# Chain rule (probability)

[!definition] Chain rule (probability) For random variables $A_k$ for $k \in \{1, 2, \ldots, n\}$ we have

$$\mathbb{P}[A_1, A_2, \ldots, A_n] = \prod_{k=1}^n \mathbb{P}[A_k \vert A_1, A_2, \ldots, A_{k-1}].$$

This follows from the definition of conditional probability .