devise
a winning blackjack card counting system and now I set to work on this
intensely. The impractical marble roller now said he could beat the
casinos at blackjack. What next?
I wrote my blackjack computer programs in the summer and fall of 1959.
Testing, then debugging followed, and then from late 1959 through
early 1960 my computer production runs produced the basic results that
gave me the five-count system in early 1960. Then during 1960 I worked
out most of the ten-count system and the ideas for the ultimate
strategy. I also made the computer runs and worked out the methodology
so that all of today's so-called "one parameter" blackjack systems
could be readily devised by anyone versed in the use of computers. In
December 1960, The Notices of the American Mathematical Society
carried the abstract of my upcoming talk, "Fortune's Formula: The Game
of Blackjack." Life would never be the same again. The intense
professional and public interest aroused by the abstract, even before
the talk, led me to seek quick publication in a scientific journal. I
chose to try the Proceedings of the National Academy of Sciences. I
needed a member of the Academy to communicate.
Claude Shannon: Genius
Shannon, then in his early forties, was and is one of the most famous
applied mathematicians in the world. As one genius among many, he was
relatively unnoticed as a graduate student--until he handed in his
master's thesis. It developed the mathematical theory of switching
electrical networks (e.g. telephone exchanges) and became the landmark
paper in the sub¬ject. After receiving his doctorate, Shannon worked
at Bell labs for several years and then became world-famous for papers
establishing the mathematical foundations of information theory.
I was able to arrange a short appointment early one chilly December
afternoon. But the secretary warned me that Shannon was only going to
be in for a few minutes, not to expect more, and that he didn't spend
time on subjects (or people) that didn't interest him (enlightened
self-interest, I thought to myself).
Feeling both awed and lucky, I arrived at Shannon's office for my
appointment. He was a thinnish alert man of middle height and build,
somewhat sharp featured. His eyes had a genial crinkle and the brows
suggested his puckish incisive humor. I told the blackjack story
briefly and showed him my paper. We changed the title from "A Winning
Strategy for Blackjack" to "A Favorable Strategy for Twenty-One" (more
sedate and respect¬able). I reluctantly accepted some suggestions for
condensation, and we agreed that I'd send him the retyped revision
right away for forwarding to the Academy.
Shannon was impressed with both my blackjack results and my method and
cross-examined me in detail, both to understand and to fmd possible
flaws. After my few minutes were up, he pointed out in closing that I
appeared to have made the big theoretical breakthrough on the subject
and that what remained to be discovered would be more in the way of
details and elaboration. And then he asked, "Are you working on
anything else in the gambling area?"
I decided to spill my other big secret and told him about roulette.
Several exciting hours later, as the wintery sky turned dusky, we
finally broke off with plans to meet again on the roulette project.
Shannon lived in a huge old three story wooden house on one of the
Mystic Lakes, several miles from Cambridge. His basement was a
gadgeteer's paradise. It had perhaps a hun¬dred thousand dollars worth
of electronic, electrical and mechanical items. There were hundreds of
categories, like motors, transistors, switches, pulleys, tools,
condensors, transformers, and on and on. Our work continued there. We
ordered a regulation roulette wheel from Reno and assembled other
equipment including (most important) a strobe light and a large clock
with a second hand that made one revolution in one second. The dial
was divided into hundredths of a second and still finer time divisions
could be estimated closely. We set up shop in "the billiard room,"
where a massive old dusty slate billiard table made a perfect solid
stable mounting for the roulette wheel.
Analyzing the Motion
My original plan was to divide the various motions of ball and rotor
into parts and analyze each one separately. They were:
* The ball is launched by the croupier. It orbits on a horizontal
track on the stator until it slows down enough to fall off this
(sloped) track towards the center (rotor). Assume at first that (a)
the wheel is perfectly level, and (b), the velocity of the ball
depends on how many revolutions it has left before falling off.
Referring to Figure 4-2, (b) means that every spin would produce the
same curve, not different ones like my half-sized wheel. Put another
way, this means that if you timed one revolution of the ball on the
stator, you could tell how many more revolutions and how much more
time until the ball left the track. If these assump¬tions turned out
to be poor, we would attempt to modify the analysis.
* Next analyze the portion of the ball orbit from the time the ball
leaves the track until it crosses from the stator to the rotor. If the
wheel is perfectly level and there are no obstacles, then it seems
plausible that this would always take the same amount of time. (We
later lamed that wheels are often significantly tilted. This tilt,
when it occurs, can affect the analysis substantially. We even¬tually
learned how to use it to our advantage.) There are, however, vanes,
obstacles, or deflectors on this portion of the wheel. The size,
number, and arrangement vary from wheel to wheel.
On average, perhaps half the time these have a significant effect on
the ball. Sometimes they knock it abruptly down into the rotor,
tending to cause it to come to rest sooner. This is typical of
"vertical" deflectors (ones approximately perpendicular to the ball's
path). Other times they "stretch out" the ball's path, caus¬ing it to
enter the rotor at a more grazing angle and to come to rest later, on
average. This is typical of "horizontal" deflectors (ones
approximately parallel to the ball's path).
* Assume the rotor is stationary (not real), and beat that situa¬tion
first. Reasoning: if you can't beat a stationary rotor, you can't beat
the more complex moving rotor. Here the uncertainty is due to the ball
being "spattered" by the frets (the dividers be¬tween the numbered
pockets). Sometimes a ball will hit a fret and bounce several pockets
on, other times it will be knocked backwards. Or it may be stopped
dead. Occasionally the ball will bounce out to the edge of the rotor
and move most of a revolution there before falling back into the inner
ring of pockets. Thus, even if we knew where the ball would enter the
rotor, the "spattering" from the frets causes considerable uncertainty
regarding where it finally stops. This tells you that there is no
possible reliable "physical" method for predicting ahead of time which
pocket the ball is going to land in, unless the wheel is grossly
defective or crooked. That makes the roulette method "used" in the
movie "The Honeymoon Machine," where the players forecasted the exact
pocket, an impossibility. It also tells you that successful physical
prediction can at most forecast with an advantage which sector of the
wheel the ball wilt end in.
* Assume now that the rotor is moving. Generally the ball and rotor
move in opposite directions; increasing the velocity of the ball
relative to the rotor. We'll assume this is always the case. I've
never seen or heard of a casino spinning ball and rotor in the same
direction. If this were done, the relative motion of ball and rotor
would be even less than with a stationary rotor and prediction would
be easier yet. With a moving rotor, the amount of ball "spattering" is
increased and predictability is further reduced. Note that this change
depends on the rotor velocity. Since that varies from time to time and
from croupier to croupier, :7 77 further complexity. It turns out that
the velocity of the rotor changes very slowly, so it is possible to
predict with high accuracy which part of the rotor will be "there" at
the predicted time and place that the ball leaves the stator.
I will now take you through a simplified version of what we first
tried to do. Later, with that overview to guide us, I'll explain some
of the modifications we had to make and describe our casino
experiences.
First, let's consider part 1, the motion of the ball on the track. The
actual function x(t), which describes the number of remain¬ing
revolutions x versus the remaining time t, is theoretically very
complex. *
Our first problem, and the key one, was to predict when and where on
the stator the ball would leave the track. This problem was key
because once we knew this, everything else except rotor velocity was a
"constant." And rotor velocity is easy to measure in advance and
incorporate into the prediction, as we shall see. Our method was to
measure the time of one ball revolution. If the time were short, the
ball was "fast" and had a long way to go. If the time were "long," the
ball was "slow" and would soon fall from the track.
We hit a microswitch as the ball passed a reference mark on the
stator. This started the electronic clock. This was at time ti (to go)
with x1 revolutions to go. (There are many such "marks" available on
all actual casino wheels.) When the ball passed the reference mark the
second time we hit the switch again, stopping the electronic clock.
That was at a time to (left to go) before the ball left the track)
with xo revolutions left. The clock measured
t2, the time T for one revolution (so x1 -- xo = /.)*
Movie Experiments
The function x(t) which we are using in this illustration is not the
actual one. The actual x(I) can be determined by a "movie experiment"
like the ones I described earlier which I did in 1959 on my half-size
wheel. To do this experiment today, get a full-size roulette wheel, a
large clock which reads accurately in hundredths of a second or
better, and a video camera or movie camera. Then take a movie of the
orbiting ball. The successive frames give values for t and x(t), which
can be plotted to get an x(t) curve like that of Figure 4-3. Several
movies should be made to see how much the x(t) curve varies from one
spin to another. This uncer¬tainty is a source of errors in
determining T, that discuss later on. These x(t) errors can be
incorporated into the theory in the same way as the timing errors.
They each cause some uncertainty in the predicted XjT) value. The data
from the movie experiment can be improved if the camera frames are
synchronized to a strobe so that the motion of both ball and clock is
"stopped" rather than blurry. I didn't do this in my original movies,
so I got a short blurry arc, instead of a ball, in each frame.
