In mathematics, the definition of the probability space is the foundation of probability theory. It was introduced by Kolmogorov in the 1930s. For an algebraic alternative to Kolmogorov's approach, see algebra of random variables. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... It has been suggested that this article or section be merged with Probability axioms. ... Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Soviet mathematician who made major advances in the fields of probability theory and topology. ... The 1930s (years from 1930–1939) were described as an abrupt shift to more radical and conservative lifestyles, as countries were struggling to find a solution to the Great Depression, also known in Europe as the World Depression. ... In the algebraic axiomatization of probability theory, one of whose main proponents was Irving Segal, the primary concept is not that of probability of an event, but rather that of a random variable. ...

Definition

A probability space is a measure space with a measure P that satisfies the probability axioms. In mathematics, a measure is a function that assigns a number, e. ... It has been suggested that this article or section be merged with Probability theory. ...

Sample space

The sample space is a nonempty set whose elements are known as outcomes or states of nature and are are often given the symbol ω. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

Events

The second item, , is a σ-algebra of subsets of Ω. Its elements are called events, which are sets of outcomes for which one can ask a probability. In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ...

Because is a σ-algebra, it contains Ω; also, the complement of any event is an event, and the union of any (finite or countably infinite) sequence of events is an event.

Probability measure

The probability measure P is a function from to the real numbers that assigns to each event a probability between 0 and 1. It must satisfy the probability axioms. It has been suggested that this article or section be merged with Probability theory. ...

It is important to note that P is a function defined on and not on Ω. The events may not be the complete power set of the sample space, that is, not every set of outcomes is an event. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...

Probability measures are often written in blackboard bold to distinguish them. When there is only one probability measure under discussion, it is often denoted by , meaning "probability of". An example of blackboard bold letters. ...

Related concepts

Conditional probability

Kolmogorov's definition of probability spaces gives rise to the natural concept of conditional probability. Every set A with non-zero probability (that is, P(A) > 0 ) defines another probability measure This article defines some terms which characterize probability distributions of two or more variables. ...

on the space. This is usually read as "probability of B given A".

Countable sample space

If Ω is countable we almost always define as the power set of Ω, i.e which is trivially a σ-algebra and the biggest one we can create using Ω. We can therefore omit and just write (Ω,P) to define the probability space. In mathematics the term countable set is used to describe the size of a set, e. ...

On the other hand, if Ω is uncountable and we use we get into trouble defining our probability measure P because is too 'huge', i.e. there will often be sets to which it will be impossible to assign a unique measure, giving rise to problems like the Banach–Tarski paradox. In this case, we have to use a smaller σ-algebra (e.g. the Borel algebra of Ω, which is the smallest σ-algebra that makes all open sets measurable). In mathematics, an uncountable set is a set which is not countable. ... The Banach–Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ...

Probability distribution

A probability distribution is a special case of the probability measure when the sample space is the reals and the events are the Borel algebra on the reals. In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ...

Random variables

A random variable X is a measurable function from the sample space Ω to another measurable space called the state space. A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

If X is a real-valued random variable, then the notation is shorthand for , assuming that "" is an event. Look up real in Wiktionary, the free dictionary. ...

Independence

Two events, A and B are said to be independent if P(A∩B)=P(A)P(B).

Two random variables, X and Y, are said to be independent if any event defined in terms of X is independent of any event defined in terms of Y. Formally, they generate independent σ-algebras, where two σ-algebras G and H, which are subsets of F are said to be independent if any element of G is independent of any element of H.

The concept of independence is where probability theory departs from measure theory. In mathematics, a measure is a function that assigns a number, e. ...

Mutual exclusivity

Two events, A and B are said to be mutually exclusive or disjoint if P(A∩B)=0. (This is weaker than A∩B=∅, which is the definition of disjoint for sets). In logic, two mutually exclusive (or mutual exclusive according to some sources) propositions are propositions that logically cannot both be true. ... In mathematics, two sets are said to be disjoint if they have no element in common. ...

If A and B are disjoint events, then P(A∪B)=P(A)+P(B). This extends to a (finite or countably infinite) sequence of events. It is not, however, true that the probability of the union of an uncountable set of events is the sum of their probabilities. For example, if Z is a normally distributed random variable, then P(Z=x) is 0 for any x, but P(Z is real)=1. The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ...

The event A∩B is referred to as A AND B, and the event A∪B as A OR B.

Examples

First example

If the space concerns one flip of a fair coin, then the outcomes are heads (H) and tails (T). The events are

• {H} heads, with probability 0.5.
• {T} tails, with probability 0.5.
• { }=∅ neither heads nor tails, with probability 0.
• {H,T} heads or tails, which is Ω, with probability 1.

If the experiment is one random number Z drawn from the standard normal distribution, then the set of outcomes is the real numbers. An example of an event would be the positive numbers, which is the event that Z is positive. Not all subsets of R would be events. Usually, the events are the Lebesgue-measurable or Borel-measurable sets of real numbers. The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a (where a < b). ...

This illustrates the fact that not all sets of outcomes are necessarily events. If Ω is a countable set, then there is no problem in allowing F to be the set of all subsets of Ω (the power set of Ω). In mathematics, a countable set is a set with the same cardinality (i. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...

Second example

If 100 voters are to be drawn randomly from among all voters in California and asked whom they will vote for governor, then the set of all sequences of 100 Californian votes would be the sample space Ω.

The set of all sequences of 100 Californian voters in which at least 60 will vote for Schwarzenegger is identified with the "event" that at least 60 of the 100 chosen voters will so vote.

Then, contains: (1) the set of all sequences of 100 where at least 60 vote for Schwarzenegger; (2) the set of all sequences of 100 where fewer than 60 vote for Schwarzenegger (the converse of (1)); (3) the sample space Ω as above; and (4) the empty set.

An example of a random variable is the number of voters who will vote for Schwarzenegger in the sample of 100.

Bibliography

• Pierre Simon de Laplace (1812) Analytical Theory of Probability

The first major treatise blending calculus with probability theory, originally in French: Théorie Analytique des Probabilités.

• Andrei Nikolajevich Kolmogorov (1950) Foundations of the Theory of Probability

The modern measure-theoretic foundation of probability theory; the original German version (Grundbegriffe der Wahrscheinlichkeitrechnung) appeared in 1933.

• Harold Jeffreys (1939) The Theory of Probability

An empiricist, Bayesian approach to the foundations of probability theory.

• Edward Nelson (1987) Radically Elementary Probability Theory

Discrete foundations of probability theory, based on nonstandard analysis and internal set theory. downloadable. http://www.math.princeton.edu/~nelson/books.html

• Patrick Billingsley: Probability and Measure, John Wiley and Sons, New York, Toronto, London, 1979.

• Henk Tijms (2004) Understanding Probability

A lively introduction to probability theory for the beginner, Cambridge Univ. Press.

• Gut, Allan (2005). Probability: A Graduate Course. Springer. ISBN 0387228330.