In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, an event is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
outcomes of an
experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...
(a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.
An event consisting of only a single outcome is called an or an ; that is, it is a
singleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...
. An event
is said to if
contains the outcome
of the
experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...
(or trial) (that is, if
). The probability (with respect to some
probability measure) that an event
occurs is the probability that
contains the outcome
of an experiment (that is, it is the probability that
).
An event defines a
complementary event
In probability theory, the complement of any event ''A'' is the event ot ''A'' i.e. the event that ''A'' does not occur.Robert R. Johnson, Patricia J. Kuby: ''Elementary Statistics''. Cengage Learning 2007, , p. 229 () The event ''A'' and ...
, namely the complementary set (the event occurring), and together these define a
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...
: did the event occur or not?
Typically, when the
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
is finite, any subset of the sample space is an event (that is, all elements of the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of the sample space are defined as events). However, this approach does not work well in cases where the sample space is
uncountably infinite
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
. So, when defining a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see ''
Events in probability spaces'', below).
A simple example
If we assemble a deck of 52
playing card
A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a fi ...
s with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any
singleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...
(an
elementary event
In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space. Using set theory terminology, an elementary event is a singleton. Elementary events an ...
), the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
(an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are
proper subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of the sample space that contain multiple elements. So, for example, potential events include:
* "Red and black at the same time without being a joker" (0 elements),
* "The 5 of Hearts" (1 element),
* "A King" (4 elements),
* "A Face card" (12 elements),
* "A Spade" (13 elements),
* "A Face card or a red suit" (32 elements),
* "A card" (52 elements).
Since all events are sets, they are usually written as sets (for example, ), and represented graphically using
Venn diagram
A Venn diagram is a widely used diagram style that shows the logical relation between set (mathematics), sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple ...
s. In the situation where each outcome in the sample space Ω is equally likely, the probability
of an event
is the following :
This rule can readily be applied to each of the example events above.
Events in probability spaces
Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard
probability distributions
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
, such as the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, the sample space is the set of real numbers or some subset of the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers
'badly behaved' sets, such as those that are
nonmeasurable
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Ze ...
. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as
joint
A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw ...
and
conditional probabilities
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occur ...
, to work, it is necessary to use a
σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice of
σ-algebra is the
Borel measurable set derived from unions and intersections of intervals. However, the larger class of
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
sets proves more useful in practice.
In the general
measure-theoretic description of
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
s, an event may be defined as an element of a selected
-algebra of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the -algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all are elements of the -algebra.
A note on notation
Even though events are subsets of some sample space
they are often written as predicates or indicators involving
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s. For example, if
is a real-valued random variable defined on the sample space
the event
can be written more conveniently as, simply,
This is especially common in formulas for a
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, such as
The
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is an example of an
inverse image
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
under the
mapping because
if and only if
See also
*
*
*
*
*
*
Notes
External links
*
Formal definitionin the
Mizar system
The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library of formalized mathematics, which can be used in ...
.
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Experiment (probability theory)
Terms in science and technology