> [!NOTE] Lemma > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $A,B\in \mathcal{F}$ such that $\mathbb{P}(A),\mathbb{P}(B)>0.$ Then $\mathbb{P}(A \mid B) = \mathbb{P}(A) \iff \mathbb{P}(A \cap B)= \mathbb{P}(A)\cdot \mathbb{P}(B)$where $\mathbb{P}(A\mid B)$ denotes the [[Conditional Probability|conditional probability]] of $A$ given $B$: that is the given [[Independence of Two Events|definitions of independence]] are equivalent for events with non-zero probability. **Proof**: We have $\begin{align} P(A\mid B) &= P(A) \\ \iff \frac{P(A \cap B)}{P(B)} &= P(A) \\ \iff P(A \cap B) &= P(A) \cdot P(B) & \text{multiplying both sides by $P(B)>0$.} \end{align}$