If an appropriate clock is not available, you can use a high quality
phonograph turntable instead. These rotate at very uniform speeds
which can be verified for your turntable with a strobe. Now get a
stiff paper disc and mark the edges in equal small units. Number these
units (much as you would a "circular" ruler) for ease in reading. Now
place a thin fixed pointer just above the disc. When the disc rotates,
you have a very accurate clock whose hand is fixed and whose face
moves. If you use a paper disc of polar coordinate graph paper (glued,
perhaps, to an old record), there will be 360 equally spaced degree
marks.
At 33'/3 r.p.m., each mark is 1/200 sec. At 45 r.p.m., each mark is
1/270 sec., and at 78 r.p.m., each mark is 1/468 sec. On a 12-inch
disc, the 360 marks will be spaced about a tenth of an inch apart so
additional marks can be used or the pictures can simply be read to a
fraction of an interval. Record test discs with equally spaced
"spokes," for use with a strobe for testing turntables, are also
available and can be used.
Timing Errors
Shannon and I used the switch which measured T to flash a strobe as
well as start and stop the clock. We discovered the lights and the
strobe flash "stopped" the ball at each of the two instants the switch
was hit. This allowed us to see how much the ball was off the
reference mark. Since we knew approximately how fast the ball was
moving, we could tell about how much in time we were early or late in
hitting the switch. This enabled us to correct the times recorded on
the clock, thereby making the data much more accurate. We also learned
from the visual feedback how to become much more accurate at timing.
Here's an illustration. Suppose the track of the wheel was 25 inches
in diameter. (I don't have any of this equipment now so I'm
remembering back over 20 years and recalling about what the sizes,
velocities, etc. seemed to be. They'll be close enough to be
representative and good enough to show you how to do it all again,
better for you if you want to.) Suppose the ball is 34 inch in
diameter and T, the time for one revolution, is 0.8 seconds. Then the
track is 78.54 inches in length, or 98.17 ball diameters. If the ball
center is one diameter away from the reference mark when the strobe
flashes, then the timing error is about 1/98.17 of Tor about 8/1000 of
a second. There will be one of these errors when the switch is first
hit and another when it is hit the second time. With practice we were
able to reduce each error to a typical (root mean square) size of one
ball diameter or about 8/1000 seconds. According to the theory of
errors, the two errors together give a typical (root mean square) size
of V2/1000 or about 11.2/1000 seconds.
These errors would be unobservable in casino play, so we couldn't
correct for them there. The critical question is how do they affect
the prediction? *
A Simple Casino Countermeasure
It should be clear that for this method to work, we have to time the
ball (and rotor) before placing our potentially winning bets. (Earlier
bets are losing, on average, so are only camouflage.) Thus, the casino
must allow us to continue to bet for a time after the ball is
launched. I have observed roulette wheels all over the world: Monte
Carlo (our final goal), Nevada, Puerto Rico, Nice, Venice, and London.
The practice has been, generally but not always, to allow bets until
the ball was almost ready to fall off the track. This was much longer
than we needed. Be warned again, though; all the casino needs to do to
prevent our method is to for¬bid bets once the ball is launched. That
simple perfect countermeasure is the Achilles heel of the system and a
major reason why I never made a total effort to implement it. (People
who use the system in casino play say the casinos don't catch on and
don't use the countermeasure. But if the player is not really careful,
I would expect the casino to catch on.)
The ball timing errors cause errors in predicting both the time and
place the ball leaves the track. Even if the spiral path of the ball
down the stator into the rotor is always the same in time and
distance, this still yields errors in predicting when and where on the
rotor the ball enters.
Error Analysis
We have a long list of sources for errors in the prediction of the
ball's final position. They are:
El Rotor timing--use 1.4 pockets to illustrate.
E2 Ball timing--use 5.5 pockets to illustrate.
E3 Variations in ball "paths" on rotor (see Figure 4-1). Error size is
unknown, call it X.
E4 Ball path down stator: error due primarily to "deflectors" and
varies with the type and placement. Use seven pockets to illustrate.
E5 Variation in distance ball travels on rotor: error due primarily to
frets between pockets "spattering" ball, plus occasional very long
paths along the rim of the rotor "outside" the pockets. Use six
pockets to illustrate.
E6 Tilted wheel. (We didn't know about this yet.)
For illustrative purposes, assume the errors approximately obey the
normal probability distribution. Then the standard deviation (typical
size) of the sum of several errors is the square root of the sum of
all the squared errors. For instance, using "pockets" as our unit,
combined errors E4 + E5 have typical size V(62 + 72) = V85 = 9.2
pockets. Now add on the timing errors: E, + E2 E5 have typical size
V(1.42 + 5.52 + 62 + 72) = V117.21 = 10.8 pockets. Thus the timing
errors in this example cause very little additional error: just 10.8 --
9.2, or 1.6 pockets.*
Of course, we haven't added in E3 yet and, if Xis big enough, it could
ruin everything. Possible variations in the ball orbit behavior on the
stator were difficult for us to measure because we found it hard to
tell at exactly what point the ball lost contact with the outer wall
of the wheel. We also learned from both our own lab experiences and
from watching in the casinos why the orbit varied somewhat. Once a
drunken, cigar-smoking bettor knocked his ash onto the track. This was
hard to clear out. it got on the ball and spread out on the track.
That immediately changed the ball's behavior. Skin oil from our
fingers or the croupier's would slowly "poison" ball and track and
seem to affect the orbit behavior.
If we or the croupier gave the ball lots of axial "spin" (in the sense
of tennis or ping pang), it could take several revolutions around the
track before this abnormal spin energy was converted to orbit energy.
(We named this effect after the famous quantum mechanics concept of "
spin-orbit coupling. ") On the other hand, the ball might be launched
with no spin or backspin, so it would skid for a while before spin and
orbit got "into synch."
Advantage Versus Error
Obviously, the greater the error, the less the advantage. If we assume
the total prediction error E is (approximately) normally distributed,
then we can construct a table showing the player's expected gain or
loss as a function of E.
Table 4-4 gives the results for a bet on the best pocket and also for
a bet on the best "octant." The best octant is a set of five pockets,
two on each side of the best pocket.
The Table shows that, when the prediction error is normally
distributed, the typical forecast error (standard deviation) must be
16 pockets or less, in order for the bettor to have an advantage. This
is 16/38, or about 0.42 revolutions. This is true both for bets on the
best pocket and the best octant. Since the best octant includes four
pockets that aren't quite as good as the best, the ad¬vantage is
somewhat less for a given typical error E. However, as we will see
later in discussing the Kelly-Breiman system for money management, it
is generally better for a small to medium-sized bankroll to bet the
best octant.
Kimmel and the Dealer's Signature
Stephen Kimmel asserted that a dealer who works eight hours a day, 50
weeks a year, tends to spin the ball and rotor in a habitual, regular
way. This would make possible accurate predictions-a bet on ten
pockets, Kimmel contended, would have a 503/4 chance of success. His
views were contained in an article "Roulette and Randomness" in the
December, 1979 issue of Gambling Times
I don't believe Kimmel's approach works. Here's why: there are three
important conditions that must remain roughly constant throughout play
for the player to take advantage of the regularity of the dealer's
signature. These conditions are (1) the rotor velocity should be
approximately the same each time the ball is spun, (2) the spinning
ball should make approximately the same number of revolutions each
time, and (3) the initial position of the rotor when the dealer
launches the ball should be approximately the same each time. This
third condition, which is not mentioned in Kinunel's article, is
crucial
By way of illustration, suppose that the rotor velocity was exactly
the same each time and that the dealer spun the ball exactly the same
number of revolutions in each instance. Suppose further that the ball
spun exactly eight revolutions and the rotor four revolutions during
this time. Given those assumptions, the ball would land about 12
revolutions beyond the point where it was launched. In other words, if
the number 13 was passing the ball as the dealer released it, the ball
would arrive 12 revolutions later, relative to the spinning rotor, at
approximately the number 13. You can see, however, that if the number
2 on the rotor was closest to the ball at the instant it was released,
the ball would then end up near that number 12 revolutions later.
If the dealer releases the ball without regard to which number on the
spinning rotor is closest to the launch point, the ball would randomly
fall on the rotor 12 revolutions later. In this case, there would be
no predictability whatsoever, even though the rotor velocity is
absolutely fixed and the number of ball revolutions constant. Any
variance in rotor velocity or number of ball revolu¬tions would
further guarantee a random outcome. Because Kim¬mel did not discuss
variations in the point of release, I do not believe in his method.
There is a better approach to this statistical analysis of roulette.
Watch a dealer and count the number of revolutions the ball makes on
the stator from the time of release until it crosses onto the rotor.
Note how constant that number of revolutions is. The results of your
observations can be statistically stated as some average number of
revolutions plus an error term.
Next, count the number of revolutions the rotor makes during the time
the ball is on the stator. This will give you another average for the
number of rotor revolutions, plus a second error term. Finally, count
how far the ball travels on the rotor after it has crossed the divider
between the rotor and stator. You can sum¬marize these results as some
average number of revolutions or pockets plus an error term.
