Chapter 4: The Science
Of Dice
A Word About
Chance
A Basic Probability Principle
Probability
Defined
Throws And Probabilities
Bad And Correct
Reasoning Points And Probabilities
Side Bets And
Probabilities
Every crapshooter wants to be a winner. And your
chances of being a winner, more often than not, are best when
you consider certain mathematical facts.
So we're going
to talk about probabilities and statistics here, as they apply
to throwing two dice. And although skilled mathematicians
figured it out, you don't have to be one to understand the
science of dice. We'll keep it as simple as possible.
First off, always remember that the second you
throw the dice, the odds are slightly against your making a
"pass" (251 to 245, or about 1.4%). Still, most players choose
to suffer this disadvantage. They like to roll the dice
themselves and set the pace of betting.
This is enough of a reason — unless they
foolishly believe that they have a magic touch with dice. In
an honest crap game with honest dice, there isn't such a
thing. We'll cover so-called "magic touches" and dice cheats
later.
Second, and from there on — the main
difference between a winner and a loser at craps lies in the
way bets are placed. The winner, as opposed to the constant
loser, knows his odds and percentages, and bets conservatively
and systematically. He rarely plays long-shots. He does play
hunches, but not often. He respects the science of dice.
All other things being equal, that's the only
logical answer to being a winner at craps. But carrying this
theory a step further — correct betting plus a sizable
bankroll can equal sizable winnings.
A Word About Chance
One of the oldest ideas is the belief that there
is an element of "chance" in the world, that things do happen
which are essentially unpredictable. How did this idea come
about? In a roundabout way. Primitive people believed "chance"
to be the doings of gods and devils. The Greeks of Homer's day
felt that "chance" took one or all of three forms:
(1) Events that the gods could alter;
(2)
Events beyond the control of gods or men;
(3) Events that
respected neither gods nor laws.
This last form was studied by the great Greek
thinkers and philosophers, and by other great minds on up
through the ages of science. And today, best corresponds to
what we mean by chance events. Of course, we use different
names for "chance events." We call them
"good luck" or "bad luck" or
"the breaks." But such ideas of luck are
merely misinterpretations of the idea of "chance."Chance is a
theory, in more technical language, a theory of probability.
And that's as technical as we need get.
By applying the
theory to simple problems in dice rolling, we can keep it
simple. Craps provides a good approach to the central ideas of
the theory of chance. You can see the laws of chance at work,
and with a minimum of effort, should have no trouble getting
the point — no pun intended.
A Basic Probability Principle
At first reading, the following few pages may
confuse all those who just can't stomach mathematics. But stay
with it. Because no other section in this book will help the
crapshooter as much as the theory of chance or probability.
The material has been adapted from Dr. Horace C. Levinson's
book, "The Science Of Chance,"
(Rinehart & Company, 1950). And we can't present it any
simpler than Dr. Levinson already has. Remember, too, that
theories of probability came from gambling houses, and there
are no better illustrations of the elementary uses of
probability than games of chance — like craps.
So let's begin. Consider one of a pair of dice,
an ordinary die with six sides. If we ask what the odds are
against throwing any one of the sides, say the six spot, you
will reply that they are evidently 5 to 1. And your reply will
be not only evident but correct.
If we ask you further why you say that the odds
are 5 to 1 against the six spot, you will perhaps
answer — the die is symmetrically constructed, and if you
throw it a large number of times there is no conceivable
reason why one of its sides should turn up more frequently
than another. Therefore each side will logically turn up on
about one sixth of the total number of throws, so that the
odds against any one of them, the six spot, for instance, are
very nearly 5 to 1.
Or you may say — at each throw there are
six possible results, corresponding to the six faces of the
die, and since the latter is symmetrically constructed, they
are all equally likely to happen. The odds against the six
spot are 5 to 1.
Instead of saying that the odds are 5 to 1
against the six spot, we can as well say that the chance of
probability of the six spot is 1 in 6, or 1/6.
So far,
that's simple enough.
Now, the first of your replies refers to the
result of making a long series of throws. What if defines is
therefore called a statistical probability. And notice the
words very nearly in your first reply.
The second reply refers to a single throw of the
die, and therefore appears to have nothing to do with
experience. What it defines is called an a priori
probability.
Each of the probabilities just defined is equal
to 1/6. This is an example of a general law known as the law
of large numbers, which tells us that these two probabilities,
when both exist, are equal. We can drop the Latin adjectives
now and speak simply of the probability.
Now we have the clue we are after. The chance or
probability of throwing the six spot is simply the number of
throws that give the six (one), divided by the total number of
possible throws (six). Thus, a basic probability principle.
