= Independent events
= Mutual independence
{synonym}
Events $A_1,\ldots,A_n$ are mutually independent when
$$
\mathbb P\left(\bigcap_{j\in J}A_j\right)
=\prod_{j\in J}\mathbb P(A_j)
$$
for every nonempty index subset $J\subseteq\{1,\ldots,n\}$. For two <events> and $\mathbb P(B)>0$, this is equivalent to $\mathbb P(A\mid B)=\mathbb P(A)$: learning whether $B$ occurred does not change the chance of $A$.
For independent uniform dice, the <events> that the first result is an <even integer> and that the second result is an <even integer> each have <probability> $1/2$; their <intersection> has <probability> $9/36=1/4$. In contrast, "first result is an <even integer>" and "first result is an <odd integer>" are <disjoint events> of positive <probability>, so they cannot be <independent events>. Learning one occurred completely rules out the other.
<Pairwise independence> is weaker than mutual independence. Take two <independent> fair <bits> $X,Y$. The three <events> $X=0$, $Y=0$, and $X=Y$ each have <probability> $1/2$ and every pair has <intersection> <probability> $1/4$. Their triple <intersection> is the single outcome $(0,0)$, of <probability> $1/4$ rather than $1/8$. Checking pairs alone misses the joint constraint.
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