Event (probability theory)

An event (also random event) is in probability theory a lot of results of a random experiment, which a probability can be assigned. For example, corresponds to a roll of the dice, the event " an even number dice " the subset {2, 4, 6} of the total set {1, 2, 3, 4, 5, 6} of all possible outcomes. It is said that an event occurred if it contains the result of the random experiment as an element.

The identical with the result set event is referred to as a safe event, as it always occurs. In contrast, refers to the identical with the empty set event as impossible event: It never enters. In the example of the dice roll the certain event is the set { 1,2,3,4,5,6 } and the impossible event the crowd.

Definition

Is a probability space, such an event is called. The events of a probability space is therefore that subsets of the result set, which lie in the σ - algebra. A set of events, which is a subset of is also called event system.

If finite or countably infinite, then one chooses for in general the power set of. In this case, events are simply arbitrary subsets of the result set.

For an event the number is, the probability of.

Set Operations with events

Is a result of a random experiment is an event and then they say, in the case also indicates that the event occurs.

Subsets and equality

If an event is a subset of another event is ( as noted ), occurs with the event always also the event. It then also says: The event attracts the event by itself. Of the probabilities is considered in this case. That is: Attracts the event after the event, then the probability of at least as large as that of.

It is precisely when and applies. Equality of events therefore means that the event is the event in the same way after the event runs like the event.

Intersection and disjointness

The intersection of two events is an event again. It occurs if and only if and both occur.

If true, that is the common occurrence of and is impossible, they say, the two events are mutually exclusive. The events and are then also called disjoint or incompatible.

Are more common events, then the section

The event that exactly occurs when all occur. The events are called pairwise disjoint if for all with.

Union

The union of two events is an event again. It occurs if and only if either or or both events occur. In other words, occurs when at least one of the two events or occurs.

Always applies to the probability of intersection and union the formula

Specifically, in the case of disjoint events.

Are general events, then the association

The event that exactly occurs when at least one of the inlet.

It is always the so-called σ - subadditivity

In the case of pairwise disjoint events in this case does equality.

For the probability of arbitrary unions of finitely many events the Siebformel applies.

Full Event System

A family of events that are pairwise disjoint and their union resulted wholly, also called complete event system or disjoint decomposition of (in general, a partition ). In this case, that for each outcome of the random experiment occurs exactly one of the events of the disjoint decomposition.

Complement, and difference

The complementary event occurs if and only if the event does not occur. It is also called counter- event and denoted by (or with ). Its probability is

For the complements of intersection and union sets the de Morgan 's formulas apply

Especially for two events is considered as well.

The difference amount is the event that exactly occurs when the event, but not at the same time the event occurs. It is

For his chance. In the special case follows.

Symmetric difference

Another set operation is the symmetric difference

Two events and. The event occurs if and only if either occurs or ( but not both), so if exactly one of the two events occurs. It is

Independent events

The two events and are called independent, if

Using the formula for the conditional probability can be as the

Write a prerequisite.

General ie, a family of events independent if for every finite subset:

The events are called pairwise independent if

Applies to all. Independent events are pairwise independent, but the converse is not true in general.

Elementary event

Sometimes the singleton events are also called elementary events. Is at most countable, then can be achieved by setting the probabilities of all the elementary events with the help of

Determine the probability of all events. This must be selected such that, and

Applies.

It should however be noted that sometimes in the literature the results are even called elementary events. These are then, however, no changes, because it is not to subsets of.

Furthermore, it must not necessarily be for the singleton in the event space. It is then not an event.