Probability

The technical processes of a game stand for experiments that generate aleatory events.

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 2-size 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 2-size 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 6-size 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 5-card 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 2-card 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 2-card combinations from the 52, less the three cards you hold and less the two cards you discarded. This sample space counts the 2-size 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.

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 long-shot 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 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 (2-size combinations of cards as value and symbol).

This is in fact the basic principle that make the probability calculus performable in gambling: 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.