> [!NOTE] Lemma > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $A,B\in \mathcal{F}$ such that $\mathbb{P}(A),\mathbb{P}(B)>0.$ The events $A$ and $B$ are [[Independence of Two Events|independent]] iff $B$ and $A$: that is, $\mathbb{P}(A \mid B) = \mathbb{P}(A) \iff \mathbb{P}(B\mid A) = \mathbb{P}(B)$where $\mathbb{P}(A\mid B)$ denotes the [[Conditional Probability|conditional probability]] of $A$ given $B.$ **Proof**: We have $\begin{align} \mathbb{P}(A \mid B) = \mathbb{P}(A) \\ \frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}=\mathbb{P}(A) \\ \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(A)} = \mathbb{P}(B) \\ \mathbb{P}(B \mid A) = \mathbb{P}(B). \end{align}$ **Proof**: