Let $A, B$ be subsets of some sample space $\Omega $ with a probability measure $P $. Assume that $\b{P}(B) > 0 $. We define the conditional probability of $A$ given $B$, denoted by $\b{P}(A|B)$ as
$$\b{P}(A|B)=\frac{\b{P}(A\cap B)}{\b{P}(B)}$$
Let $B_1, B_2, …, B_n \subset \Omega $. We use the following notation to denote the conditional probability of $A$ given $\bigcap_{ k=1 }^n B_k $:
$$\b{P}(A | B_1, B_2, …, B_n) = \b{P}(A | \bigcap_{ k=1 }^n B_k)$$
Let $n$ be a positive integer. Let $A_1, A_2, …, A_n$ be sets in some sample space $\Omega$, and let $P$ be a probability measure on $\Omega$. Assume that $\forall k \in \set{ 1, 2, …, n }, \b{P}(A_k) \neq 0 $. Then
$$\b{P}(\bigcap ^ n _ {k = 1} A_k) = \b{P}(A_1) \b{P}(A_2 | A_1) \b{P}(A_3 | A_1, A_2) … \b{P}(A_n | A_1, …, A_{n - 1})$$