Events and sample spaces in gambling
The technical processes of a game stand for
experiments that generate aleatory events. Such events occur wherever
the games are played, at home, in a
casino or
online casino.
Throwing the dice in craps is an experiment that
generates events such as occurrences of certain numbers on the dice,
obtaining a certain sum of the shown numbers, obtaining numbers with
certain properties (less than a specific number, higher that a specific
number, even, uneven, and so on). The sample space of such an experiment
is {1, 2, 3, 4, 5, 6} for rolling one die or {(1, 1), (1, 2), ..., (1,
6), (2, 1), (2, 2), ..., (2, 6), ..., (6, 1), (6, 2), ..., (6, 6)} for
rolling two dice. The latter is a set of ordered pairs and counts 6 x 6
= 36 elements.
The events can be identified with sets, namely
parts of the sample space.
For example, the event occurrence of an
even number is represented by the following set in the experiment of
rolling one die: {2, 4, 6}.
Spinning the roulette wheel is an experiment whose
generated events could be the occurrence of a certain number, of a
certain color or a certain property of the numbers (low, high, even,
uneven, from a certain row or column, and so on). The sample space of
the experiment involving spinning the roulette wheel is the set of
numbers the roulette holds: {1, 2, 3, ..., 36, 0, 00} for the American
roulette, or {1, 2, 3, ..., 36, 0} for the
European. The event
occurrence of a red number is represented by the set {1, 3, 5, 7, 9,
12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36}. These are the
numbers inscribed in red on the roulette wheel and table.
Dealing cards in blackjack is an experiment that
generates events such as the occurrence of a certain card or value as
the first card dealt, obtaining a certain total of points from the first
two cards dealt, exceeding 21 points from the first three cards dealt,
and so on.
In card games we encounter many types of
experiments and categories of events. Each type of experiment has its
own sample space. For example, the experiment of dealing the first card
to the first player has as its sample space the set of all 52 cards (or
104, if played with two decks). The experiment of dealing the second
card to the first player has as its sample space the set of all 52 cards
(or 104), less the first card dealt. The experiment of dealing the first
two cards to the first player has as its sample space a set of ordered
pairs, namely all the 2size arrangements of cards from the 52 (or 104).
In a game with one player, the event the player
is dealt a card of 10 points as the first dealt card is represented
by the set of cards {10♠, 10♣, 10♥, 10♦, J♠, J♣, J♥, J♦, Q♠, Q♣, Q♥, Q♦,
K♠, K♣, K♥, K♦}.
The event the player is dealt a total of five
points from the first two dealt cards is represented by the set of
2size combinations of card values {(A, 4), (2, 3)}, which in fact
counts 4 x 4 + 4 x 4 = 32 combinations of cards (as value and symbol).
In 6/49 lottery, the experiment of drawing six
numbers from the 49 generate events such as drawing six specific
numbers, drawing five numbers from six specific numbers, drawing four
numbers from six specific numbers, drawing at least one number from a
certain group of numbers, etc. The sample space
here is the set of all
6size combinations of numbers from the 49.
In classical poker, the experiment of dealing the
initial five card hands generates events such as dealing at least one
certain card to a specific player, dealing a pair to at least two
players, dealing four identical symbols to at least one player, and so
on. The sample space in this case is the set of all 5card combinations
from the 52 (or the deck used). Dealing two cards to a player who has
discarded two cards is another experiment whose sample space is now the
set of all 2card combinations from the 52, less the cards seen by the
observer who solves the probability problem.
For example, if you are in play in the above
situation and want to figure out some odds regarding your hand, the
sample space you should consider is the set of all 2card combinations
from the 52, less the three cards you hold and less the two cards you
discarded. This sample space counts the 2size combinations from 47.
All these isolated examples are not the most
representative from the respective games. They are presented as an
introduction to what mathematics in games of chance means, namely
particular probability models, in which probability theory can be
applied to obtain the probabilities of the events we are interested in.
Probability models in gambling
A probability model starts from an experiment and a
mathematical structure attached to that experiment, namely the field of
events. The event is the main unit probability theory works on. In
gambling/online
gambling, there are many categories of events, all of which can be
textually predefined. In the previous examples of gambling experiments
we saw some of the events that experiments generate. They are a minute
part of all possible events, which in fact is the set of all parts of
the sample space. For a
specific game, the various types of events can
be:
– Events related to your own play or to opponents’ play;
– Events related to one person’s play or to several persons’ play;
– Immediate events or longshot events.
Each category can be further divided into several other subcategories,
depending on the game referred to. From a mathematical point of view,
the events are nothing more than subsets and the field of events is a
Boole algebra.
The complete mathematical model is given by the
probability field attached to the experiment, which is the triple
sample space—field of events—probability function. For any basic
application in a game of chance, the probability model is of the
simplest type—the sample space is finite, the field of events is the set
of parts of the sample space, implicitly finite, too, and the
probability function is given by the definition of probability on a
finite field of events. From this definition and the axioms of a Boole
algebra flow all the properties of probability that can be applied in
the practical calculus in gambling. Any predictable event in gambling,
no matter how complex, can be decomposed into elementary events with
respect to the union of sets.
