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The 'PROBABILITY BASED STRATEGY'
A theoretical essay from Romanian mathematician
Probability Theory is the only rigorous theory modeling
hazard, so any gambling strategy must be probability
This statement is the subject of my article below,
which is also part of my last book, "Probability Guide
of Gambling". This guide holds a large collection
of probability results and strategies, covering thousands
of gaming situations from all major games including
dice, slots, baccarat, roulette, blackjack, poker,
electronic poker, lottery and sports bets.
It is not a classical scientific study, but a manifold
application in 'practical guide' form. It is structured
so gamblers with minimal mathematical background can
skip the mathematical parts and directly find the
results they need. You can have your own electronic
version of this guide at its dedicated website
where you can find all details about it, including
structure, examples of how the guide helps in gambling
decisions and sample
In this guide, the presented numerical results are
accompanied where considered necessary by recommendations
to choose certain gaming variants or even certain
games. These recommendations are built on the 'probability
based strategy' and are stated within the context
of the personal criteria of each player, regarding
the goal of the game, the process of game and the
player's risk profile.
What does a probability based strategy actually
"A strategy using criteria of evaluation and comparison
of probabilities of various gaming events, in making
decisions for accomplishing the proposed goal",
might do for a general definition.
In most cases the declared goal of the gambler is
of cash, but also fun, entertainment or experiencing
certain emotions that are specific to competition
or risk might be his or her goals.
Here we will show why the probability based strategy
is optimal in situations where the goal of the player
is the winning cash.
As we said, a probability based strategy is using
decisions made as result of evaluating the probability
figures. These are decisions such as making a particular
playing decision at a certain moment of the game and
also of choosing a certain game to begin with.
Probability Theory is the only rigorous theory that
models the hazard. But it only offers measurements
of this hazard and not certainties about punctual
events. The certainty offered by the "Law of Large
Numbers" (see page 35 of my book) is one of the limit,
approximation and existence type. This theorem does
not provide precise information about occurrence of
expected events (for example, it cannot tell how many
times we have to throw a die to surely get a 5), but
even the limit behavior gives additional information.
The basic element of probability based strategy is
choosing the gaming variant that offers the highest
probability of occurrence of expected event, in condition
of identical ulterior advantages. Assume that the
player has reached a decision situation, at a certain
game. His or her options are the gaming variants A
and B, both providing the same ulterior advantages,
in case of occurrence of expected event.
Why is it good for the player to choose the gaming
variant that offers the higher probability of expected
event, as long as Probability Theory does not provide
him or her any certainty about it? If we answer this
question, the choice of probability based strategy
as being optimal will be justified.
Let's denote by P(A) the probability of occurrence
of expected event in experiment A (playing variant
A) and by P(B) the probability of occurrence of expected
event in experiment B (playing variant B) and assume
that P(A)> P(B). We will first provide a motivation
for choosing the gaming variant A, in case the gambler
is a regular player of this game.
Let's come back to the "Law of Large Numbers". This
theorem states that, in a sequence of independent
experiments, the relative frequency of occurrence
of a certain event is converging towards the probability
of that event.
Let's consider experiment A as being part of a sequence
of experiments, namely the sequence of experiments
of playing variant A by the same gambler, at the same
type of game, every time he or she reaches that respective
situation (in ulterior games).
Similarly, let's consider experiment B as being part
of the sequence of analogue experiments where the
gambler chooses variant B (hypothetically). Let's
denote by a(n) the number of occurrences of expected
event after n experiments of A type and by b(n) the
number of occurrences of expected event after n experiments
of B type.
We will show that a sufficiently big number N exists,
such that a(n) > b(n), for any n > N. The demonstration
is obvious: We assume the opposite, namely that for
any N, exists n > N such that a(n) < b(n) or a(n)
= b(n). In the same conditions, results a(n)/n < b(n)/n
or a(n)/n = b(n)/n. By passing to the limit in this
inequality and using the law of large numbers (a(n)/n
--> P(A), b(n)/n --> P(B)), we get that P(A)
< P(B) or P(A) = P(B) and that is false (we initially
assumed that P(A) > P(B)). Therefore, a(n) > b(n)
from a certain rank upward, within the sequence of
experiments. This means that the number of favorable
events will be bigger (cumulatively) in case of sequence
of experiments of A type, from a certain rank upward.
