> [!NOTE] Lemma > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $A,B\in \mathcal{F}.$ The events $A$ and $\Omega\setminus B$ are [[Independence of Two Events|independent]] iff $\Omega \setminus A$ and $B$ are independent, where $\Omega \setminus A$ denotes a [[Set Difference|set difference]]. **Proof**: Follows from [[Independence of Events is Symmetric|symmetry of independence]] and applying [[Independent of Event iff Independent of Complement]] twice: $A$ and $B$ are independent iff $\Omega \setminus A$ and $B$ are independent iff $B$ and $\Omega\setminus A$ are independent iff $\Omega\setminus B$ and $\Omega \setminus A$ are independent. **Proof**: Note that by [[De Morgan's Laws for Union]], $\Omega \setminus A \cap \Omega \setminus B = \Omega \setminus (A \cup B).$Thus by [[Probability of Complement of Event]] and [[Probability Measure is Strongly Additive]], $\begin{align} \mathbb{P}(\Omega \setminus A \cap \Omega \setminus B) &= \mathbb{P}( \Omega \setminus (A \cup B)) \\ &= 1 - \mathbb{P}(A \cup B) \\ &= 1 - \mathbb{P}(A) - \mathbb{P}(B) + \mathbb{P}(A \cap B) \\ & = 1 - \mathbb{P}(A) - \mathbb{P}(B) + \mathbb{P}(A)\cdot \mathbb{P}(B) \\ &= (1-\mathbb{P}(A))\cdot (1- \mathbb{P}(B)) \\ &= \mathbb{P}(\Omega \setminus A) \cdot \mathbb{P}(\Omega \setminus B). \end{align}$