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The mathematics of gambling are a collection of probability applications encountered in games of chance and can be included in game theory. From a mathematical point of view, the games of chance are experiments generating various types of aleatory events, the probability of which can be calculated by using the properties of probability on a finite space of events. The technical processes of a game stand for experiments that generate aleatory events. Here are a few examples:.
A probability model starts from an experiment and a mathematical structure attached to that experiment, namely the space field of events. The event is the main unit probability theory works on. In 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. Each category can be further divided into several other subcategories, depending on the game referred to. These events can be literally defined, but it must be done very carefully when framing a probability problem. From a mathematical point of view, the events are nothing more than subsets and the space of events is a Boolean algebra.
Among these events, we find elementary and compound events, exclusive and nonexclusive events, and independent and non-independent events. These are a few examples of gambling events, whose properties of compoundness, exclusiveness and independency are easily observable. These properties are very important in practical probability calculus. 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 space 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 space of events:. Combinatorial calculus is an important part of gambling probability applications. In games of chance, most of the gambling probability calculus in which we use the classical definition of probability reverts to counting combinations.