Let $X_1, \ldots, X_n$ be random variables taking values in countable sets $E_1, \ldots, E_n$. Then $X_1, \ldots, X_n$ are independent if and only if for every $(x_1, \ldots, x_n) \in E_1 \times \cdots \times E_n$,
\begin{align*}
\mathbb P(X_1=x_1, \ldots, X_n=x_n)
= \prod_{i=1}^n \mathbb P(X_i=x_i).
\end{align*}