> [!NOTE] Lemma > Let $(\Omega,\mathcal{F},\mathbb{P})$ be a [[Uniform Probability Space with Finite Sample Space|uniform probability space]]. Then for all $A\in \mathcal{F}$ $\mathbb{P}(A) = \frac{|A|}{|\Omega|}$where $|A|$ denotes the [[Cardinality|cardinality]] of $A.$ **Note:** $A$ is finite by [[Subset of Finite Set is Finite]]. **Proof**: Since $\mathbb{P}$ is [[Uniform Probability Measure|uniform]], for all $\omega_{1},\omega_{2}\in \Omega,$ $\mathbb{P}(\{ \omega_{1} \})=\mathbb{P}(\{ \omega_{2} \}).$ Let $p\in [0,1]$ such that for all $\omega\in \Omega,$ $p=\mathbb{P}(\{ \omega \}).$Since $\mathbb{P}$ is a [[Probability Measure|probability measure]], $\begin{align} 1&= \mathbb{P}(\Omega) \\ &=\sum_{\omega\in\Omega} \mathbb{P}\{\omega \} &\text{by finite additivity} \\ &= \sum_{\omega\in \Omega} p \\ &= p \sum_{\omega\in \Omega} 1 = p |\Omega| \end{align}$using [[Cardinality of Union of Disjoint Sets|Cardinality of Union of Disjoint Sets]] in the last line. Therefore $p= \frac{1}{|\Omega|}.$By finite additivity, $\begin{align} P(A) &= \sum_{\omega\in A} \mathbb{P}\{ \omega \} \\ & = \sum_{\omega\in A} p = p\sum_{\omega\in A} 1 \\ &= p |A| \\ &=\frac{|A|}{|\Omega|} \end{align}$ # Applications **Examples**: > [!Example] > Consider an urn with $50$ balls numbered $1$ to $50.$ Assume that they are drawn uniformly at random. After defining a suitable probability space, determine the probability that the first ball drawn shows a number divisible by $12.$ > >**Solution**: Define $(\Omega,\mathcal{F},\mathbb{P})$ as follows: $\Omega=\{ 1,2,3,\dots,50 \},\mathcal{F}=\mathcal{P}(\Omega)$ and $\mathbb{P}$ the uniform probability measure. Then the event in question is $E=\{ 12,24,36,48 \}.$Thus $\mathbb{P}(E) = \frac{|E|}{|\Omega|}= \frac{4}{50}=\frac{2}{25}.$