On the number of bases of almost all matroids

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.