Independence

"Are these two risks correlated?" is the single question that separates a diversified book from a concentrated one, and its cleanest possible answer is "no — they're independent." This note makes that word precise: independence is the special case of conditioning where learning that $B$ occurred changes nothing about $A$'s probability. The definition looks simple ($\mathbb{P}(A\cap B)=\mathbb{P}(A)\mathbb{P}(B)$), but the note exists because intuition about it is unreliable at every turn: disjoint events are never independent (a common first guess is exactly backwards), independence and conditional independence given a third event are logically unrelated, and — most consequentially for risk aggregation — pairwise independence of a collection of events does not imply their mutual independence. Every one of these traps is illustrated with a worked counterexample below, because in this topic the counterexample is the teaching content.

Learning objectives. By the end of this note you should be able to: