> [!NOTE] Definition 1 (Uniform Probability Space) > A uniform probability space is a [[List|triple]] $(\Omega,\mathcal{F},\mathbb{P})$ where: > > $\Omega,$ called the [[Sample Space|sample space]], is a non-empty [[Finite Set|finite set]] of all possible outcomes of the experiment; > > $\mathcal{F}$ is the [[Zermelo Frankel set theory (ZFC)|power set]] of $\Omega,$ $\mathcal{P}(\Omega)$; > > $\mathbb{P}:\mathcal{F}\to[0,1]$ is a uniform [[Probability Measure|probability measure]]: that is $\mathbb{P}(\emptyset)=0$; $\mathbb{P}(\Omega)=1$; $\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)$ for all $A,B\in \mathcal{F}$ such that $A \cap B = \emptyset$; $\mathbb{P}(\{ w \})=\mathbb{P}(\{ \tilde{w} \})$ for all $w,\tilde{w}\in \Omega.$ > > [!NOTE] Definition 2 (Uniform Probability Space) > A uniform probability space is [[Probability Space|probability space]] $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathbb{P}$ is a [[Uniform Probability Measure|uniform]] and $\Omega$ is finite. # Properties By [[Probability of Events in Uniform Probability Space With Finite Sample Space]], for all $A\in \mathcal{F},$ $\mathbb{P}(A)={|A|}/{|\Omega|}.$