Editing Probability space (section)

{{Short description|Mathematical concept}}
{{About|the mathematical concept|the novel|Probability Space (novel)}}
{{Probability fundamentals}}
In [[probability theory]], a '''probability space''' or a '''probability triple''' <math>(\Omega, \mathcal{F}, P)</math> is a [[space (mathematics)|mathematical construct]] that provides a formal model of a [[randomness|random]] process or "experiment". For example, one can define a probability space which models the throwing of a {{not a typo|[[dice|die]]}}.

A probability space consists of three elements:<ref>Loève, Michel. Probability Theory, Vol 1. New York: D. Van Nostrand Company, 1955.</ref><ref>Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press.</ref>
# A ''[[sample space]]'', <math>\Omega</math>, which is the set of all possible [[Outcome (probability)|outcomes]] of a random process under consideration.
# An '''event space''', <math>\mathcal{F}</math>, which is a set of [[event (probability theory)|event]]s, where an event is a subset of outcomes in the sample space.
# A ''[[probability measure|probability function]]'', <math>P</math>, which assigns, to each event in the event space, a [[probability]], which is a number between 0 and 1 (inclusive).

In order to provide a model of probability, these elements must satisfy [[probability axioms]].

In the example of the throw of a standard die,

# The sample space <math>\Omega</math> is typically the set <math>\{1, 2, 3, 4, 5, 6\}</math> where each element in the set is a label which represents the outcome of the die landing on that label. For example, <math>1</math> represents the outcome that the die lands on 1.
# The event space <math>\mathcal{F}</math> could be the [[power set|set of all subsets]] of the sample space, which would then contain simple events such as <math>\{5\}</math> ("the die lands on 5"), as well as complex events such as <math>\{2, 4, 6\}</math> ("the die lands on an even number").
# The probability function <math>P</math> would then map each event to the number of outcomes in that event divided by 6 – so for example, <math>\{5\}</math> would be mapped to <math>1/6</math>, and <math>\{2, 4, 6\}</math> would be mapped to <math>3/6 = 1/2</math>.

When an experiment is conducted, it results in exactly one outcome <math>\omega</math> from the sample space <math>\Omega</math>. All the events in the event space <math>\mathcal{F}</math> that contain the selected outcome <math>\omega</math> are said to "have occurred". The probability function <math>P</math> must be so defined that if the experiment were repeated arbitrarily many times, the number of occurrences of each event as a fraction of the total number of experiments, will most likely tend towards the probability assigned to that event.

The Soviet mathematician [[Andrey Kolmogorov]] introduced the notion of a probability space and the [[axioms of probability]] in the 1930s. In modern probability theory, there are alternative approaches for axiomatization, such as the [[algebra of random variables]].