For example, if we consider the event player 1
is dealt a pair in a Texas Hold’em game before the flop, this event
is the union of all combinations of (xx) type, x being a value
from 2 to A. Each such combination (xx) is in turn a union of the
elementary events (x♣ x♠), (x♣, x♥), (x♣, x♦), (x♠, x♥), (x♠, x♦) and
(x♥, x♦), all of which are equally possible. The entire union counts
13C(4, 2) = 78 elementary events (2size combinations of cards as value
and symbol).
There are also applications in gambling involving
events related to the longrun play, whose suitable probability models
are chosen from classical probability distributions such as Bernoullian,
Poisson, Polynomial, or Hypergeometric distribution.
Probability calculus in gambling
Probability calculus actually means to use of all
the properties of the probability in order to obtain explicit formulas
of the probabilities of the measured events and apply these formulas in
the given circumstances for obtaining a final numerical result.
The basic principle that makes the probability
calculus performable in gambling is that any compound event can be
decomposed into equally possible elementary events, then the probability
properties and formulas can be applied to it to find its numerical
probability. Besides the basic properties of probability, the formulas
from the classical distributions are of great help for some complex
gaming events.
In most probability computations in gambling, the
application of the formulas reverts to combinatorial calculus, which is
an essential tool for this type of applications.
The hardest task of the gaming mathematician
performing probability calculus is to provide explicit formulas in
algebraic form, which express the sought probabilities.
Expected value
The mathematical model of a game of chance involves
not only probability, but also other statistical parameters and
indicators, of which the expected value is the most important.
In gambling, probabilities are associated with
stakes in order to predict an average future gain or loss. This
predicted future gain or loss is called expectation or
expected value (EV) and is the sum of the
probability of each possible outcome of the experiment multiplied by its
payoff (value). Thus, it represents the average amount one expects to
win per bet if bets with identical odds are repeated many times.
For example, an American roulette
wheel has 38 equally possible outcomes. Assume a bet placed on a single
number pays 35–to–1 (this
means that you are paid 35 times your bet and your bet is returned, so
you get back 36 times your bet). So, the expected value of the profit
resulting from a one dollar repeated bet on a single number is
,
which is about –$0.05. Therefore one expects, on average, to lose over
five cents for every dollar bet.
The mathematical expectation
(expected value) is defined as follows: Definition: If X is a discrete
random variable with values and
corresponding probabilities ,
,
the sum or sum of series (if convergent)
is
called mathematical expectation, expected value or mean of variable X.
So, the mathematical expectation is a weighted mean, in the sense of the
definition given above. It terms of gambling, this value means the
amount (positive or negative) a player should expect, if performing same
type of experiment (game or gaming situation) in identical conditions
and by making the same bet, via mathematical probability. The expected
value being negative is the sign for that bet being profitable for the
house, by ensuring its house edge. In practice, expected value is
a statistical parameter assigned to every bet that has a computable
probability and a payout, even though one cannot run that bet infinitely
many times. Together with probability, expected value stands as
criterion for decisions in games and betting where the bets have
specific payouts.
The role of probability in gambling strategies
A strategy only makes sense if related to both game
and player. That is because it is the player who builds and applies a
strategy, according to his/her own goals. Among all the criteria used
within a strategy, there are subjective personal criteria related to
player's profile, but also objective criteria, of which probability is
the most important. Acting in a certain way in a particular gaming
situation basing on the evaluation and comparison of odds/probabilities
means to make a decision based on probability as the most objective
measure of possibility we have. These might be decisions related to
gaming situations during the game, but also of choosing a certain game
or another, quitting a game for another or even not playing at all. Even
objective, the criteria standing at the base of such decisions still can
have a subjective component, which is the threshold of afforded risk.
This
parameter is an average probability to which each player refers when
making decisions on his/her next action and is the level of the
probability of failure with which the player is comfortable staying in
the game. A general criterion using the threshold of afforded risk in a
gaming strategy would be: “If the odds for my opponents (or house) to
beat (gain an advantage over) me at moment t are higher than p,
then I will quit (stand, fold, etc.).” The threshold of afforded risk is
p and the value of p is different for each player and even
can change several times during a game, for the same player.
A probabilitybased strategy consists (and
is defined as) only of decisions resulting from the evaluation and
comparison of probability results. Mathematics proved that a
probabilitybased strategy is theoretically optimal among other types of
strategies, in the respect of gaining advantage during the game, at its
end, and on the longrun play.
There exist strategies for any game of chance,
either played against with opponents or the house, and all of them can
be probabilitybased. While in games like poker strategy includes the
interaction with the opponents and applies to each round (and
probability is essential in expressing the strength of a hand), in
simple games like lottery or slots the only strategy is the strategy of
choosing (choosing what numbers to play and how frequent to play in
lottery, choosing which game to play, how many paylines to enable, and
choosing not to play a particular game in slots, etc.) and this strategy
can also be based on probability.
Sources 
All these subjects of mathematics of gambling are discussed in depth in
the book PROBABILITY GUIDE TO GAMBLING, which also holds a great
collection of precalculated probability results for the major
casino
games (Slots, Roulette, Blackjack, Baccarat, Draw Poker and Texas Hold'em Poker),
lottery and sport betting. For each game we have a whole book dedicated
to its mathematics. See the Books
section for details. You can also consult our Articles
section for subjects on mathematics applied in gambling, written by
specialists. 

