Mutual independence of events

Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $I$ be an index set, and let $A_i \in \mathcal{F}$ be an event for each $i \in I$. The events $(A_i)_{i \in I}$ are mutually independent if for each finite nonempty subset $J \subseteq I$, $$P\left(\bigcap_{j \in J} A_j\right) = \prod_{j \in J} P(A_j).$$

Remarks

For two events this is the independence of two events. For three events $A, B, C$ the definition asks for four equations: one for each of the pairs $\{A, B\}$, $\{A, C\}$, $\{B, C\}$ and one for the triple $\{A, B, C\}$. Mutual independence implies pairwise independence. The converse is false: see pairwise independent events that are not mutually independent. If we replace some of the events $A_i$ by their complements, the family stays mutually independent.

Explanations

L1 human

For many events, independence must hold for every group of them, not only for every pair. For three tosses of a fair coin, the events "toss $k$ gives heads" for $k = 1, 2, 3$ are mutually independent, and all three happen with probability $1/8$.