structural:summarization
Two events

Two events A {\displaystyle A} and B {\displaystyle B} are independent (often written as A ⊥ B {\displaystyle A\perp B} or A ⊥ ⊥ B {\displaystyle A\perp \!\!\!\perp B}, where the latter symbol often is also used for conditional independence) if and only if their joint probability equals the product of their probabilities:

A ∩ B ≠ ∅ {\displaystyle A\cap B\neq \emptyset } indicates that two independent events A {\displaystyle A} and B {\displaystyle B} have common elements in their sample space so that they are not mutually exclusive (mutually exclusive iff A ∩ B = ∅ {\displaystyle A\cap B=\emptyset }). Why this defines independence is made clear by rewriting with conditional probabilities P (A ∣ B) = P (A ∩ B) P (B) {\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}} as the probability at which the event A {\displaystyle A} occurs provided that the event B {\displaystyle B} has or is assumed to have occurred:

and similarly

Thus, the occurrence of B {\displaystyle B} does not affect the probability of A {\displaystyle A}, and vice versa. In other words, A {\displaystyle A} and B {\displaystyle B} are independent of each other. Although the derived expressions may seem more intuitive, they are not the preferred definition, as the conditional probabilities may be undefined if P (A) {\displaystyle \mathrm {P} (A)} or P (B) {\displaystyle \mathrm {P} (B)} are 0. Furthermore, the preferred definition makes clear by symmetry that when A {\displaystyle A} is independent of B {\displaystyle B}, B {\displaystyle B} is also independent of A {\displaystyle A}.