Editing Probability space (section)

== Related concepts ==

=== Probability distribution ===
{{Main article|Probability distribution}}

=== Random variables ===
{{Main article|Random variable}}
A random variable ''X'' is a [[measurable function]] ''X'': Ω → ''S'' from the sample space Ω to another measurable space ''S'' called the ''state space''.

If ''A'' ⊂ ''S'', the notation Pr(''X'' ∈ ''A'') is a commonly used shorthand for <math>P(\{\omega \in \Omega: X(\omega) \in A\})</math>.

=== Defining the events in terms of the sample space ===
If Ω is [[countable]], we almost always define <math> \mathcal{F}</math> as the [[power set]] of Ω, i.e. <math> \mathcal{F} = 2^\Omega</math> which is trivially a σ-algebra and the biggest one we can create using Ω. We can therefore omit <math> \mathcal{F}</math> and just write (Ω,P) to define the probability space.

On the other hand, if Ω is [[uncountable]] and we use <math> \mathcal{F} = 2^\Omega</math> we get into trouble defining our probability measure ''P'' because <math> \mathcal{F}</math> is too "large", i.e. there will often be sets to which it will be impossible to assign a unique measure. In this case, we have to use a smaller σ-algebra <math> \mathcal{F}</math>, for example the [[Borel algebra]] of  Ω, which is the smallest σ-algebra that makes all open sets measurable.

=== Conditional probability ===
{{Main article|Conditional probability}}
Kolmogorov's definition of probability spaces gives rise to the natural concept of conditional probability.  Every set {{mvar|A}} with non-zero probability (that is, {{math|''P''(''A'') > 0}}) defines another probability measure
<math display="block"> P(B \mid A) = {P(B \cap A) \over P(A)} </math>
on the space. This is usually pronounced as the "probability of ''B'' given ''A''".

For any event {{math|''A''}} such that {{math|''P''(''A'') > 0}}, the function {{math|''Q''}} defined by {{math|1=''Q''(''B'') = ''P''(''B''&nbsp;{{!}}&nbsp;''A'')}} for all events {{mvar|B}} is itself a probability measure.

=== Independence ===
{{Main article|Statistical independence}}
Two events, ''A'' and ''B'' are said to be independent if {{math|1=''P''(''A'' ∩ ''B'') = ''P''(''A'') ''P''(''B'')}}.

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

=== Mutual exclusivity ===
{{Main article|Mutual exclusivity}}
Two events, {{math|''A''}} and {{math|''B''}} are said to be mutually exclusive or ''disjoint'' if the occurrence of one implies the non-occurrence of the other, i.e., their intersection is empty. This is a stronger condition than the probability of their intersection being zero.

If {{math|''A''}} and {{math|''B''}} are disjoint events, then {{math|1=''P''(''A'' ∪ ''B'') = ''P''(''A'') + ''P''(''B'')}}. This extends to a (finite or countably infinite) sequence of events. However, the probability of the union of an uncountable set of events is not the sum of their probabilities. For example, if {{mvar|Z}} is a [[normal distribution|normally distributed]] random variable, then {{math|1=''P''(''Z'' = ''x'')}} is 0 for any {{mvar|x}}, but {{math|1=''P''(''Z'' ∈ '''R''') = 1}}.

The event {{math|''A'' ∩ ''B''}} is referred to as "''A'' and ''B''", and the event {{math|''A'' ∪ ''B''}} as "''A'' or ''B''".