Probability space
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In mathematics, a probability space is the foundation of probability theory. It enables a precise definition of probability.
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Definition
A probability space (Ω, F, P) is formally a measure space with total measure one (i.e. P(Ω)=1).
The first item, Ω, is a nonempty set, whose elements are sometimes known as outcomes or states of nature. There is a state of nature for any possibility. An element of Ω is often given the symbol ω.
The second item, F, is a set, whose elements are called events. The events are subsets of Ω. The set F has to fulfil certain conditions, specifically it has to be a σ-algebra. Together, Ω and F form a measurable space. An event is a set of outcomes for which one can ask its probability.
The third item, P, is called the probability measure, or the probability. It is a function from F to the real numbers, assigning each event a probability between 0 and 1. It must satisfy certain conditions, namely it must be a measure and P(Ω)=1.
Probability measures are often written in blackboard bold to distinguish them, e.g. <math>\mathbb P</math> or <math>\mathbb Q</math>. When there is only one probability measure under discussion, it is often denoted by Pr, meaning "probability of".
Examples
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.
- { }=Template:0/ 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.
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).
Other concepts
Random variables
A random variable is a function from Ω to another set, often the real numbers. Specifically, it must be a measurable function. This means, for example, that if X is a real random variable, then the set of outcomes for which X is positive, {ω∈Ω:X(ω)>0} is an event.
It is common to abbreviate {ω∈Ω:X(ω)>0} to {X>0} and further to write P(X>0) instead of P({X>0}).
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.
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=Template:0/, which is the definition of disjoint for sets).
If A and B are disjoint events, then P(A∪B)=P(A)+P(B). This extends to a (finite or 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 distibuted random variable, then P(Z=x) is 0 for any x, but P(Z is real)=1.
The event A∩B is referred to as A AND B, and the event A∪B as A OR B.de:Wahrscheinlichkeitsraum eo:Probablo-spaco fr:Espace probabilisé is:Líkindamál it:Misura di probabilità he:מרחב הסתברות ja:確率空間 no:Sannsynlighetsrom pl:Przestrzeń probabilistyczna pt:Espaço probabilidade ru:Вероятностное пространство