The mathematics of gambling

The mathematics of gambling is a collectionț and applyț specific probabilistic games of

luckș and I appearț lineapplied mathematics. DFrom a mathematical point of view, gambling is an experiment.
that generates various types of random events, the probability of which can be calculated using
propertyț the probabilitiesț on a finite field of events.

Experiments, events, probability fields

The technical processes of a game represent experiments that generate random events. Here are some
example:

Throwing dice in the game ofcrapsit is an experiment that generates events such as appearanceț ia
certain numbers, forț the collection of a certain sum of the numbers that appeared, occurrenceț of a number with certain
propertyț i (smaller than a certain number, greater than a certain number, even, odd, etc.).
Mulț the possible results indicateș the data of such an experiment is {1, 2, 3, 4, 5, 6} for
the throw of a die or {(1, 1), (1, 2), ..., (1, 6), (2, 1), (2, 2), ..., (2, 6), ..., (6, 1), (6, 2), ..., (6, 6)}
for throwing two dice. The last is a setț ordered pairsș There are 6 x 6 = 36 elements.
Events can be identified with mulț specifically hairț and I have a lotț possible results. Towards
example, the event appearsț the outcome of an even number in the dice-throwing experiment is represented
of the femaleț time {2, 4, 6}.

The turning of the wheelț A roulette is an experiment whose generated events can be occurrences.ț of a
a certain number, of a certain color or a certain propertyț and the numbers (small, large, odd,
odd, from a certain column, etc.). Manyț the time of possible resultsș the experiment of
rotation of the wheelț the rule is softț the time of the numbersș there are on the roulette: {1, 2, 3, ..., 36, 0, 00} for the roulette

American or {1, 2, 3, ..., 36, 0} for European roulette. These are the numbers inscribed on the wheel.
rouletteș on the game table. The event appearsț for a number roș you are represented by the mulț ima {1, 3,
5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36}.
 Shareț I reachț The dealer's ace at blackjack is an experiment that generates events such as
apariț the one of a certain bookț and a certain value for the first card received, obț the nurturing of a certain
total value from the first two booksț the primitives, afterș scored 21 points from the first three gamesț I received,
etc. In card gamesț we encounter many types of experimentsș and categories of events. Each type of
an experiment is its own set of possible results. For example, the distribution experiment of
first bookț the first player's right is muchț the time of possible results all the mulț the name of the 52
lambț I (or 104, if two packs are used). The experiment of distributing the second cardț i
the first player has the right to multiplyț the time of possible results manyț the subtitle of the 52 booksț I or
104), I canț in the first card distributed. The experiment of distributing the first two cardsț the first
player has the right manyț the time of possible results a mulț ordered pairs, namely all
arrangements of 2 cardsț in the 52 (or 104).

In a single-player blackjack game, the event player is dealt first.ș a straight 10-point card
the first card is represented by mulț the time of booksț i {10♠, 10♣, 10♥, 10♦, J♠, J♣, J♥, J♦, Q♠, Q♣, Q♥, Q♦,
K♠, K♣, K♥, K♦}. The event playerț a total of five points from the first two booksț i
primitive is represented by the multitudeț Combining {(A, 4), (2, 3)}ț ii with two elements from the setț time
the values of the carț which actually counts 4 x 4 + 4 x 4 = 32 combinationsț of booksț I (in valueș the symbol).

The 6/49 lottery, the experiment of drawing 6 numbers from the 49 generates events such as
apariț it is aș given numbers, occurrenceț it is five fromș given numbers, appearț to the skyț in a number
from a given set of numbers, etc. In this experiment, manyț the possible results tend to be manyț time
to everyone combinesț a set of 6 numbers from 49.

In classic poker, the experiment of dealing initial handsț five cardsț and generates events
distributeț the allocation of a certain value to a certain player, distributionț of a pair at the lastț in doi
players, distributeț there are four booksț I have identical symbols the sameț in a player's, etc. In this case,
badț the range of possible results is manyț the time of all combinesț five cardsț in the matters of the package
used. The distribution of two booksț the night of a player who discarded two cards is another experiment, it
to whom muchț The time of possible results is now manyț the time of all combinationsț two cardsț from the things of
the package, lessț in carsț the views of the observer who solves the probability problemț ii.
Combinations

