Probability Theory

Probability theory is concerned with determining the likelihood that a certain event will occur during a given random experiment. In this sense, an experiment is any situation which involves observation or measurement. Random experiments are those which can have different outcomes regardless of the initial conditions and will be heretofore referred to simply as experiments.

The results obtained from an experiment are known as the outcomes. When a die is rolled, the outcome is the number found on the topside. For any experiment, the set of all outcomes is called thesample space. The sample space, S, of the die example, is denoted by S= which represents all of the possible numbers that can result from the roll of a die. We usually consider sample spaces in which all outcomes are equally likely.

The sample space of an experiment is classified as finite or infinite. When there is a limit to the number of outcomes in an experiment, such as choosing a single card from a deck of cards, the sample space is finite. On the other hand, an infinite sample space occurs when there is no limit to the number of outcomes, such as when a dart is thrown at a target with a continuum of points.

While a sample space describes the set of every possible outcome for an experiment, an event is any subset of the sample space. When two dice are rolled, the set of outcomes for an event such as a sum of 4 on two dice is represented by E =.

In some experiments, multiple events are evaluated and set theory is needed to describe the relationship between them. Events can be compounded forming unions, intersections, and complements. The union of two events A and B is an event which contains all of the outcomes contained in event A and B. It is mathematically represented as A ∪ B. The intersection of the same two events is an event which contains only outcomes present in both A and B, and is denoted A ∩ B. The complement of event A, represented by A', is an event which contains all of the outcomes of the sample space not found in A.

Looking back at the table we can see how set theory is used to mathematically describe the outcome of real world experiments. Suppose A represents the event in which a 4 is obtained on the first roll and B represents an event in which the total number on the dice is 5.

The compound set A ∪ B includes all of the outcomes from both sets,

The compound set A ∩ B includes only events common to both sets,. Finally, the complement of event A would include all of the events in which a 4 was not rolled first.