Independence of Events

Two events are independent when knowing one occurred gives no information about the other.

Definition

Events A and B are independent if P(A ∩ B) = P(A) P(B). Equivalently, when P(B) > 0, independence means P(A | B) = P(A): conditioning on B leaves the probability of A unchanged.

Independence is not disjointness

These are opposite ideas. Disjoint events with positive probability are strongly dependent: if one occurs the other cannot, so P(A | B) = 0 ≠ P(A). Independence requires overlap of exactly the product size.

Pairwise versus mutual independence

For three or more events, pairwise independence is weaker than mutual independence. Mutual independence requires the product rule to hold for every subset, not just for pairs. It is possible to construct events that are pairwise independent yet fail the joint condition, so the distinction is not academic.

Conditional independence

A and B are conditionally independent given C if P(A ∩ B | C) = P(A | C) P(B | C). This is the assumption behind naive Bayes classifiers and graphical models, and it can hold even when A and B are marginally dependent — or fail even when they are marginally independent.

Why it matters

Independence assumptions make joint distributions tractable: instead of specifying a full joint table, you multiply marginals. But an unwarranted independence assumption understates the probability of correlated failures. In reliability modeling, treating two components as independent when they share a common cause underestimates the chance they fail together, which is exactly the case that matters most.