In order for this approach to work, it is necessary that the square
root of the sums of the squares of the error terms be less than 17
pockets. The proof of this appears in Table 4-4 which shows what the
rate of return is, given various root mean square errors. That table
demonstrates that a positive return is possible only when that root
mean square error is less than 17 pockets.
Now for the improved method. In the unlikely event that the root mean
square error is less than 17 pockets, then--and only then--you have a
chance to win. The key lies in using the position of the rotor when
the ball is launched as your starting point for predicting where the
ball will fall out on the wheel.
For example, suppose you find that for a certain dealer the ball
travels eight revolutions with a root mean square error of five
pockets. Suppose also that during this time, the rotor travels four
revolutions, with a root mean square error of six pockets. And suppose
still further that once the ball is on the rotor, it travels 13
pockets with a root mean square error of eight pockets. Given these
suppositions, you can predict that the ball will travel eight
revolutions plus four revolutions plus 13 pockets from the launch
position, or 13 pockets beyond that point. The root mean square error
is the square root of five squared plus six squared plus eight
squared. This turns out to be 11.2 pockets, well within the required
error of less than 17 pockets. In this case, the prediction system
would work.
However, I think you will find that when you collect this data, the
errors at each stage are several times as large as I have used in this
example. My own observation is that the dealer error in the number of
revolutions for the ball spin is about 20 pockets for the more
consistent dealers; it is much larger with a less consistent one. I
also noticed that the rotor velocity is not nearly as constant as
Kimmel would like. That is because the dealer gives it an extra kick
every few spins to rebuild its velocity.
It is also true that the deflecting vanes on the sides of the rotor
add considerable randomness to the outcome, as do the frets or spacers
between the pockets. The upshot is that I don't believe that any
dealer is predictable enough to cause a root mean square error of less
than 17 pockets. I'm willing to examine proof to the contrary, but I
would be very surprised if anyone could ever pro duce it.
If a dealer dutifully practiced spinning the ball a fixed number of
revolutions, and if a motor drive spun the rotor at a constant
velocity, and if we have a very good way of deciding exactly which
number is opposite the ball just as it is released, it might be barely
possible to gain a small prediction advantage. I consider even that
very unlikely.
In closing, I'll give you the perfect casino countermeasure to the
strategy of the dealer's signature, pretending for the moment that the
strategy worked. First, the casino halts the betting before the dealer
spins the ball. Second, the dealer closes his eyes or looks away from
the wheel when he releases the ball so that he has no knowledge of
which number on the rotor is closest to the ball when it is launched.
Then, for the reasons explained above, the result will be perfectly
random.
THE WHEEL OF FORTUNE
In the last chapter, I described a system for winning at roulette
based on physical prediction. That system was developed largely in
1961 and 1962 in collaboration with Claude Shannon at MIT. One by-
product was an even simpler system for physical predic¬tion of the
Wheel of Fortune. A story about me and blackjack card-counting in Life
magazine, March 27, 1964, reported on this in a section entitled
"Beating the Wheel of Fortune with the Big Toe."
While I was at the Fifth Annual Conference on Gambling and Risk Taking
at Caesars Tahoe in October of 1981, I collected data on a Wheel of
Fortune at Caesars. I wanted to see whether their wheels could still
be predicted in the same way.
My Casio C-80 watch has a digital stop watch feature which times to
1/100 of a second. I used it to time one revolution of the wheel and
then recorded how many revolutions it went. I col¬lected the data in
Table 5-1 at the Wheel of Fortune nearest to Caesars' cashier cage.
To see how predictable the Wheel was, I looked for amathematical curve
which would best fit these data points. A curve which worked well was
R = A times T to the B power where A = 121.545 and B = --2.11153. In
the equation, T is the time for the wheel to make one revolution and R
is the number of addi¬tional revolutions which it then travels.
Intuitively, if T is short, the wheel did one revolution quickly so it
will go far and R will be large. But if T is long, the wheel was slow
and will stop soon so R will be small.
The letter p in the third column of the Table ("raw data") stands for
"pegs." The wheel has pegs separating the payoff numbers. As the wheel
rotates, the pegs push past a flexible "flap¬per." This gradually
slows the wheel. When the wheel stops, the winning number is the one
with the flapper between its pegs.
The raw data column gives 3.5 + 22p for observation number 1. This
means that the wheel traveled 3.5 revolutions plus 22 pegs or further
numbers. Since there are 54 numbers in all, it went 3.5 + 22/54 or
3.907 revolutions in all. That is shown under "decimal" in column 4.
The prediction P is made from the equation. The "error" P-R is the
amount the prediction is off from what actually happened. Strictly
speaking, what I am calling a prediction is only a fit to the data.
The fit approaches a "true" fit more closely as more data is included.
However, there is generally a difference between the "true" fit and
the actual fitted equation.
New data tends to cluster around this slightly different unknown true
fit, so it will tend to deviate from the actual fit to the data by
this extra amount. Thus, we expect future data to be predicted by the
equation not quite as well as the data in Table 5-1.
The error P --R has a standard deviation ("typical size") of .0587
revolutions, or 3.2 numbers. The true curve location (stan¬dard
deviation of the curve) is probably within .0169 revolutions or 0.9
numbers, on average. Considering this and the greatest positive and
negative values in the column, error in "pegs" sug¬gests that the
prediction will almost always be within five "pegs" or positions of
the actual outcome wheel. They are listed in order, clockwise, as seen
by the player. Each number gives the profit per unit bet. Thus, a
player who bets on 2 wins $2 for each $1 bet. The number marked 40A,
and called Caesars, pays 40 to 1 and the number 40B, called Cleo, also
pays 40 to 1. A bet on one of them does not win if the other one comes
up.
There are 24 "ones" in Table 5-2. Thus, if each of the 54 numbers
comes up once, "one" wins 24 times and loses 30 times for a loss of 6
units in 54 unit bets, or an expected loss rate of -- 6/54 = -- 1/9
=11.1%%. Similar calculations lead to Table 5-3. For the player who
doesn't predict, the house edge is enormous. This is a game to avoid.
Now let's see what the player advantage might be from predic¬tions.
Suppose for the sake of discussion that the final wheel posi¬tion is
always within five numbers of the predicted wheel position. For any
prediction in the eleven number strip centered around 40A, we should
bet on 40A. In 54 spins where each final position occurred once, we
will place 11 bets on 40A and win one of them for a gain of 40 --10 =
30 units.
The discussion is the same for 40B. For any prediction in either of
the eleven number strips surrounding each 20, twenty-two numbers in
all, we bet on 20. In twenty-two bets we expect to 20 units twice and
lose one unit twenty times for a net gal twenty units. This leaves 54
-44 or ten predicted positions where we need instructions.
There are four lOs in this left-over set of ten positions. Suppose we
bet the 10 each time one of these positions is predicted. It seems
plausible to suppose that we would win ten units four times and lose 1
unit six times for a net gain of 40 6 =34 units. (Actually, since the
lOs in this case are either the predicted number or within one
position of the predicted number, we expect to do better still.
Finally, in 54 unit bets we net 30 units from 40A, 30 units from 40B,
20 units from the two 20s, and 34 units from the four 10s, for a total
of 114 units/54 units or a 211%% rate of return.
It may be possible to improve both the timing procedure and the method
of exploiting predictability. This would improve the results.
We see now that the Caesars wheel can be predicted well enough so that
we can beat it if the casino will let us put down bets after the wheel
has been set in motion.

Card Games
Casino card games such as baccarat and blackjack differ significantly
from casino games such as craps, roulette, and slot machines in that
they are not indepen¬dent trial processes-that is, the cards that
already have been played do affect the odds on subsequent hands.
Consider for a moment the game of blackjack, where the cards used on a
round are put aside and successive rounds are dealt from an
increasingly depleted pack. The cards are reshuffled before a round if
the remain¬ing unused cards would be insufficient to complete a round
or earlier, usually at the casino's discretion. What the early
research on blackjack (contained in Beat the Dealer) showed and what
has been confirmed repeated¬ly in the intervening 23 years is that the
end pack pro¬vides favorable situations often enough to give the
player an overall advantage.
While it is foolish to keep a record of past decisions at craps in
order to determine which numbers are "hot" or "cold" (the dice have no
memory), an ability to keep track of which cards have been played and
knowledge of their relationship to the player's expectation can be
beneficial, as long as the cards are not reshuffled after every hand.
The ability to keep track of the cards played does not alone guarantee
gambling success at a particular game. Indeed, one of the chief tasks
of this section will be to examine the usefulness of card counting
strategies in baccarat, considering the bets offered and the nature of
the game,
In Chapter two we will comment on blackjack systems, as well as
statistical methods useful in detecting casino cheating. The latter
subject is important to those who play the game seriously, because
cheating in¬cidents can erode any small edge the player may gain
through the use of basic strategy and card counting.
