Independence of two events

Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B \in \mathcal{F}$ be events. The events $A$ and $B$ are independent if $$P(A \cap B) = P(A) \, P(B).$$

Remarks

If $P(B) > 0$, then $A$ and $B$ are independent if and only if $P(A \mid B) = P(A)$, with the conditional probability. If $A$ and $B$ are independent, then so are $A$ and $B^c$, $A^c$ and $B$, and $A^c$ and $B^c$. An event of probability $0$ or $1$ is independent of each event. An event $A$ is independent of itself if and only if $P(A) \in \{0, 1\}$. Independence is not the same as disjointness. Disjoint events $A$ and $B$ with $P(A) > 0$ and $P(B) > 0$ are never independent.

Explanations

L1 human

Two events are independent when knowing that one happened tells you nothing about the other. Roll a fair die two times. The events "the first roll is $6$" and "the second roll is $6$" are independent. Both happen with probability $\frac16 \cdot \frac16 = \frac{1}{36}$.