Event (probability theory)

In probability theory, an event (also a random event ) is part of a set of results from a random experiment to which a probability can be assigned. For example, the event “throwing an even number” is assigned to the subset of the total amount of all possible results (the result space ). It is said that an event occurs when it contains the result of the random experiment as an element. ${\displaystyle \{2,4,6\}}$ ${\displaystyle \{1,2,3,4,5,6\}}$

The event that is identical to the result set is called a certain event , since it always occurs. In contrast, the event identical to the empty set is called an impossible event : it never occurs. In the example of the die roll , the sure event is the set and the impossible event is the set . ${\displaystyle \Omega }$ ${\displaystyle \{1,2,3,4,5,6\}}$ ${\displaystyle \varnothing }$

definition

If there is a probability space , then an event is called. The events of a probability space are thus those subsets of the result set that lie in the σ-algebra , the so-called event system . ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle A\in \Sigma }$ ${\displaystyle \Omega }$ ${\displaystyle \Sigma }$

The events are those sets to which one later wants to assign a probability by means of a probability measure . In the more general framework of measure theory , the events are also called measurable quantities . ${\displaystyle A\in \Sigma }$ ${\displaystyle P(A)}$

Examples

Finite result set

The result set is given

${\displaystyle \Omega =\{1,2,3\}}$,

provided with the event system

${\displaystyle \Sigma :=\{\Omega ,\emptyset ,\{1\},\{2,3\}\}}$.

Then, for example, the sets and the sets are events because they are contained in the event system. The crowd is not an event. Although it is a subset of the result set, it is not contained in the event system. Since the event system is a σ-algebra, the result set and the empty set are always events. ${\displaystyle \{1\}}$ ${\displaystyle \{2,3\}}$ ${\displaystyle \{2\}}$ ${\displaystyle \Omega }$ ${\displaystyle \emptyset }$

Discrete result set

For arbitrary discrete result sets , i.e. those with at most a countable infinite number of elements, the power set is usually used as the event system. Then every subset of the result set is an event, since the power set is exactly the set of all subsets. ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {P}}(\Omega )}$

Real result sets

For real result sets, one usually uses Borel's σ-algebra as the event system. Here, for example, are all open intervals, i.e. sets of the form with , events. In fact, these systems of sets are so large that almost anything that can be meaningfully defined is an event. However, there are sets that are not events, such as the Vitali sets . ${\displaystyle (a,b)}$ ${\displaystyle a<b}$

Set operations with events

If a result of a random experiment and an event, then one also says in the case : the event occurs . ${\displaystyle \omega \in \Omega }$ ${\displaystyle A\in \Sigma }$ ${\displaystyle \omega \in A}$ ${\displaystyle A}$

Subsets and equality

If an event is a subset of another event (noted as ), then the event always occurs with the event . One then also says: the event leads to the event . The following applies to the probabilities in this case . That means: if the event leads to the event , then the probability of is at least as great as that of . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\subseteq B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle P(A)\leq P(B)}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle A}$

It applies exactly if and applies. The equality of events means that the event entails the event in the same way as the event entails the event . ${\displaystyle A=B}$ ${\displaystyle A\subseteq B}$ ${\displaystyle B\subseteq A}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle A}$

Intersection and disjointness

The intersection of two events is again an event. It occurs exactly when and both occur. ${\displaystyle A\cap B}$ ${\displaystyle A}$ ${\displaystyle B}$

If it is true, i.e. the joint occurrence of and is impossible, then we say that the two events are mutually exclusive . The events and are then also called disjoint or incompatible . ${\displaystyle A\cap B=\varnothing }$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$

If events are more general , then the cut is made ${\displaystyle A_{1},A_{2},\ldots }$

${\displaystyle \bigcap _{n=1}^{\infty }A_{n}}$

the event that occurs exactly when all occur. The events are called pairwise disjoint if applies to all with . ${\displaystyle A_{n}}$ ${\displaystyle A_{m}\cap A_{n}=\varnothing }$ ${\displaystyle m,n\in \mathbb {N} }$ ${\displaystyle m\neq n}$

Union

The union of two events is also an event. It occurs exactly when either or or both events occur. In other words: occurs when at least one of the two events or occurs. ${\displaystyle A\cup B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\cup B}$ ${\displaystyle A}$ ${\displaystyle B}$

The formula always applies to the probability of intersection and union

${\displaystyle P(A\cap B)+P(A\cup B)=P(A)+P(B)\,.}$

Special is in the case of disjoint events . ${\displaystyle P(A\cup B)=P(A)+P(B)}$

If more general events are, then the union ${\displaystyle A_{1},A_{2},\ldots }$

${\displaystyle \bigcup _{n=1}^{\infty }A_{n}}$

the event that occurs exactly when at least one of the occurs. ${\displaystyle A_{n}}$

The so-called σ-subadditivity always applies

${\displaystyle P\left(\bigcup _{n=1}^{\infty }A_{n}\right)\leq \sum _{n=1}^{\infty }P(A_{n})\,.}$

In the case of pairwise disjoint events, equality applies here.

