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For questions involving the notion of independence of events, of independence of collections of events, or of independence of random variables. Use this tag along with the tags (probability), (probability-theory) or (statistics). Do not use for linear independence of vectors and such.

For events: Two events $$A$$ and $$B$$ are independent if $$P(A\cap B)=P(A)P(B)$$ More generally, a family $$\mathscr F$$ of events is independent if, for every finite number of distinct events $$A_1$$, $$A_2$$, $$\ldots$$, $$A_n$$ in $$\mathscr F$$, $$P\left(\bigcap_{i=1}^nA_i\right) =\prod_{i=1}^nP(A_i)$$

Two collections of events (for example, two $$\sigma$$-algebras) $$\mathscr F$$ and $$\mathscr G$$ are mutually independent (or simply, independent) if every $$A$$ in $$\mathscr F$$ and every $$B$$ in $$\mathscr G$$ are independent.

More generally, some collections $$\mathscr F_i$$ of events, indexed by some finite or infinite set $$I$$, are mutually independent (or simply, independent) if, for every finite subset $$\\{i_1,i_2,\ldots,i_n\\}$$ of $$I$$ and every event $$A_k$$ in $$\mathscr F_{i_k}$$, the family $$\\{A_1,\ldots,A_n\\}$$ is independent.

For random variables: Two random variables $$X$$ and $$Y$$ (defined on the same probability space) are independent if their $$\sigma$$-algebras $$\sigma(X)$$ and $$\sigma(Y)$$ are (mutually) independent.

In particular, 2 events $$A$$ and $$B$$ are independent if and only if the indicator random variables $$1_A$$ and $$1_B$$ are independent.

More generally, a family $$\mathscr X$$ of random variables (defined on the same probability space) is independent if, for every finite sub-family $$\\{X_1,X_2,\ldots,X_n\\}$$ of $$\mathscr X$$, the $$\sigma$$-algebras $$\sigma(X_{1})$$, $$\sigma(X_{2})$$, $$\dots$$, $$\sigma(X_{n})$$ are (mutually) independent.