INTRODUCTORY STATEMENT
The casino patron who decides to "try his luck" at the tables and the
horse player who wagers at the racetrack confront what seem to be
formidable adversaries. The casinos hope to have the dvantage on
every bet offered and, at the track, the pari-mutuel takeout of 17-25%%
on every bet assures all but a few will wind up losers.
As soon as he enters the casino, the player must make several
important decisions, the first being: What game do I play? Even after
this choice is made, most games offer additional options: Do I play
individual numbers or the even-money bets in roulette? Do I stand with
a pair of eights in blackjack or should I hit or split the pair?
Should I bet pass or the one-roll propositions in craps?
The horse player is offered a number of choices as well. He is usually
faced with a field of six to 12 horses. He can play one or more horses
to win, place, or show, in addition to combining any number of horses
in the exotic or "gimmick" wagers. All of these choices have "right"
answers, if the player seeks to maximize his return or minimize his
loss. They all can be at least partially solved through the use of
mathematical theory. The intelligent player must have a basic
understanding of the mathematics behind the game or games he plays if
he is to survive financially or actually profit. There are situations
where the player has the advantage. The most-publicized example, of
course, is casino blackjack. The game has become tougher in recent
years due to casino countermeasures, but blackjack can still be
profitable for the sophisticated player. There could be several other
favorable games, as the reader will soon discover.
A familiarity with basic probability will allow the alert gambler to
discover those positive expectancy games and exploit them where they
exist. A vast knowledge of mathematics is not required. Some of the
finest poker players in the country never went to college, but they do
have a sense of what makes a good poker hand and what their chances of
having the best hand are after all the cards have been dealt.
Mathematical Expectation
I have already made reference to the concept of mathematical
expectation. This principle is central to an understanding of the
chapters to follow.
Imagine for a moment a coin toss game with an unbiased coin (a coin we
assume will produce 50%% heads and 50%% tails). Sup¬pose also that we
are offered an opportunity to bet that the next flip will be heads and
the payoff will be even money when we win (we receive a $1 profit in
addition to the return of wager). Our mathematical expectation in this
example is:
(.5X1) + (.5)( - 1) =
The mathematical expectation of any bet in any game is com¬puted by
multiplying each possible gain or loss by the probability of that gain
or loss, then adding the two figures. In the preceding example, we
expect to gain nothing from playing this game. This is known as a fair
game, one in which the player has no advantage or disadvantage.
Now suppose the payoff was changed to 3/2 (a gain of $1.50 in addition
to our $1 bet). Our expectation would change to:

(.5X1 .53) + (.6X - 1) = +.25
Playing this game 100 times would give us a positive expectation of
$25.
The two examples presented thus far are admittedly simple, but often
this type of analysis is all that is needed to evaluate a
prop¬osition. Consider the "dozens" bet in roulette. Our expectation
for a $1 bet is:
(12/38X2) + (26/38X -1) = - .0526
As another example, suppose that on the first hand of four-deck
blackjack the player bets $12, he is dealt 6,5, and the dealer. then
shows an ace up. The dealer asks the player if he wants insurance.
This is a separate $6 bet. It pays $12 if the dealer's hole card is a
ten-value. It pays -$6 otherwise. A full four-deck pack has 64 tens
and 144 non-tens. Assuming the deck is "randomly" shuffled (this means
that all orderings of the cards are equally prob¬able), the chances
are equally likely that each of the 205 unseen cards is the dealer's
hole card. Thus the player's expectation is:
(641205X12) + (141/205)( - 6) = - 78/205
or about - $.38. The player should not take insurance.
Different betting amounts have different expectations. But the
player's expectation as a percent of the amount bet is always the same
number. In the case of betting on the Red in roulette, this is 18/38 -
20/38 = -2/38 = -1/19 or about -5.26%%. Thus, the expectation of any
size bet on Red at American double-zero roulette is - 1/19 or about -
5.26%% of the total amount bet. So to get the expectation for any size
bet on Red, just multiply by -- 5.26%%. With one exception, the other
American double-zero roulette bets also have this expectation per unit
bet. The player's expectation per unit is often simply called the
player's disadvan-tage. What the player loses, the house wins, so the
house advan¬tage, house percentage, or house expectation per unit bet
by the
player is +5.26%%.
A useful basic fact about the player's expectation is this: the
expectation for a series of bets is the total of the expectations for
the individual bets. For instance, if you bet $1 on Red, then $2, then
$4, your expectations are -$2138, -$4/38, and - $8/38. Your total
expectation is -$14/38 or (a loss of) about -$.37. Thus, if your
expectation on each of a series of bets is -5.26%% of the amount bet,
then the expectation on the whole series is -5.26%% of the total of all
bets. This is one of the fundamental reasons why "staking systems"
don't work: a series of negative expectation bets must have negative
expectation.
Repeated Trials
Expectation is the amount you tend to gain or lose on average when you
bet. It, however, does not explain the fluctuations from expectation
that occur in actual trials.
Consider the fair game example mentioned earlier in the chapter. In a
series of any length, we have an expectation of 0. In any such series
it is possible to be ahead or behind. Your total profit or loss can be
shown to have an average deviation from expecta¬tion of about SIN. Let
D = T -E be the difference of deviation be¬tween what you actually
gain or lose (I), and the expected gain or loss (E). Therefore, for
100 bets, the average deviation from E =0 is about $10 (in fact, the
chances are about 68%% that you'll be within $10 of even; they're about
96%% that you'll be within $20 of even). For ten thousand $1 bets it's
about $100 and for a million $1 bets it's about $1,000. Table 2-1
shows what happens. For instance, the last line of Table 1-1 says that
if we match coins one million times at $1 per bet, our expected gain
or loss is zero (a "fair" game). But on average, we'll be about $1,000
ahead or behind. In fact, we'll be between +$1,000 and -$1,000 about
68%% of the time. (For a million $1 bets, the deviation D has
approximately a normal probability distribution with mean zero and
standard deviation $1,000.) We call the total of the bets in a series
the "action;' A. For one series of one million $1 bets, the action is
$1,000,000. However (fifth column) D/A =0.001, so the deviation as a
percent of the action is very small. And about 68%% of the time T/A is
between -- .001 and +.001 so as a per¬cent of the action the result is
very near the expected result of zero. Note that the average size of
D, the deviation exiation from the pected result E, grows--contrary to
popular belief. However, the average size of the percentage of
deviation, D/A, tends to zero, in agreement with a correct version of
the "law of averages."
For $1 bets on Red at American roulette, the corresponding results
appear in Table 1-2. Notice that in the last column the spread in T/A
gets closer and closer to E/A = --.0526. This is where we get the
statement that if you play a "long time" you'll lose about 5.26%% of
the total action. Note, too, in column 4 that there appears to be less
and less chance of being ahead as the number of trials goes on. In
fact, it can be shown that in all negative expecta¬tion games the
chance of being ahead tends to zero as play continues.
Using the concept of action, we can now understand the famous "law of
averages." This says, roughly, that if you make a long series of bets
and record both the action (A) and your total profit or loss (1"),
then the fraction T/A is approximately the same as the fraction E/A
where E is the total of the expected gain or loss for each bet. Many
people misunderstand this "law." They think that it says the E and T
are approximately the same after a long series of bets. This is false.
In fact, the difference between E and T tends to get larger as A gets
bigger.
Now, the ordinary player probably won't make a million $1 bets. But
the casino probably will see that many and more. From the casino's
point of view, it doesn't matter whether one player makes all the bets
or whether a series of players does. In either case, its profit in the
long run is assured and will be very close to 5.26%% of the action.
With many players, each making some of the 1,000,000 bets, some may be
lucky and win, but these will generally be compensated for by others
who lose more than the expected amount. For instance, if each of
10,000 players take turns making a hundred $1 bets, Table 1-2 tells us
that about 68%% of the time their result will be between + $4.74 and
$15.26
About 16%% of the time the player wins more than $4.74 ("lucky") and
about 16%% of the time the player loses more than $15.26 ("unlucky").
But players cannot predict or control which group they'll be in.
This same "law of averages" applies to more complicated sequences of
bets. For instance, suppose you bet $10 on Red at roulette (E = - .
53), then bet $103 on "players" at Baccarat (E = - $1.06), then bet
$10 on a hand in a single-deck blackjack game where the ten-count is
15 tens, 15 others (E = +$.90). The total E is $.53 -$1.06 + $.90 = -
$.69. The total A is $10 + $100 + $10 = $120. If you make a long
series of bets and record E and A as well as your gains and losses for
each one, then just as in the coin matching example (Table 1-1) and
the roulette example (Table 1-2), the fraction D/A tends to zero so T/
A tends to E/A. That means that over, say, a lifetime, your total
losses as a percent of your total action will tend to be very close to
your total expecta-tion as a percent of your action.
If you want a good gambling life, make positive expectation bets. You
can, as a first approximation, think of each negative expectation bet
as charging your account with a tax in the amount of the expectation.
Conversely, each positive expectation bet might be thought of as
crediting your account with a profit in the amount of the expectation.
