> [!NOTE] Definition 1 (Independent Events) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $A,B\in \mathcal{F}.$ The events $A$ and $B$ are *independent* iff $\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B).$ > [!NOTE] Definition 2 (Independent Events) > 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 *independent* iff $\mathbb{P}(A \mid B) = \mathbb{P}(A)$where $\mathbb{P}(A\mid B)$ denotes the [[Conditional Probability|conditional probability]] of $A$ given $B.$ **Note**: By [[Independence of Events is Symmetric]], $\mathbb{P}(A \mid B)= \mathbb{P}(A)$ iff $\mathbb{P}(B\mid A) = \mathbb{P}(A).$ Furthermore, by [[Equivalence of Definitions of Independent Events]], $\mathbb{P}(A \cap B)= \mathbb{P}(A)\cdot P(B)$ iff $\mathbb{P}(A\mid B)= \mathbb{P}(A).$ # Properties By [[Independent of Event iff Independent of Complement]], $A$ and $B$ are independent iff $\Omega\setminus A$ and $B$ are independent. By [[Disjoint Events are Independent iff Probability of one is Zero]], $A \cap B = \emptyset \land \mathbb{P}(A \cap B)=\mathbb{P}(A)\cdot \mathbb{P}(B)$ iff $\mathbb{P}(A) = 0 \lor \mathbb{P}(B)=0.$ # Applications **Generalisations**: A set of events are [[Pairwise Independent Set of Events|pairwise independent]] iff each pair of events is independent. The set is [[Mutually Independent Set of Events|mutually independent]] iff the probability of the intersection of all subsets of events is the product of their individual probabilities.