The sieving formula holds for the probability of arbitrary associations of finitely many events .

Complete event system

A family of events that are pairwise disjoint and the union of which results in a whole is also called a complete event system or disjoint decomposition of (in general: a partition of ). In this case, for each result of the random experiment exactly one of the events of the disjoint decomposition occurs. ${\displaystyle \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

Complement and difference

The complementary event occurs exactly when the event does not occur. It is also called a counter-event and is designated with (alternatively also with ). Its probability is ${\displaystyle \Omega \setminus A}$ ${\displaystyle A}$ ${\displaystyle {\overline {A}}}$ ${\displaystyle A^{\mathsf {c}}}$

${\displaystyle P({\overline {A}})=1-P(A)\,.}$

De Morgan's formulas apply to the complements of intersection and union sets

${\displaystyle {\overline {\bigcap _{n=1}^{\infty }A_{n}}}=\bigcup _{n=1}^{\infty }{\overline {A_{n}}}\,,}$

${\displaystyle {\overline {\bigcup _{n=1}^{\infty }A_{n}}}=\bigcap _{n=1}^{\infty }{\overline {A_{n}}}\,.}$

The same applies to two events as well . ${\displaystyle {\overline {A\cap B}}={\overline {A}}\cup {\overline {B}}}$ ${\displaystyle {\overline {A\cup B}}={\overline {A}}\cap {\overline {B}}}$

The difference set is the event that occurs exactly when the event occurs , but not at the same time as the event . It applies ${\displaystyle A\setminus B}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle A\setminus B=A\cap {\overline {B}}\,.}$

The following applies to its probability . In special cases it follows . ${\displaystyle P(A\setminus B)=P(A)-P(A\cap B)}$ ${\displaystyle B\subseteq A}$ ${\displaystyle P(A\setminus B)=P(A)-P(B)}$

Symmetrical difference

Another set operation is the symmetric difference

${\displaystyle A\bigtriangleup B=\left(A\setminus B\right)\cup \left(B\setminus A\right)=(A\cup B)\setminus (A\cap B)}$

two events and . The event occurs exactly when either or occurs (but not both), i.e. when exactly one of the two events occurs. It applies ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\bigtriangleup B}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle P(A\bigtriangleup B)=P(A)+P(B)-2P(A\cap B)\,.}$

Independent events

→ Main article : Stochastically independent events

The two events and are called independent of each other if ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle P(A\cap B)=P(A)\cdot P(B).}$

Using the formula for the conditional probability, this can be expressed as

${\displaystyle P(A)=P(A\mid B)}$

write, provided . ${\displaystyle P(B)>0}$

More generally, a family of events is said to be independent if for every finite subset : ${\displaystyle (A_{i})_{i\in I}}$ ${\displaystyle J\subseteq I}$

${\displaystyle P\left(\bigcap _{j\in J}A_{j}\right)=\prod _{j\in J}P(A_{j})\,.}$

The events are called pairwise independent if

${\displaystyle P(A_{i}\cap A_{j})=P(A_{i})\cdot P(A_{j})}$

applies to all . Independent events are pairwise independent, but the reverse is generally not true. ${\displaystyle i,j\in I}$

Act of God

The single-element events are sometimes also referred to as elementary events . Is at most countable , then by defining the probabilities of all elementary events with the help of ${\displaystyle \{\omega \}\subseteq \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \rho (\omega )=P(\{\omega \})}$

${\displaystyle P(A)=\sum _{\omega \in A}\rho (\omega )}$

determine the probability of all events . They must be chosen so that as well as ${\displaystyle A\subseteq \Omega }$ ${\displaystyle \rho (\omega )}$ ${\displaystyle 0\leq \rho (\omega )\leq 1}$

${\displaystyle \sum _{\omega \in \Omega }\rho (\omega )=1}$

applies.

It should be noted, however, that the results themselves are sometimes referred to in the literature as natural events. However, these are then not events because they are not subsets of . ${\displaystyle \omega \in \Omega }$ ${\displaystyle \Omega }$

Furthermore, the one-element set does not necessarily have to be in the event space. It is then not an event. ${\displaystyle \omega \in \Omega }$ ${\displaystyle \{\omega \}}$ ${\displaystyle \Sigma }$

literature

• Norbert Henze : Stochastics for Beginners: An Introduction to the Fascinating World of Chance. 10th edition, Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-03076-6 , doi : 10.1007 / 978-3-658-03077-3 . Pp. 5–9, 283 ( excerpt (Google) )
• Rainer Schlittgen : Introduction to Statistics. 9th edition. Oldenbourg Wissenschaftsverlag, Oldenbourg 2000, ISBN 3-486-27446-5

Individual evidence

1. ↑ Klaus D. Schmidt: Measure and probability. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89729-3 , p. 195.