If you only pay tax, you go broke.. If you only collect credits, you
get rich.

BlackJack
Blackjack, or twenty-one, is a card game played throughout the world.
The casinos in the United States currently realize an annual net
profit of roughly one billion dollars from the game. Taking a price/
earnings ratio of 15 as typical for present day com¬mon stocks, the
United States blackjack operation might be com¬pared to a $15 billion
corporation.
To begin the game a dealer randomly shuffles the cards and players
place their bets. The number of decks does not materially affect our
discussion. It generally is one, two, four, six or eight. There are a
maximum and minimum allowed bet.
The players' hands are dealt after they have placed their bets. Each
player then uses skill in his choice of a strategy for improving his
hand. Finally, the dealer plays out his hand according to a fixed
strategy which does not allow skill, and bets are settled. In the case
where play begins from one complete randomly shuffled deck, an
approximate best strategy (i.e., one giving greatest expected return)
was first given in 1956 by Baldwin, Cantey, Maisel, and McDermott.

Though the rules of blackjack vary slightly, the player follow¬ing the
Baldwin group strategy typically has the tiny edge of + .10 %%. (The
pessimistic figure of --.62 %% cited in the Baldwin's group's work was
erroneous and may have discouraged the authors from further analysis.)
These mathematical results were in sharp contrast to the earlier and
very different intuitive strategies generally recommended by card
experts, and the associated player disadvantage of two or three
percent. We call the best strategy against a complete deck the basic
strategy. Determined in 1965, it is almost identical with the Baldwin
group strategy and it gives the player an edge of +.13 against one
deck and -- .53 against four decks.
If the game were always dealt from a complete shuffled deck, we would
have repeated independent trials. But for compelling practical
reasons, the deck is not generally reshuffled after each round of
play. Thus as successive rounds are played from a given deck, we have
sampling without replacement and dependent trials. It is necessary to
show the players most or all of the cards used on a given round of
play before they place their bets for the next round. Knowing that
certain cards are missing from the pack, the player can, in principle,
repeatedly recalculate his optimal strategy and his corresponding
expectation. (The strategies for various card counting procedures, and
their expectations, were determined directly from probability theory
with the aid of computers. The results were reverified by independent
Monte Carlo calculations.)
Blackjack Systems
All practical winning strategies for the casino blackjack player,
beginning with my original work in 1961, are based on this knowledge
of the changing composition of the deck. In practice each card is
assigned a point value as it is seen. By convention the point value is
chosen to be positive if having the card out of the pack significantly
favors the player and negative if it significantly favors the casino.
The magnitude of the point value reflects the magnitude of the card's
effect but is generally chosen to be a small integer for practical
purposes. Then the cumulative point count is taken to be proportional
to the player's expectation.
To a surprising degree, the player's best strategy and cor-responding
expectation depend only on the fractions of each type of card
currently in the pack and only change slowly with the size of the
pack. Thus the better systems "normalize" by dividing the cumulative
point count by the total number of as yet unseen cards. Most point
count systems are initialized at zero cumulative total for the full
pack, and the normalized cumulative count is taken to indicate the
change in player expectation from the value for the full pack.
The original point count systems, the prototypes for the many
subsequent ones, were my five count, ten count, and "ultimate
strategy." An enormous amount of effort by many investigators has
since been expended to improve upon these count systems. Some of these
systems are shown in Table 2-1 (courtesy of Julian Braun).
The idea behind these point count systems is to assign point values to
each card which are proportional to the observed effects of deleting a
"small quantity" of that card. Table 2-2 (courtesy of Julian Braun,
private correspondence) shows this for one deck and for four decks,
under typical Las Vegas rules. One must com¬promise between simplicity
(small integer values) and accuracy. My "ultimate strategy" is a point
count based on moderate integer values which fits quite closely the
data available in the early 1960s. Until recently all the other count
systems were simplifica¬tions of the "ultimate."
System 1 (Table 2-1) does not normalize by the number of re-maining
cards. Thus the player need only compute and store the cumulative
point count. Normalization gives the improved results of system 2, but
requires the added effort of computing the number of remaining cards
and of dividing the point count by the number of remaining cards when
decisions are to be made. In practice the player can estimate the
unplayed cards by eye and use it with system 1 and get almost the
results of system 2 with much less effort. Systems 2, 3, 5 and 7 all
divide by the number of remaining cards.

Systems 4, 6 and 8, which are also normalized, have the first new
idea. They assign a point count of zero to the ace for strategy
purposes. This is consistent with the evidence: in most instances that
have been examined, the optimal strategy seems to be relative¬ly
unaffected by changes in the fraction of aces in the pack. However,
the player's expectation is generally affected by aces more than by
any other card (Table 2-2). Therefore these systems keep a separate
ace count. Then the deviation of the fraction of aces from the normal
1/13 is incorporated for calculating the player's expectation for
betting purposes.*
The (c) column in Table 2-1 still remains to be explained. It is a
numerical assessment of a particular system's closeness to an ideal
system based on the change in expectation values contained in Table
2-2. The calculation of the (c) value eliminates the necessity of
simulating a large number of hands (say a million) to evaluate a
strategy. The computation of these numbers requires some advanced
mathematical background, so its explanation is left to the appendix.
Cheating: Dealing Seconds
Various card counting systems give the blackjack player an advantage,
provided that the cards are well shuffled and that the game is honest.
But many methods may be used to cheat the player. I have been
victimized by most of the more common techniques and have catalogued
them in Beat the Dealer.
One of the simplest and most effective ways for a dealer to cheat is
to peek at the top card and then deal either that card or the one
under it, called the second. A good peek can be invisible to the
player. A good second deal, though visible to the player, can be done
so quickly and smoothly that the eye generally will not detect it.
Although the deal of the second card may sound different from the deal
of the first one, the background noise of the casinos usually covers
this completely. Peeking and second dealing leave no evidence. Because
these methods are widespread, it is worth knowing how powerful they
are.
Does even a top professional blackjack counter have a chance against a
dealer who peeks and deals seconds? Consider first the simple case of
one player versus a dealer with one deck. This is an extreme example,
but it will illustrate the important ideas. I shuffle the deck and
hold it face up in order to deal practice hands. Because I can see the
top card at all times, dealing from a face-up deck is equivalent to
peeking on each and every card. I will deal either the first or second
card, depending on which gives the dealer the greatest chance to win.
I will think out loud as an imaginary dealer might, and the principles
I use will be listed as they occur. The results for a pass through one
deck are listed in Table 2-3 (pp. 20-21). There were nine hands and
the dealer won them all.
On hands one, two, four, six, eight and nine, the dealer wins by
busting the player. Because there is only one player, it does not
matter what cards the dealer draws after the player busts. When there
are two or more players, the dealer may choose a different strategy.
If, for example, the dealer wishes to beat all the players but doesn't
want to peek very often, an efficient approach is simply to peek when
he can on each round of cards until he finds a good card for himself
on top. He then retains this card by deal¬ing seconds until he comes
to his own hand, at which time he deals the top card to himself. That
strategy would lead to the dealer having unusually good hands at the
expense of the collective player hands; because some good hands have
been shifted from the players, the player hands would be somewhat
poorer than average.
A player could detect such cheating by tallying the number of good
cards (such as aces and 10s) which are dealt to the dealer as his
first two cards and comparing that total with the number of aces and
lOs predicted by theory. In Peter Griffin's book, The Theory of
Blackjack, he describes how he became suspicious after losing against
consistently good dealer hands. Griffin writes that he " . . .
embarked on a lengthy observation of the frequency of dealer up cards
in the casinos I had suffered most in. The result of my sample, that
the dealers had 770 aces or lOs out of 1,820 hands played, was a
statistically significant indication of some sort of legerdemain."
Griffin's tally is overwhelming evidence that something was peculiar.
The odds against such an excess of ten-value cards and aces going to
the dealer in a sample this size are about four in ten thousand.
Another approach the dealer might select is to beat one player at the
table while giving everybody else normal cards. To do this, the dealer
peeks frequently enough to give himself the option of dealing a rust
or second to the unfortunate player each time that player's turn to
draw a card comes up. Dealing stiffs to a player so that he is likely
to bust is, as we see from the chart in Table 2-3, so easy to do that
the player has little chance.
If all dealers peeked and dealt seconds according to the cheating
strategy indicated in Table 2-3, I estimate that with one player
versus the dealer, the dealer would generally win at least 95 percent
of the time. With one dealer against several players, the dealer would
win approximately 90 percent of the time. Anyone who is interested can
get a good indication of what the actual numbers are by dealing a
large number of hands and recording the results.
The deadliest way a dealer can cheat is to win just a few extra hands
an hour from the players. This approach is effective because it is not
extreme enough to attract attention, or to be statistically
significant and therefore detectable over a normal playing time of a
few hours. For example, the odds in blackjack are fairly close to even
for either the dealer or the player to win a typical hand. Suppose
that by cheating the dealer shifts the advantage not to 100 percent
but to just 50 percent in favor of the house. What effect does this
have on the game?