And if you think that we've used a lot of words to get
here-remember, reasoned in such a way, you'll never forget
it.
Fundamental Definition of Probability
So we come to the fundamental definition of
probability. The probability of an event is defined as the
number of cases favorable to the event, divided by the total
number of possible cases, provided that the latter are equally
likely to occur.
Don't fret, we'll explain.
This practical rule tells us how to go about
finding the probabilities or chances in a large variety of
problems, in particular those relating to games of chance.
To illustrate, let's apply this rule to another
problem with one die: What is the chance of throwing either a
five spot or a six spot?
First list the possible cases. With one die
there are always six, the number of its sides, in other
words.
Next pick out the favorable cases, which are the
five spot and the six spot - there are two favorable
cases.
The probability we are looking for is therefore
the fraction whose numerator is 2 and whose denominator is 6;
it is 2/6 or 1/3. Note: Every probability that can be
expressed in numbers takes the form of a fraction.
Throws and Probabilities
Now let's get into the game and throw two dice
as in our game of craps. What is the chance of throwing a
total of 7? Well, first we must know the total number of
possible cases. In other words, we must count the number of
distinct throws that can be made with two dice. Clearly there
are thirty-six.
But before going any further, let's give each of
our dice a coat of paint, one blue, the other red. Why?
Because unless we have some way of distinguishing the two dice
from each other - we'll fall into a stupid error as the novice
once did — and some still do.
Now to list the favorable cases, those throws
that total 7. If we turn up the ace on the blue die and the
six spot on the red die, the total is 7, and we have the first
of the favorable cases. But we also obtain a total of 7 if we
turn up the ace on the red die and the six spot on the blue
die. Thus there are two favorable combinations involving the
ace and the six spot. Get the idea?
This simple fact, so obvious when the dice are
thought of as painted different colors, was once a real
stumbling block to the novice with a slow mind. And believe it
or not, it still is. If we continue the listing of the
favorable cases, we can write the complete list in the form of
a short table:
BLUE
RED
1
6
6
1
2
5
5
2
3
4
4
3
Each line of this table indicates one
favorable combination, so that there are six favorable cases.
As the total possible cases number thirty-six, our fundamental
rule tells us that the probability of throwing 7 with two dice
is 6/36 or 1/6.
This result is correct, provided that each of
the thirty-six possible cases is equally likely. To see that
this is indeed the case, we remember that each of the six
faces of a single die is as likely to turn up as another, and
that throwing two dice once comes to the same thing as
throwing one die twice, since the throws are independent of
each other.
Various Throws and Probabilities
Let's be elementary again. Craps is played with
two ordinary dice. The man who has possession of them wins
immediately if the total of the spots on his first throw is 7
or 11. He loses immediately if this total is 2, 3, or 12 but
continues to throw the dice. If the total is any one of the
remaining six possible points, he neither wins nor loses on
the first throw, but continues to roll the dice until he has
either duplicated his own first throw, or has thrown a total
of 7. The total shown by the dice on his first throw is called
the crapshooter's point. If he throws his point first, he
wins. If he throws 7 first, he loses and is required by the
rules of the game to give up the dice.
You might ask first whether the player
throwing the dice has the odds with him or against him. We
gave you the answer before, but we'll answer again in another
way. As it is well known that when craps is played in gambling
houses, the house never throws the dice. And as we know that
the odds are always with the gambling house, we may feel
reasonably certain in advance — that our calculations
will show that the odds are against the player with the dice.
Other questions, indeed, can be asked concerning odds in favor
of or against making various throws. They will be answered as
we go along.
Total of
Throw
Probability
2 or
12 1/36
3
or
11 2/36
or 1/18
4 or
10
3/36 or 1/12
5 or
9
4/36 or 1/9
6 or
8 5/36
7
6/36 or 1/6
By this table and the rules of the game, we
know that the probability that a player throwing the dice will
win on his first throw is 8/36 or 1/9. The chance that he will
win or lose on the first throw is therefore 3/9 or 1/3. The
probability that the first throw will not be decisive is
2/3.
We have now taken care
of the cases where the first throw yields one of the following
points — 2, 3, 7,11,12. Suppose now that some other total
results from the first throw, a 6 for instance. Six, then,
becomes the player's point. He must throw a 6 before a 7 in
order to win. What are the chances? The chance of throwing 7
is 6/36, and the chance of throwing 6 is 5/36.
Before we answer, let's
put in right here an example of reasoning, which well applies
in this case and others.
Bad and Correct Reasoning
One might be tempted to reason as
follows: The ratio of the chances of throwing a 7 to those of
throwing a 6 is 6 to 5. Therefore the probability that a 6
will appear before a 7 is 5/11. The probability that 7 will
appear first is 6/11.
