Editing Probability space (section)

== Discrete case ==

[[Discrete probability theory]] needs only [[countable set|at most countable]] sample spaces <math>\Omega</math>. Probabilities can be ascribed to points of <math>\Omega</math> by the [[probability mass function]] <math>p:\Omega\to[0,1]</math> such that <math display="inline">\sum_{\omega\in\Omega} p(\omega)=1</math>. All subsets of <math>\Omega</math> can be treated as events (thus, <math>\mathcal{F}=2^\Omega</math> is the [[power set]]). The probability measure takes the simple form
{{NumBlk||<math display="block"> P(A) = \sum_{\omega\in A} p(\omega) \quad \text{for all } A \subseteq \Omega.</math>|{{EquationRef|⁎}}}}
The greatest σ-algebra <math>\mathcal{F}=2^\Omega</math> describes the complete information. In general, a σ-algebra <math>\mathcal{F}\subseteq2^\Omega</math> corresponds to a finite or countable [[partition of a set|partition]] <math>\Omega=B_1\cup B_2\cup\dots</math>, the general form of an event <math>A\in\mathcal{F}</math> being <math>A=B_{k_1}\cup B_{k_2}\cup\dots</math>. See also the examples.

The case <math>p(\omega)=0</math> is permitted by the definition, but rarely used, since such <math>\omega</math> can safely be excluded from the sample space.