If we assume that the player plays 100 hands, a typical total for an
hour's playing time, and we also assume that the player bets an
average of two units per hand, then being cheated once per 100 hands
reduces the player's win by one unit on the average. A pro¬fessional
player varying his bet from one to five units would prob¬ably win
between five and 15 units per hour. The actual rate would depend upon
casino rules, the player's level of skill, and the power and variety
of winning methods that he employed. Let's take a typical professional
playing under good conditions and assume that his win rate is ten
units per hour and his average bet size is two units. Given those
assumptions, being cheated ten times per hour or one-tenth of the time
would cancel his advan¬tage. Being cheated more than ten percent of
the time would prob¬ably turn him into a loser.
Cheating in the real world is probably more effective than in the
hypothetical example just cited, because the calculations for that
example assume cheating is equally likely for small bets and big bets.
In my experience, the bettor is much more likely to be cheated on
large bets than on small ones. Therefore, the dealer who cheats with
maximum efficiency will wait until a player makes his top bet. Suppose
that bet totals five units. If the cheat shifts the odds to 50 percent
in favor of the house, the expected loss is 2-1/2 units, and just four
cheating efforts per 100 hands will cancel a professional player's
advantage. A cheating rate of five or ten hands per 100 will put this
player at a severe disadvantage.
We can see from this that a comparatively small amount of cheating
applied to the larger hands can have a significant impact on the
game's outcome. This gives you an idea of what to look for when you
are in the casinos and think that something may be amiss.
Missing Cards: The Short Shoe
I have heard complaints that cards have been missing from the pack in
some casino blackjack games. We'll discuss how you might spot this
cheating method.
In 1962, I wrote on page 51 of Beat the Dealer, "Counting the . . .
cards . . . is an invaluable asset in the detection of cheating
because a common device is to remove one or more cards from the deck."
Lance Humble discusses cheating methods for four-deck games dealt from
a shoe in his International Blackjack Club newsletter. He says, "The
house can take certain cards such as tens and aces out of the shoe.
This is usually done after several rounds have been dealt and after
the decks have been shuffled several times. It is done by palming the
cards while they are being
shuffled and by hiding them on the dealer's person. The dealer then
disposes of the cards when he goes on his break." But cheating this
way is not limited to the casino. Players have been known to remove
"small" cards from the pack to tilt the edge their way. The casino can
spot this simply by taking the pack and counting it; the player
usually has to use statistical methods.
In the cheating trade, the method is known as the short shoe. Let's
say the dealer is dealing from a shoe containing four decks of 52
cards each. In 52 cards, there should be 16 ten-value cards: the tens,
jacks, queens and kings. Logically, in four decks of 208 cards, there
should be 64 ten-value cards. I'll call all of these "tens" from now
on. Casinos rarely remove the aces--even novice players sometimes count
these.
Suppose the shift boss or pit boss takes out a total of ten tens; some
of each kind, of course, not all kings or queens. The shoe is
shortened from 64 tens to 54 tens, and the four decks from 208 cards
to 198 cards.
The loss of these ten tens shifts the advantage from the player to the
dealer or house. The ratio of others/tens changes from the normal
144/64 = 2.25 to 144/54 =2.67, and this gains a little over one
percent for the house. How can you discover the lack of tens without
the dealer knowing it?
Here is one method that is used. If you're playing at the black¬jack
table, sit in the last chair on the dealer's right. Bet a small fixed
amount throughout a whole pack of four decks. After the dealer puts
the cut card back only, let's say, ten percent of the way into the
four shuffled decks and returns the decks into the shoe, then ready
yourself to count the cards. Play your hand mechanically, only
pretending interest in your good or bad fortunes. What you're
interested in finding out is the number of tens in the whole four-deck
shoe.
Let's say the shift boss has removed ten tens. (Reports are that they
seem to love removing exactly ten from a four-deck shoe.) When the
white cut card shows at the face of the shoe, let's say that the
running count of tens has reached 52. That means mathematically that
if all 64 ten-value cards were in the shoe, then, of the remaining 15
cards behind the cut card, as many as 12 of them would be tens, which
mathematically is very unlikely. This is how one detects the missing
ten tens because the dealer never shows their faces but just places
them face down on top of the stack of discarded cards to his right,
which he then proceeds to shuffle face down in the usual manner
preparatory to another four-deck shoe session.
Although at first the running count is not easy to keep in a real
casino situation, a secondary difficulty is estimating the
approx¬imate number of cards left behind the cut card after all the
shoe has been dealt. To practice this, take any deck of 52 cards and
cut off what you think are ten, 15 or 20 cards, commit yourself to
some definite number, and then count the cards to confirm the
closeness of your estimate. After a while, you can look at a bunch of
cards cut off and come quite close to their actual number.
In summary, count the number of tens seen from the beginning of a
freshly shuffled and allegedly complete shoe. When the last card is
seen and it is time to reshuffle the shoe, subtract the number of tens
seen from the number that are supposed to be in the shoe-64 for a four-
deck shoe--to get the number of unseen ten-value cards which should
remain. If 54 ten-value cards were seen, there should be ten tens
among the unused cards. Then estimate the number of unseen cards. You
have to be sure to add to the estimated residual stack any cards which
you did not see during the course of play, such as burned cards. Step
four is to ask whether the number of unseen ten-value cards is
remarkably large for the number of residual cards. If so, consider
seriously the possibility that the shoe may be short. For instance,
suppose there are 15 unseen cards, ten of which are supposed to be ten-
values. A computation shows that the probability that the last 15
cards of a well-shuffled four-deck shoe will have at least ten ten-
value cards is 0.003247 or about one chance in 308.
Thus the evidence against the casino on the basis of this one shoe
alone is not overwhelming. But if we were to count down the same shoe
several times and each time were to find the remaining cards
suspiciously ten-rich, then the evidence would become very strong.
Suppose that we counted down the shoe four times and that each time
there were exactly 15 unseen cards. Suppose that the number of unseen
tens, assuming a full four decks, was nine, 11, ten, and 13
respectively. Then referring to Table 2-4, the pro¬babilities to six
decimal places are H(9) = .014651 to have nine or more unseen tens,
and for at least 11, ten, and 13 respectively, the chances are H(11)
= .000539, H(10) = .003247, and H(13) = .000005. These correspond to
odds of about 1168, 1/1,855, 11308 and 1/200,000 respectively. The
odds against all these events hap¬pening together is much greater
still. In this example, the evidence strongly suggests that up to nine
ten-value cards are missing. There can't be more than nine missing, of
course, because we saw all but nine on one countdown.
If the casino shuffles after only 104 cards are seen, it is not so
easy to tell if ten ten-value cards were removed. A mathematical proof
of this is contained in the appendix. *
This discussion should make it clear that the method suggested is
generally not able to easily spot the removal of ten-value cards
unless the shoe is counted several times or is dealt down close to the
end.
One of the interesting ironies of the short shoe method of cheating
players is that neither the shift boss nor the pitboss--the latter
bringing the decks of cards to the dealer's table--need tell the dealer
that his shoe is short. Thus, the dealer doesn't necessarily have to
know that he's cheating. After all, he's just dealing. It's an open
question how many dealers know that they're dealing from a short shoe.
Reports are that the short shoe is a frequent method that casinos use
in cheating at blackjack using more than one deck. The tables with
higher minimums (say $25) are more tempting candidates for short shoes
than those with the lower minimums.
An experienced card counter can improve the method by count¬ing both
tens and non-tens. Then he'll know exactly how many unseen cards there
are, as well as unseen tens. Table 2-4 can then be used with greater
confidence. In practice, you don't need to count through a shoe while
bet ting (and thus losing money in the process) to find out that the
casino is cheating. If you suspect foul play, count while standing
behind the player to the dealer's right.
You might easily catch a short shoe by simply counting all the cards
that are used, whether or not you see what they are. Then if the
remaining cards, at the reshuffle, are few enough so you can
accurately estimate their number, you can check the total count. For
instance, you count 165 cards used and you estimate that 31 ± 3 cards
remain. Then there were 196 ± 3 cards rather than the 208 expixted, so
the shoe is short.
A casino countermeasure is to put back a 4, 5 or 6 for each ace or ten-
value card removed. Then the total number of cards remains 208, and
the casino gets an even greater advantage than it would from a short
shoe.
Cards do get added to the deck, and there's a spooky coin-cidence to
illustrate this. On page 51 of Beat the Dealer, I wrote in 1962, "One
might wonder at this point whether casinos have also tried adding
cards to the deck. I have only seen it done once. It is very risky.
Imagine the shock and fury of a player who picks up his hand and sees
that not only are both his cards 5s, but they are also both spades."
