How Small Can the Overlap Be?

Two events $A$ and $B$ live in the same sample space and satisfy

$P(A) = 0.6 \qquad P(B) = 0.5$

Nothing else is known about them — in particular they are not assumed to be independent, and they are not assumed to be mutually exclusive.

Across every probability model consistent with these two values, what is the smallest possible value of $P(A \cap B)$?