Law of total probability for an event and its complement

Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B \in \mathcal{F}$ with $0 < P(B) < 1$. Let $B^c = \Omega \setminus B$. Then $$P(A) = P(A \mid B) \, P(B) + P(A \mid B^c) \, (1 - P(B)).$$

Remarks

This is the law of total probability for the partition $\{B, B^c\}$, together with $P(B^c) = 1 - P(B)$. The formula is the denominator of Bayes' theorem in the case of two hypotheses.

Explanations

L1 human

Either $B$ happens or it does not. So the chance of $A$ is its chance when $B$ happens times $P(B)$, plus its chance when $B$ does not happen times $1 - P(B)$.