  For a matroid $M$ of rank $r$ on $n$ elements, let $b(M)$ denote the fraction
of bases of $M$ among the subsets of the ground set with cardinality $r$. We
show that $$\Omega(1/n)\leq 1-b(M)\leq O(\log(n)^3/n)\text{ as }n\rightarrow
\infty$$ for asymptotically almost all matroids $M$ on $n$ elements. We derive
that asymptotically almost all matroids on $n$ elements (1) have a
$U_{k,2k}$-minor, whenever $k\leq O(\log(n))$, (2) have girth $\geq
\Omega(\log(n))$, (3) have Tutte connectivity $\geq \Omega(\sqrt{\log(n)})$,
and (4) do not arise as the truncation of another matroid.
  Our argument is based on a refined method for writing compressed descriptions
of any given matroid, which allows bounding the number of matroids in a class
relative to the number of sparse paving matroids.