And then 15 years later in 1977, a player in a one-deck game did get a
hand with two of the same card-the 5 of spades. Walter Tyminski's
casino gaming newsletter, Rogue et Noir News, reported on page 3 of
the June 15, 1977 issue, "What would you do if the player at your
right in a single blackjack game had two 5 of spades? Nicholas Zaika,
a bail bondsman from Detroit, had that experience at the Sahara in Las
Vegas on May 24 at a $5 minimum table.
"Zaika wasn't in the best of humor because he had reportedly lost
$594,000 at other Sahara tables, by far the largest loss he has ever
experienced. Zaika had the blackjack supervisor check the cards and
there were 53 cards in the deck, the duplicate be-in the 5 of
spades ...The gamer has engaged the services of Las Vegas attorney
George Grazadei to pursue claims he feels that he has against the
casino..
"The Sahara denies any wrongdoing and says that it is cooperating
fully with the investigation . . . Players aren't likely to introduce
an extra 5 because the presence of the extra 5 favors the house and
not the player." Suppose instead of just counting tens used and total
cards used, you kept track of how many aces, 2s, 3s, queens, kings,
and so on were used. This extra information should give the player a
better chance of detecting the short shoe. The ultimate proof would be
to count the number of each of the 52 types of cards which have been
used. Mathematical readers might wish to investigate effec¬tive
statistical or other ways of using information for detecting shoes in
which the numbers of some of the cards have been changed.

Baccarat
The games of baccarat and chemin de fer are well known gambling games
played for high stakes in several parts of the world. Baccarat is said
to be a card game of Italian origin that was in¬troduced into France
about 1190 A.D. Two forms of the game developed. One form was called
baccarat and the other was called chemin de fer. The most basic
difference between these two games is simply that three hands are
dealt in baccarat (called baccarat en banque in England) and two hands
are dealt in chemin de fer (called baccarat-chemin de fer in England
and Nevada).
The cards ace through nine are each worth their face value and the
cards ten, jack, queen and king are each worth zero points. A hand is
evaluated as the sum modulo ten of its cards, i.e., on¬ly the last
digit of the total is counted. The object of the game is to be as
close to eight or nine as possible with two cards, or as close to nine
as possible with at most three cards if one does not have eight or
nine on his first two cards. Then the high hand wins.
The games of baccarat and chemin de fer became popular in public
casinos all over Europe, as well as in private games, about 1830. At
the present time, one or both of these games are well known in London,
southern France, the Riviera, Germany and the United States. A form of
chemin de fer, which we shall call Nevada baccarat, has been played in
a few Nevada casinos since 1958.
The rules, structure and format of the three games have strong
similarities. I studied Nevada baccarat with William E. Walden most
intensively because the casinos where it is played were readi¬ly
accessible. Our techniques can be carried over to the other forms of
baccarat and chemin de fer.
We were originally motivated by the observation that baccarat and
chemin de fer have several points of resemblance to the game of
blackjack, or twenty-one. The fact that practical winning strategies
for twenty-one have been discovered suggested that there might also be
practical winning strategies for baccarat and chemin de fer. In
contrast to the situation in twenty-one, we found that there are no
current practical winning strategies for the main part of the game,
i.e., for the money Banker and Player bets.
Rules and Procedures
To begin the Nevada baccarat game, eight decks of cards are shuffled
and a joker is placed face up near the end. The cards are then put
into a wooden dealing box called a shoe. The first card is exposed,
and its value is noted, face cards being counted as tens. Then this
number of cards is discarded, or "burned."
The table has twelve seats, occupied by an assortment of customers and
shills. A shill is a house employee who bets money and pretends to be
a player in order to attract customers or stimulate play. We refer to
them indiscriminately as "players." There are two principal bets,
called "Banker" and "Players." Any player may make either of these
bets before the beginning of any round of play, or "coup."
To begin the evening's play, two of the players are singled out. One
is termed The Banker and the other is termed The Player. The seats are
numbered counterclockwise from one to twelve. Player number one is
initially The Banker, unless he refuses. In this case the opportunity
passes counterclockwise around the table until someone accepts. The
Player is generally chosen to be that player, other than The Banker,
who has the largest bet on the Player. We have not noticed an occasion
when there were no bets on The Player. When we played, there were
shills in the game and they generally bet on The Player (except when
acting as The Banker, when they generally bet on The Banker).
The Banker retains the shoe and deals as long as the bet
"Banker" (which we also refer to as a bet on The Banker) does not
lose. When the bet "Players" (which we also refer to as a bet on The
Player) wins, the shoe moves to the player on the right. This player
now becomes The Banker. If the coup is a tie, the players are allowed
to alter their bets in any manner they wish. The same Banker then
deals another coup.
To begin a coup, The Banker and The Player are dealt two cards each.
As we noted above, the cards ace through nine are each worth their
face value and tens and face cards are each worth zero points. Only
the last digit in the total is counted.
After The Banker and The Player each receive two cards, the croupier
faces their hands. If either two-card total equals 8 or 9 (termed a
natural 8 or a natural 9, as the case may be), all bets are settled at
once.
If neither The Player nor The Banker have a natural, The Player and
The Banker then draw or stand according to the set of rules in Table
3-1.
The high hand wins. If the hands are equal, there is a tie and no
money changes hands. Players are then free to change their bets in any
desired manner. If the coup being played is complete when the joker is
reached, the shoe ends and the cards are reshuffled. Otherwise the
coup is first played out to completion. Then the shoe ends and the
cards are reshuffled. However, the casino may reshuffle the cards at
any time between coups.

The Main Bets:
Two main bets against the house can be made. One can bet on either The
Banker or The Player. Winning bets on The Player are paid at even
money. Winning bets on The Banker are paid 0.95 of the amount bet. The
five percent tax which is imposed on what otherwise would have been an
even-money pay-off is called "vigorish." For eight complete decks, the
probability that The Banker wins is 0.458597, and the probability of a
tie is 0.095156.
The basic idea of the calculation of these numbers is to consider all
possible distinct six-card sequences. The outcome for each sequence is
computed and the corresponding probability of that sequence is
computed and accumulated in the appropriate register. Numerous short
cuts, which simplify and abbreviate the calculation, were taken.
The house advantage (we use advantage as a synonym for mathematical
expectation) over The Player is 1.2351 percent. The house advantage
over The Banker is 0.458597X5 percent-1.2351 percent or 1.0579
percent, where 2.2930 percent is the effective house tax on The
Banker's winnings. If ties are not counted as trials, then the figures
for house advantage should be multiplied by 1/0.904844, which give a
house advantage per bet that is not a tie, over The Banker of 1.1692
percent and over The Player of 1.3650 percent. The effective house tax
on The Banker in this situation is 2.5341 percent.
We attempted to determine whether or not the abnormal com¬positions of
the shoe, which arise as successive coups are dealt, give rise to
fluctuations in the expectations of The Banker and The Player bets
which are sufficient to overcome the house edge. It turns out that
this occasionally happens but the fluctuations are not large enough
nor frequent enough to be the basis of a practical winning strategy.
This was determined in two ways. First, we varied the quantity of
cards of a single numerical value. The results were negative.
We next inquired as to whether, if one were able to analyze the end-
deck perfectly (e.g. the player might receive radioed instruc¬tions
from a computer), there were appreciable player advantages on either
bet a significant part of the time. We selected 29 sets of 13 cards
each, each set drawn randomly from eight complete decks. There were
small positive expectations in only two instances out of 58. Once The
Player had a 3.207o edge and once The Banker had a 0.1010 edge.
We next proved, by arguments too lengthy and intricate to give here,
that the probability distributions describing the conditional
expectations of The Banker and The Player spread out as the number of
unplayed cards decreases. Thus there are fewer advan¬tageous bets of
each type, and they are less advantageous, as the number of unplayed
cards increases above 13. The converse occurs as the number of
unplayed cards decreased below 13.
The observed practical minimum ranged from eight to 17 in one casino
and from 20 up in another. The theoretical minimum, when no cards are
burned, is six. Thus the results for 13 unplayed cards seem to
conclusively demonstrate that no practical win¬ning strategy is
possible for the Nevada game, even with a com¬puting machine playing a
perfect game.
To see why, consider the accompanying Table 3-2 (pp. 34-35), based on
Table 2 of Walden's thesis.
From this Table we see the effect of removing one of any card from the
eight-deck baccarat pack. Proceeding in the way we developed the
theory of blackjack, we get relative point values which are listed in
the next to the last column. The last column gives a simpler
approximate point count system.
We would now like to know how powerful a point count system in
baccarat is compared with point count systems in black¬jack. To do
this we compute the root mean square (RMS) value of the column called
"Change in Advantage of Banker Bet."
We do this by squaring each of those numbers, counting the square for
zero-value cards four times because there are four times as many. Then
we add these squares, divide by 13 and take the square root. The
resulting root mean square or RMS value is .0064%%. That measures how
fast the deck shifts from its base start¬ing value for a full pack.
Taking one card of a given rank (we think of there being 13 ranks)
changes the fraction of any of these 13 ranks by an amount
32/416-31/415 which equals .00222. If we divide the RMS value by this
value we get .0288 as a measure of how rapidly this advan¬tage of the
two bets shifts from the starting value as the composi¬tion of the
deck changes.