Gambling is also a good field for exemplification ofcombinations,to permuteyesarrangements,

which are encountered everywhere: combinations of cards in a player's hand, on the table, or awaited; combinations
from numbers to the simultaneous throwing of several dice; combinations of numbers atlottosaucebingocombinations
from symbols to slots; permutations and arrangements in sports betting races and so on. Calculation
combinatorics is an important part of probabilistic applications in gambling. In these games,
most probabilistic calculations that use the classical definition of probability come down to counting
combinations.
For example, in a classic poker game, the event at least one player has four of a kind can be
identified with the set of all combinations of the type (xxxxy), where x and y are distinct card values.
This set has 13C(4,4)(52-4)=624 combinations, so it is too large to be able to be
held here. Possible combinations are (3♠ 3♣ 3♥ 3♦ J♣) or (7♠ 7♣ 7♥ 7♦ 2♣). These can be
identify with elementary events among those that form the measured event.

Strategy and mathematical expectation

Gambling is not just a basis for pure applications of probabilistic calculations, and gaming situations
they are not just isolated events whose numerical probability is established through mathematical methods–
they are also the games whose unfolding is influenced by human actions. In gambling games,
The human element has a decisive character. The player is not only interested in mathematical probabilities.
but of different gaming events, it also has expectations regarding the results of the game, as long as there is a
permanent interaction between the game and the player. In order to achieve favorable results from this
In interactions, players take into account all possible information, including statistics, to develop strategies.
for fun. As long as people rely on past statistical results to obtain a probability
subjective right degree of confidence, there is also the reverse psychological process – predicting statistical outcomes
future based on a given probability. Such predictive behavior is fully manifested in
gambling games, where the probabilities are associated with the bets placed in the game, in order to predict
average gains or losses in the future. Such a gain or loss predicted based on probabilities can
namemathematical expectationis the average value and is the sum of the products of the probability of each
possible outcome and its specific gain. Thus, the mathematical expectation represents the average sum that a
the player expects to win it for a specific bet repeated multiple times. A game or a situation of
a game in which the mathematical expectation for the player is zero (there is no gain, nor net loss) is called
correct game. The correct attribute does not refer here to the technical processes of the game, but to the balance of chances between
house and player.
The probabilistic model

A probabilistic model is based on an experiment and a mathematical structure attached to that experiment.
namely the field of events. The event is the structural unit with which the theory works.
of probabilities. In gambling, there are many categories of events and all can be predefined.
textual. In the previous examples of experiments in the field of gambling, I have become familiar with
a few events that these experiments generate. They represent a tiny part of the set
of all events, which is actually the set of parts of the set of possible outcomes. For a game
specific, the events can be of various types: – events regarding one's own game or the games of the opponents; –
events regarding single-player or multiplayer games; - immediate events or
long-term events. Each category can be further divided into many other subcategories, in
the function of the game it refers to. Mathematically speaking, these events are nothing more than
submultiples, and the event field is aBoolean algebraIAmong these events, we find events
elementary and compound, compatible and incompatible, independent and non-independent.

In the dice throwing experiment:

The event {3, 5} (whose textual definition is the occurrence of 3 or 5) is composed,

since {3, 5} = {3} U {5};
The events {1}, {2}, {3}, {4}, {5}, {6} are elementary;
–Evenimentele {3, 5}ș i {4} are incompatible or exclusive because they cannot be produced.
simultaneously
The events {1, 2, 5} and {2, 5} are compatible because their intersection is not empty;

In the experiment of rolling two dice one after the other, the event of getting a 3 on the first die
the appearance of 5 on the second die is independent, as the occurrence of the first does not depend
of the production of the second and vice versa.
In the experiment of distributing the two individual cards in Texas Hold'em Poker:

The event of dealing the cards (3♣, 3♦) to a player is an elementary event;
The event of distributing two valuable books to a player is composite because
the union of the events (3♣, 3♠), (3♣, 3♥), (3♣, 3♦), (3♠, 3♥), (3♠, 3♦) and (3♥, 3♦);
Event player 1 primeș the player 2 receives a pair of
compatible description (can be produced simultaneously);
Evenevents player 1 receives two larger cup connectors than player 2.
You receive two larger cup connectors that are incompatible (only one can be used)
produce);
–Event Player 1 receives (7, K) and Player 2 receivesș The (4, Q) are not independent.
(the production of the second depends on the production of the first)