THE PROBABILITY-BASED STRATEGY
Probability Theory is the only rigorous theory modeling the hazard, so
any gambling strategy must be probability-based.
This statement is the subject of this article, which is 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: dice, slots, baccarat,
roulette, blackjack, poker, electronic poker, lottery and sport bets.
It is not a classical scientific study, but a manifold application in
a practical guide form. It is so structured for gamblers with no
minimal mathematical background to skip the mathematical parts and
directly pick the results they need. You can have your own electronic
version of this guide at its dedicated web site
<www.probability.go.ro> , where you can find all details about it,
including structure, examples of how the guide helps in gambling
decisions and paragraph samples.
In this guide, the presented numerical results were accompanied -
where considered as necessary - by recommendations of choosing certain
gaming variants or even certain games.
These recommendations were built on the probability-based strategy and
were stated under reserve of the personal criterions of player,
regarding the goal of the game, the process of game and assuming the
risks.
What actually means a probability-based strategy?
The following enunciation: "The strategy using criterions of
evaluation and comparison of probabilities of various gaming events,
in making decisions for accomplishing the proposed goal" might stand
for a general definition.
In most of cases, the declared goal of gambler is the immediate cash
winning, but also fun, entertainment or living certain emotions that
are specific to competition or risk might stand for his/her goals.
Here we will show why the probability-based strategy is optimum,
obviously in situations where the goal of game is the cash winning.
As we said, a probability-based strategy is using decisions made as
result of evaluating the probability figures.
These are decisions of choosing a particular gaming variant at a
certain moment of the game, but also of choosing a certain game.
The Probability Theory is the only rigorous theory that models the
hazard.
But it only offers measurements of this hazard and not certainties
about the punctual events.
The certainty offered by the "Law of Large Numbers" (see page 35 of my
book) is one of limit, approximation and existence type.
This theorem does not provide precise information about occurrence of
expected events (for example, it not tell how many times we have to
throw a die for surely get a 5), but even the limit behavior means
additional information.
The basic element of probability-based strategy is the decision of
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/her choosing options are the gaming variants A and B, both
providing same ulterior advantages, in case of occurrence of expected
event.
Why is good for the player to choose the gaming variant that offers
higher probability of expected event, as long as Probability Theory
does not provide him/her any certainty about it?
If we answer to this question, the choice of probability-based
strategy as being optimum will be justified.
Let's denote by P(A) the probability of occurrence of expected event
in the experiment A (playing variant A) and by P(B) the probability
of occurrence of expected event in the experiment B (playing variant
B) and assume that P(A)> P(B).
We will first provide a motivation of choosing the gaming variant A,
in case the gambler is a regular player of respective game.
Let's come back to the "Law of Large Numbers". This theorem says that,
in a sequence of independent experiments, the relative frequency of
occurrence of certain event is converging to 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 the variant
A by the same gambler, at the same type of game, every time he/she
reaches respective situation (in ulterior games).
Similarly, let's consider experiment B as being part of the sequence
of analogue experiments where 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 does establish 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 situation of a sequence of experiments
of 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 not have information about N, which can be any big and
eventually never reached, the motivation remains a tendency one and
might have no practical coverage, except the cases in which the
difference P(A) - P(B) is significant.
Despite al these, it remains a motivation that justifies the choice of
probability-based strategy.
But how we justify the choice of probability-based strategy in
situation where the gambler is not a regular player of respective
game, eventually he/she plays it only once, so the time tendency is
not a motivation any more?
We have here a punctual gaming situation, in which gambler has to make
a decision of choosing one of variants A and B, in condition of
identical ulterior advantages.
Why should he/she choose variant A, which offers a higher probability
for the expected event?
A similar situation, not apparently related to gambling, is the
following:
"You are in a phone-call box and you must urgently communicate
important information to one of yours neighbors (let's say you left
your front door opened). You have only one coin, so you can make one
single call. 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 two numbers will you call?
Isn't 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 optimum choice, based on
probability criterion?
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 regular player case), to include respective
experiment, we still can introduce it in such a sequence, 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 respective gaming moment.
Similarly, let's consider the sequence of experiments of B type,
chronologically performed until respective moment.
For the two previously defined sequences of experiments we can do the
same deduction as in 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 only at theoretic
level.
So, the answer to the question generated by the decision of 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, 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 optimum.
The above presented situation of isolated game has a much smaller
practical coverage than in case of a regular player, except the cases
in which the difference P(A) - P(B) is enough big.
The entire above demonstration was done in theoretical situation where
the favorable event expected after each gaming variant offers same
advantage to player.
This advantage could be immediate cash winning, a superior position
within game or other strategic advantages.
A probability-based strategy can be optimum only related to player's
goals and these could be various.
Thus, the probability criterions do not always make the decisions, but
also personal and subjective criterions of the player.
As known, the probabilities do not provide certainties about hazard.
The probability-based strategy is optimum, but it still not guarantees
for sure winning. In other words, such strategy is optimum when trying
to "push the luck", but it not bring the luck 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, as well
as throwing the die 10 times does not assure it.
Theoretically, is possible to throw the die 1000 times and not get one
single 1, although it is improbable.
The only thing we surely know is that the frequency of occurrences of
1 is getting closer to 1/6, as the number of throws is growing.
There is an obvious difference between terms "possible" and
"probable".
Probability is a measure, mathematically rigorously defined, when
"possible" is a much complex and undefined philosophical category.
Probability is modeling a minute part of the possible.
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 can not rule over the hazard, but we
can "push the luck" by using the probability laws, even if this often
results in looses.
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