This is an example of bad reasoning which gives the
correct result. It is bad reasoning because, as we have stated
it, the conclusion does not follow from the principles of
probability. Also, we have assumed that either a 6 or a 7 is
certain to appear if we roll the dice long enough.
So let's make this
reasoning correct. The player's point is 6. The probability
that he will throw a neutral point (all throws except 6 and 7)
is 25/36. The chance that he will make two consecutive neutral
throws is (25/36)2, and the probability that he will make this
throw x times in succession is (25/36) x.
For the game to
continue indefinitely, it would be necessary for the player to
throw an indefinitely large number of neutral throws in
succession. But the probability of doing so is (25/36) x,
which becomes smaller and smaller as x increases. Since we can
make it as small as we please by taking x large enough, we can
legitimately consider the probability that the game will not
end as 0.
Read that over
again. It's not as complicated as it might
sound.
Now, with all neutral
throws thus eliminated, there remains to be considered only
the throws of 6 and 7. We can now conclude that out of the
eleven possible cases that give 6 or 7, five favor the 6 and
six favor the 7. Therefore, the probability of throwing a 6
before throwing a 7 is 5/11. Correct result with correct
reasoning.
Points and Probabilities
It is easy to make the
corresponding calculation for each of the six possible points.
The probability is the same for the point 6 as it is for 8,
the same for 5 as for 9, and so on, just as in the preceding
table. The calculation for each of the possible points gives
the following results:
Probability of Throwing Indicated
Point
Before Throwing a Seven
Points
Probability
4
(or
10) 3/9
5
(or
9) 4/10
6
(or
8) 5/11
We wish to know the
probability, before the first throw in the game is made, that
the crapshooter will win on each of the points just listed. In
order to win in this manner he must of course, neither win nor
lose on his first throw.
We can find what we wish
by combining the two preceding tables as follows:
Probability of Winning
On Indicated
Point
Points
Probability
4 (or
10) 3/36 X
3/9 = 1/36
5 (or
9) 4/36
X 4/10 = 2/45
6 (or
8) 5/36
X 5/11 = 25/396
This simply means that
the probability that the crapshooter will win on a point
specified in advance, say point 5, is 2/45, and the
probability that he will win on point 9 is also
2/45.
To find the total
probability that the crapshooter will win, we add to his
probability of winning on the first throw, which is 2/9, the
sum of the three probabilities shown in this table, each
multiplied by 2. This gives a probability of 244/495, or
0.49293.
This is the sort of
result we expected to begin with. The odds are against the
crapshooter, although by only a very small margin. In fact, in
the long run, the crapshooter loses (or the gambling house
wins, if there is one) only 1.41 per cent of the amounts
staked. Compare this to the loss of 2.7 per cent on the
numbers in roulette.
Side Bets and Probabilities
A great deal of the
interest in the usual crap game comes from the side bets of
various sorts. And many of these bets consist in giving odds
that one total will appear before another.
The fair odds on all
such bets (regardless of what's marked on the average crap
table layout), including those totals that cannot be a
player's point, by the rules of the game, are shown in the
following table.
The table gives the odds
against throwing one or the other of the totals given in the
left-hand column before throwing one or other of the totals in
the top line.
The listing together of
two numbers, such as 4 and 10, is done only to abbreviate. The
odds as given apply to either 4 or 10, not to both on the same
series of throws.
Odds Against
Throwing
Before
Throwing
3 (or 11) 4 (or 10) 5 (or
9) 6 (or 8) 7
2 (or
12)................ 2 to
1 3 to
1 4 to
1 5 to 1
6 to 1
3 (or 11)................................ 3 to
2 4 to
2 5 to 2
6 to 2
4 (or
10)................................................ 4 to
3 5 to 3
6 to 3
5 (or
9).............................................................. 5
to 4 6 to 4
6 (or
8)............................................................................6
to 5
Thus, the correct odds
against throwing a 4 before throwing a 6, for instance, are 5
to 3. And so on. So if a player ever says to the shooter, "Two
to one you get a 6 before a 4" — and this and like bets
are often foolishly made — it's a bad bet. In the long
run he will lose 1/24 of the amount staked, as he will in
varying degrees with all other such odds given-odds that are
out-of-line with the facts above.
As we said at the start,
this section on craps is the most important in the book. It
will profit you to reread it — maybe many times. In other
words, know your odds in the game of craps and expect returns
accordingly. Of course, you'll continue tempting
"chance" and fooling with
"luck" — as many crap-shooters do and always
will. But it's the wise ga mbler, and most often the winning
gambler, who keeps such hunches to a minimum. In the long run,
the theory of probability runs the game.
| 