Now we are going to compare this with the situation in black¬jack.
Table 3-3, for one-deck blackjack, can be treated in the same way to
see how fast the advantage changes in blackjack.
The Table is from Peter Griffin's book, Theory of Blackjack Revised,
page 44. We get RMS value of 0.467%%. The correspond-ing change in the
fraction of a single rank when one card is drawn
is 4/52-3/51 or .0181. The ratio of RMS value to this value is .258%%.
If we divide this by the corresponding result in baccarat we get 8.97,
which tells us that as the true count in blackjack varies the change
in player advantage or disadvantage shifts nine times as fast in
blackjack as it does in barrarat
Note that dividing the RMS value by the "change in fraction" of a
single pack adjusts the one-deck blackjack figures and the eight-deck
baccarat figures so they are comparable. If we had used, for example,
an eight-deck blackjack table instead, we still would have had a final
ratio of about nine times.
This allows us to translate how well a point count in baccarat works
compared with one in blackjack. In baccarat we start out with more
than a 1%% disadvantage and with eight decks. Imagine
a blackjack game with an eight-deck pack and a 1070 disadvan-tage. Now
imagine play continues through the blackjack deck. The blackjack deck
advantage from a -- 1%% starting level might, on very rare occasions,
shift 9%% to a + 8%% advantage.
As often as this happens in blackjack would be the approx¬imate
frequency with which we would get 1 /9th as much shift in
b cot-at; meaning from a -- 1%% advantage to a 0%% advantage or break
even for the banker bet. Since there are two bets, banker and player,
the player bet would also be break even or better about as often.
The conclusion is that you might expect to break even or better in
eight-deck baccarat about twice as often as you would expect to have
an 8%% edge in eight-deck blackjack. How often would you have a 1070
advantage in eight-deck baccarat? About twice as often as you would
get a 17%% edge in blackjack. The obvious con¬clusion is that
advantages in baccarat are very small, they are very rare and the few
that occur are nearly always in the last five to 20 cards in the pack.
The Tie Bet
In addition to wagers on the Player or Banker hands, the casinos offer
a bet on "ties." In the event the Banker and Player hands have the
same total, this bet gains nine times the amount bet. Otherwise the
bet is lost. The probability of a tie is 9.5156%%, hence the
expectation of the bet is --4.884 %% .
It is clear, however, that the probability and thus the expecta¬tion
of a tie depends on the subset of unplayed cards. For in¬stance, in
the extreme and improbable event that the residual deck
consists solely of ten-value cards, the probability of a tie is equal
to one and the expectation is nine. Thus card counting strategies are
potentially advantageous.
Using computer simulation, random subsets of different sizes were
selected from a complete 416-card (eight deck) pack. The results were
disappointing from a money-making perspec¬tive--the advantages which
occur with complete knowledge of the used cards are limited to the
extreme end of the pack and are generally not large. Practical card
counting strategies are at best marginal, and at best precarious, for
they are easily eliminated by Muffling the deck with 26 cards
remaining.

Other Games
While we have so far concerned ourselves solely with casino games,
some of the principles we have used are equally applicable to other
gambling situations. In this section, we apply mathematical theory to
horse race betting and backgammon.
Horse racing is the number one spectator sport in America and a large
amount of its success in this regard can be attributed to the wagering
opportunities. The racegoer becomes a participant in the spectacle.
While we offer no surefire system, we do suggest an approach that
shows promise for the gambler-investor.
Backgammon is an exceedingly complicated game from a mathematical
point of view. Because of the possibility of repeated restarts by the
counters, the game is potentially infinite. This impedes analysis, but
we offer several insights into the end game and the doubling cube,
parts of the game where optimal strategies can be computed.

Horse Racing - Pferdewetten
While so far I have limited myself to discussing casino games, the
concepts presented apply equally to other gambling games, such as
horse racing. At the racetrack, one is offered a variety of different
wagering possibilities. The player can wager on one or more horses to
win, place, or show, as well as combine any number of horses in the
various exotic bets (daily double, exactas, quinellas, etc.) The goal
of the gambler at the racetrack is to isolate in each race those bets
that yield a positive return, after considering the pari-mutuel
takeout and breakage. One approach with applications in horse racing
involves the use of a technique called hedging. Hedging, often used in
the securities and financial markets, involves taking two or more
investment positions simultaneously. The risks should cancel out and
an excess rate of expected return should remain.
Why "Hedge?"
In the securities and finance markets, to hedge is to take two or more
investment positions simultaneously. The risks should cancel out and
an excess rate of expected return should remain.
In a real horse race (or any pari-mutuel contest for that mat¬ter),
the true probabilities are not known. If we knew the true
probabilities or had better estimates than the pari-mutuel pool
offers, we might find horses with a positive expectation. Then we
could simply bet directly on those horses instead of develop¬ing the
following method for the daily double.
There is a plausible argument which upholds the pari-mutuel pool's
estimate of the true horse winning probabilities: "If there were a
method of predicting horse winning probabilities, and these
probabilities differed enough from the pari-mutuel pool's estimate to
give the predictor an advantage, then he would place bets and by so
doing would cause the pari-mutuel pool odds to shift in such a way as
to reduce that advantage. With many bettors and much information and
available computing power the overall ef¬fect is to reduce such
advantages so they are small or even become disadvantages."
In other words, "If you could beat the casinos at blackjack, then they
would change the game so you couldn't. Thus, there isn't any system
for beating them."
If we assume that pari-mutuel pool probabilities are true
prob¬abilities then the horse hedge system does not improve our edge
over the track take! You might think that makes the horse hedge idea
useless, but this is not true. Consider the daily double pool: The
payoffs should be consistent with the probabilities in the individual
race win pool; but in general, they aren't consistent. Thus, we have a
chance to use the probabilities based on the individual race win
pools.
The Daily Double
Let's apply the horse hedge idea to the daily double bet. The same
idea, with some modifications, also applies to exactas, pick six and
similar bets and to exactas, quinellas and trifectas in jai alai.
For a little background on the daily double, I quote from the book
Science in Betting: The Players and the Horses, by E. R. Da Silva, and
Roy M. Dorcus:
In daily double betting, any horse in the first race can be combined
with any horse in the second race, and to win the bettor must
successfully select the winners of both races. Some bettors combine
all of the horses in the first two races. If there are ten horses in
each race, in order to cover all possible combinations of horses, one
would have to buy one hundred tickets at $2.00 each. If by chance long-
odds horses won both races, it would be possible to make a profit on
that single daily double. However, such a situation is not common
throughout a week or a season. One daily double $2.00 ticket at Del
Mar recently paid $2,878.60, another paid $685 and there were in
addition three others during this season which paid over $200--yet
actual returns for this gPason were only $6,808. The average number of
horses was eleven in the first race and ten in the second. To have
com¬bined all these horses in all the daily doubles for this season
would have cost $200 per race, and since there were forty-two days in
this season, the total cost would have been $9,240, producing a loss
of several thousand dollars. Notice that they consider betting equal
amounts on each horse. From one season at Del Mar, they found that
$9,240 in total bets were returned; $6,808 for a payback fraction of
0.74 or a loss of 26%% of the amount bet. Thus, betting equal amounts
on each combination did not work.
Illustrated in Table 6-1 is the horse hedge method for daily doubles
in a real race. The Table lists the winning probabilities based on
odds for the first race at Del Mar on August 13, 1980. The horses are
listed according to post position in the first col-umn. The second
column has the handicapping odds given in the L.A. Times on the
morning of race day. The third column is ob¬tained from these odds by
taking the right hand number in the second column and dividing by the
sum of the two numbers.
For example, 30-1 gives a probability of 1/31 = 0.0323. For the horse
in the 13th post position, 7-2 gives a probability of 2/9 = 0.2222.
When there is no track take, the probabilities calculated this way
must add up to 1.00.
When there is a track take, the probabilities calculated from the
final payoff odds at race time will equal more than 1.00. In fact,
they add to 1/K, where K is the fraction of the pool, which is
returned to the bettors. This rule is not quite exact due to the
irregular effects of breakage, but the effects are generally small and
not worth discussing.
In order to correct for probabilities that do not add up to 1.00, we
add them, deducting horses which may have been scratched. We then use
the final total and divide it into the preliminary pro-babilities so
that it equals 1.00. (Corrected probabilities appear in column four.)
Column five gives the final odds on various horses. Column six has
corresponding uncorrected probabilities and column seven lists
corrected probabilities. Notice that column six adds to 1.2691; by
dividing this into 1.00 we get 0.7880 which corresponds to a track
take of 21.30%% for this particular race.
Column three equaled 1.7237 before deducting the horses which were
later scratched, making the track take too large. The sum for Del Mar
is typically about 1.20; therefore the handicapper's setting of the
odds was not consistent. On average, the odds were set too low in this
race for the horses. When four horses were scratched, the odds on the
rem