Conditional Probability
Definition
The conditional probability of event $A$ given event $B$ is
$$ \mathbb{P}(A \mid B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}, \quad \mathbb{P}(B) > 0. $$Chain Rule
For events $A_1, \dots, A_n$:
$$ \mathbb{P}(A_1 \cap \cdots \cap A_n) = \mathbb{P}(A_1)\,\mathbb{P}(A_2 \mid A_1)\,\mathbb{P}(A_3 \mid A_1, A_2) \cdots \mathbb{P}(A_n \mid A_1, \dots, A_{n-1}). $$Law of Total Probability
If $\{B_i\}$ is a partition of the sample space:
$$ \mathbb{P}(A) = \sum_i \mathbb{P}(A \mid B_i)\,\mathbb{P}(B_i). $$