Mutual vs Pairwise Independence

Probability · Axiom Academy

LESSON Mutual vs Pairwise Independence Understanding the subtle but crucial difference between two types of independence Events A₁, A₂, ..., Aₙ are pairwise independent if every pair of events is independent. This means checking independence for each of the C(n,2) pairs. 2. Mutual Independence (The Stronger Condition) Events A₁, A₂, ..., Aₙ are mutually independent if every subset of events satisfies the product rule. This requires checking 2ⁿ - n - 1 conditions, far more than pairwise! Consider flipping two fair coins. Let A = "first coin is heads", B = "second coin is heads", C = "exactly one head appears". These events are pairwise independent but NOT mutually independent! Pairwise independence is weaker than mutual independence. When we say "events are independent" in practice, we usually mean mutually independent. The counterexample shows we must be careful with three or more events.

This is the written version of the interactive lesson above. See the full Probability course.