independent and dependent events

Distinguishing between events:

Two events $A, B$ are independent if and only if

P (A ∣ B) = P (A) and P (B ∣ A) = P (B)

i.e., the occurrence of either event does not affect the occurrence of the other.

By the definition of conditional probability, this is equivalent to saying

P (A \cap B) = P (A) \cdot P (B)

when events $A, B$ are independent.

For independent events $A, B$ , we may write $A ⊥ B$ .

Two events are dependent if they are not independent.

A set of events ${E_{1}, E_{2}, \dots}$ is independent if for any finite subset ${E_{k_{1}}, E_{k_{2}}, \dots, E_{k_{n}}}$ of size $n$ ,

\begin{aligned} P (⋂_{i = 1}^{n} E_{k_{i}}) & = \prod_{i = 1}^{n} P (E_{k_{i}}) \\ P (E_{k_{1}} \cap E_{k_{2}} \cap \dots \cap E_{k_{n}}) & = P (E_{k_{1}}) \cdot P (E_{k_{2}}) \dots P (E_{k_{n}}) \end{aligned}