Although no mathematical result establishes which
this number N is (we only know it exists), the above
demonstration is offering a motivation (purely theoretical)
for choosing the gaming variant A, namely the following:
"We know there is a level N from which the cumulative
number of favorable events is bigger in the situation
of a sequence of experiments of A type. If we 'alter'
this sequence by introducing experiments of B type,
the fulfillment of the N experiments will be delayed."
This is the only theoretical motivation that uses
probability properties. Because we do not have information
about N, which can be any size and eventually never
reached, the motivation remains a theoretical one
and might have no practical coverage, except the cases
in which the difference P(A) - P(B) is significant.
Despite all these, it remains a motivation that
justifies the choice of probability based strategy.
But how do we justify the choice of probability based
strategy in a situation where the gambler is not a
regular player of the particular game, he or she plays
it only once so the time tendency is not a motivation
We have here a punctual gaming situation, in which
the gambler has to make a decision to choose one of
variants A and B, in a condition of identical ulterior
advantages. Why should he or she choose variant A,
which offers a higher probability for the expected
(A similar situation, not apparently related to gambling,
is the following:)
You are in a phone box and you must urgently communicate
important information to one of your neighbors. (Let's
say you left your front door open).
You have only one coin, so you can make one single
You have two neighboring houses.
Two persons are living in one of them and three
persons are living in the other.
Both their telephones have answering machines.
Which one of the two numbers will you call? Isn't
it true that you "feel" that you must choose the house
with 3 persons, because the probability for someone
to be at home is higher? But how do we rigorously
explain this optimal choice, based on probability
criteria? Coming back to the assumed gaming situation,
the "Law of Large Numbers" will still provide us a
theoretical motivation for the choice.
Although we don't have here a sequence of tangible
experiments, not even expected ones (as in the regular
player case) to include respective experiment, we
still can introduce it in such a sequence, so as for
application of the "Law of Large Numbers" to be available.
Let's consider the sequence of independent experiments
of all experiments of A type performed in time by
other players, in the same gaming situation, chronologically,
until the respective gaming moment. Similarly, let's
consider the sequence of experiments of B type, chronologically
performed until the respective moment.
For the two previously defined sequences of experiments
we can do the same deduction as in the first case
(regular player), based on the law of large numbers,
and the result will be: a sufficiently big number
N of experiments exists, such that for any n > N,
we have a(n) > b(n) or a(n) = b(n). Therefore, the
motivation of "not altering" the sequence of experiments
of A type still stands valid, even if only at the
So, the answer to the question generated by the your
required choice is the following: "Although we don't
know where the rank N (provided by the law of large
numbers) is standing and also if experiment A offers
a favorable result, it is proper to choose the gaming
variant A at least for the reason of other eventual
ulterior similar gaming situations to form an "unaltered"
sequence of experiments of A type."
As we said, the motivation is a purely theoretical
one, but it accomplishes the proposed goal, namely
to show that the probability based strategy is optimal.
The above presented situation of an isolated game
has a much smaller practical coverage than in the
case of a regular player, except the cases in which
the difference P(A) - P(B) is big enough.
The entire demonstration
above was done in a theoretical situation where
the favorable event expected after each gaming variant
offers the same advantage to the player. This advantage
could be the immediate winning of cash,
a superior position in the game, or other strategic
A probability based strategy can be seen as optimal
only after taking into account the player's goals
and these could be various. Thus, the probability
criteria do not always determine decisions, but also
the personal and subjective criteria of the player.
As is well known, probabilities do not provide certainties
about hazard. The probability based strategy is optimal,
but it still does not guarantee winning.
In other words, such a strategy is the optimum strategy
to adopt when trying to "push your luck", but it does
not bring you luck made to order.
Although the probability of getting 1 after one die
throw is 1/6, if we throw the die 6 times, this does
not assure us getting a 1, just as throwing the die
10 times does not assure it. Theoretically, is possible
to throw the die 1,000 times and not get a single
1, although it is improbable. The only thing we know
for sure is that the frequency of occurrences of 1
is getting closer to 1/6, as the number of throws
There is an obvious difference between the terms
"possible" and "probable". Probability is a measure,
mathematically rigorously defined, whereas "possible"
is a much more complex and difficult to define philosophical
category. Probability is modeling a minute part of
A hypothetical rigorous definition of "possible"
should include another "zero degree" philosophical
term, namely the reality. This dimension difference
between the two terms is the most relevant expression
of the fact that we cannot rule over the hazard, but
we can "push our luck" by using the laws of probability,
even if this often results in losses.
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