Gaming Mathematics - The Probability Model

The Probability Model

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.

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. 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.

In the experiment of rolling a die:

• Event {3, 5} (whose literal definition is occurrence of 3 or 5) is compound because {3, 5}= {3} U {5};
• Events {1}, {2}, {3}, {4}, {5}, {6} are elementary;
• Events {3, 5} and {4} are incompatible or exclusive because their intersection is empty; that is, they cannot occur simultaneously;
• Events {1, 2, 5} and {2, 5} are nonexclusive, because their intersection is not empty;
• In the experiment of rolling two dice one after another, the events obtaining 3 on the first die and obtaining 5 on the second die are independent because the occurrence of the second event is not influenced by the occurrence of the first, and vice versa.

In the experiment of dealing the pocket cards in Texas Hold’em Poker:

• The event of dealing (3♣, 3♦) to a player is an elementary event;
• The event of dealing two 3’s to a player is compound because is the union of events (3♣, 3♠), (3♣, 3♥), (3♣, 3♦), (3♠, 3♥), (3♠, 3♦) and (3♥, 3♦);
• The events player 1 is dealt a pair of kings and player 2 is dealt a pair of kings are nonexclusive (they can both occur);
• The events player 1 is dealt two connectors of hearts higher than J and player 2 is dealt two connectors of hearts higher than J are exclusive (only one can occur);
• The events player 1 is dealt (7, K) and player 2 is dealt (4, Q) are non-independent (the occurrence of the second depends on the occurrence of the first, while the same deck is in use).

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: