The multiplication rule is just a rearranged version of the Bayes rule.
$
\begin{align}
\mathbf P(A \vert B) &= \frac{\mathbf P(A \cap B)}{\mathbf P(B)} \\[8pt]
\mathbf P(A \cap B) &= \mathbf P(A \vert B) * \mathbf P(B) \\[8pt]
\mathbf P(A \cap B) &=\mathbf P(B \vert A) * \mathbf P(A)
\end{align}
$
This extend to the intersection of 3 or any $n$ events:
$
\begin{align}
\mathbf P(A \cap B \cap C)&=
\mathbf P(A) * \mathbf P(B \vert A) *
\mathbf P(C\vert A \cap \mathbf B) \\
\mathbf P(A_1 \cap \dots \cap A_n)&=
\mathbf P(A_1)*\prod_{i=2}^n \mathbf P(A_i\vert A_{i-1} \cap \dots \cap A_1)
\end{align}
$
It can also be viewed as a stepwise decision process, with increasing conditions at each step.
![[probability-multiplication.png|center|400]]