  We use counting arguments to show that asymptotically almost all sparse
paving matroids contain an $H$-minor, where $H$ falls into one of several
simple classes of matroids. Furthermore the result holds for all $H$ in a
larger class of matroids, if we restrict to asymptotically almost all sparse
paving matroids of fixed rank $r$ (where $r$ is necessarily no smaller than the
rank of $H$).
