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 ...
, a probability space or a probability triple
is a
mathematical construct that provides a formal model of a
random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
process or "experiment". For example, one can define a probability space which models the throwing of a
die
Die, as a verb, refers to death, the cessation of life.
Die may also refer to:
Games
* Die, singular of dice, small throwable objects used for producing random numbers
Manufacturing
* Die (integrated circuit), a rectangular piece of a semicondu ...
.
A probability space consists of three elements:
[Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press.]
# A
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 ...
,
, which is the set of all possible
outcomes.
# An event space, which is a set of
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of e ...
s
, an event being a set of outcomes in the sample space.
# A probability function, which assigns each event in the event space 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 ...
, which is a number between 0 and 1.
In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article.
In the example of the throw of a standard die, we would take the sample space to be
. For the event space, we could simply use the
set of all subsets of the sample space, which would then contain simple events such as
("the die lands on 5"), as well as complex events such as
("the die lands on an even number"). Finally, for the probability function, we would map each event to the number of outcomes in that event divided by 6 — so for example,
would be mapped to
, and
would be mapped to
.
When an experiment is conducted, we imagine that "nature" "selects" a single outcome,
, from the sample space
. All the events in the event space
that contain the selected outcome
are said to "have occurred". This "selection" happens in such a way that if the experiment were repeated many times, the number of occurrences of each event, as a fraction of the total number of experiments, would most likely tend towards the probability assigned to that event by the probability function
.
The Soviet mathematician
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
introduced the notion of probability space, together with other
axioms of probability
The Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probabili ...
, in the 1930s. In modern probability theory there are a number of alternative approaches for axiomatization — for example,
algebra of random variables
The algebra of random variables in statistics, provides rules for the symbolic manipulation of random variables, while avoiding delving too deeply into the mathematically sophisticated ideas of probability theory. Its symbolism allows the treat ...
.
Introduction
![Dice measure](https://upload.wikimedia.org/wikipedia/commons/7/74/Dice_measure.svg)
A probability space is a mathematical triplet
that presents a
model
A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
for a particular class of real-world situations.
As with other models, its author ultimately defines which elements
,
, and
will contain.
* 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 the set of all possible outcomes. An
outcome is the result of a single execution of the model. Outcomes may be states of nature, possibilities, experimental results and the like. Every instance of the real-world situation (or run of the experiment) must produce exactly one outcome. If outcomes of different runs of an experiment differ in any way that matters, they are distinct outcomes. Which differences matter depends on the kind of analysis we want to do. This leads to different choices of sample space.
* The
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
is a collection of all the
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of e ...
s we would like to consider. This collection may or may not include each of the
elementary
Elementary may refer to:
Arts, entertainment, and media Music
* ''Elementary'' (Cindy Morgan album), 2001
* ''Elementary'' (The End album), 2007
* ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977
Other uses in arts, entertainment, an ...
events. Here, an "event" is a set of zero or more outcomes; that is, 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. An event is considered to have "happened" during an experiment when the outcome of the latter is an element of the event. Since the same outcome may be a member of many events, it is possible for many events to have happened given a single outcome. For example, when the trial consists of throwing two dice, the set of all outcomes with a sum of 7
pips may constitute an event, whereas outcomes with an odd number of pips may constitute another event. If the outcome is the element of the elementary event of two pips on the first die and five on the second, then both of the events, "7 pips" and "odd number of pips", are said to have happened.
* The
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
is a
set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R a ...
returning an event's
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 ...
. A probability is a real number between zero (impossible events have probability zero, though probability-zero events are not necessarily impossible) and one (the event happens
almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
, with almost total certainty). Thus
is a function
The probability measure function must satisfy two simple requirements: First, the probability of a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
union of mutually exclusive events must be equal to the countable sum of the probabilities of each of these events. For example, the probability of the union of the mutually exclusive events
and
in the random experiment of one coin toss,
, is the sum of probability for
and the probability for
,
. Second, the probability of the sample space
must be equal to 1 (which accounts for the fact that, given an execution of the model, some outcome must occur). In the previous example the probability of the set of outcomes
must be equal to one, because it is entirely certain that the outcome will be either
or
(the model neglects any other possibility) in a single coin toss.
Not every subset of the sample space
must necessarily be considered an event: some of the subsets are simply not of interest, others cannot be
"measured". This is not so obvious in a case like a coin toss. In a different example, one could consider javelin throw lengths, where the events typically are intervals like "between 60 and 65 meters" and unions of such intervals, but not sets like the "irrational numbers between 60 and 65 meters".
Definition
In short, a probability space is a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
such that the measure of the whole space is equal to one.
The expanded definition is the following: a probability space is a triple
consisting 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 ...
— an arbitrary
non-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 t ...
,
* the
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
(also called σ-field) — a set of subsets of
, called
events
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of ev ...
, such that:
**
contains the sample space:
,
**
is closed under
complements: if
, then also
,
**
is closed under
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
unions: if
for
, then also
*** The corollary from the previous two properties and
De Morgan’s law
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
is that
is also closed under countable
intersections: if
for
, then also
* the
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...