independence
Two events are independent when knowing one happened tells you nothing about whether the other will. Two coins tossed in two different rooms: learning the first came up heads leaves the chance of the second exactly where it was. Their fates simply do not touch. Independence is the formal name for 'these things have nothing to do with each other'.
Precisely, A and B are independent when P(A and B) = P(A) × P(B) — the chance of both is just the product of the separate chances. Equivalently, the conditional probability equals the unconditional one: P(A | B) = P(A), confirming that learning B changes nothing. If two coins are independent and each is fair, the chance of two heads is 1/2 × 1/2 = 1/4. Beware: independence is not the same as being mutually exclusive — exclusive events are actually highly dependent, because learning one happened tells you the other definitely did not.
Independence is the load-bearing assumption that makes insurance work and also its most dangerous one. If a thousand policyholders' losses are independent, the law of large numbers smooths them into a predictable average and a modest premium covers the pool. But when losses are dependent — an earthquake, a pandemic, a market crash hitting everyone at once — independence collapses, claims pile up together, and a portfolio that looked safe can fail. Much of risk theory and reinsurance exists precisely because real-world risks are not as independent as the simplest models pretend.
House fires in different cities are roughly independent, so a national insurer can pool them. But homes on the same coastline are not — one hurricane can ignite many claims at once, breaking the independence the premium assumed.
Independence is the assumption pooling rests on — and catastrophes are what break it.
Independent is not the same as mutually exclusive: exclusive events are strongly dependent (one happening rules the other out), whereas independent events have zero such influence.