Conditional probability

The conditional probability of $A$ given $B$ is $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ for $P(B)>0$. It restricts the world to cases where $B$ happened and asks how often $A$ also happens there.

Rearranging gives the product rule $P(A\cap B)=P(A\mid B)P(B)$, and summing over a partition gives the law of total probability: $P(A)={\sum }_{i}P(A\mid B_i)P(B_i)$.

Supervised classifiers are, at heart, models of a conditional distribution $p(y\mid \mathbf{x})$: the probability of each label given the features.