Home Statistical Dictionary Independent Events
Independent Events
Two events whose joint probability equals the product of their individual probabilities, so that knowing one occurred provides no information about whether the other occurred.
In Plain English
Two events are independent when learning that one happened doesn't change how likely you'd think the other is, the result of one coin flip tells you nothing about the result of the next one. Mathematically this has a very clean test: independent events' joint probability is just the product of their separate probabilities.
Definition
Events and are independent if , equivalently, whenever (conditioning on doesn't change 's probability). Independence generalizes to a collection of events being mutually independent if this product rule holds for every subcollection, a stronger condition than merely pairwise independence, and it is distinct from, in fact nearly opposite to, the concept of mutually exclusive events.
Formula
Common Misconceptions
- Pairwise independence (every pair of events in a collection is independent) does not imply mutual independence (the product rule holding for the whole collection simultaneously), classic counterexamples exist where every pair is independent but the full collection is not, an important subtlety in problems involving three or more events.
Properties
- Independence is the mathematical formalization of 'learning about one event tells you nothing about the other', its equivalent conditional-probability form, , makes this interpretation especially direct.
- Pairwise independence among a collection of three or more events does not guarantee mutual independence (the full product rule holding across all of them simultaneously), a genuinely counterintuitive subtlety that has produced several classic textbook counterexamples.
- Independence and mutual exclusivity are frequently confused but are nearly opposite concepts, mutually exclusive events (with positive probability) are always dependent, since one occurring guarantees the other did not, the exact opposite of the 'no information gained' hallmark of independence.
At a Glance