A range of problems in Diophantine geometry have been unified in a series of conjectures going under the name of the Zilber-Pink conjectures on so-called "unlikely intersections". Here, if X is a smooth irreducible algebraic variety and Y and Z are two subvarieties, then we say that the intersection of Y with Z is unlikely if dim(Y) + dim(Z) < dim(X). For a simple example of Zilber-Pink conjecture, take X = {\mathbb{G}_m}^g to be a Cartesian power of the multiplicative group over the complex numbers, Y \subseteq X to be an irreducible subvariety which is not contained in a translate of a proper algebraic subgroup, then the union of X \cap T over all algebraic subgroups T \leq X with dim(T) + dim(Y) < g is not Zariski dense in X. We consider a complementary problem: how large should the union of such intersections be when they are likely? Using methods from o-minimality and differential algebra, we show that after accounting for some obvious geometric obstructions, the set of likely intersections will be dense in Y for the usual Euclidean topology. In this lecture, I will survey the o-minimal approach to Diophantine geometry and Hodge theory, connect this with differential algebra, and then formulate and explain a precise version of the likely intersections theorem. (This is a report on joint work with Sebastian Eterovic available at arXiv:2211.10592.)