Solution
= Solution
The <events> $A\cap B$ and $A\cap B^c$ are disjoint and their <union> is $A$. Additivity of <probability> gives
$$
\mathbb P(A)=\mathbb P(A\cap B)+\mathbb P(A\cap B^c).
$$
Both conditioning <events> have positive <probability> because $0<\mathbb P(B)<1$. Apply the definition of <conditional probability> to each term to obtain the <law of total probability>:
$$
\boxed{\mathbb P(A)=\mathbb P(A\mid B)\mathbb P(B)+\mathbb P(A\mid B^c)\mathbb P(B^c).}
$$