  Using a one-way Monte Carlo algorithm from several different starting points,
we get an approximation to the distribution of toric threefold bases that can
be used in four-dimensional F-theory compactification. We separate the
threefold bases into "resolvable" ones where the Weierstrass polynomials
$(f,g)$ can vanish to order (4,6) or higher on codimension-two loci and the
"good" bases where these (4,6) loci are not allowed. A simple estimate suggests
that the number of distinct resolvable base geometries exceeds $10^{3000}$,
with over $10^{250}$ "good" bases, though the actual numbers are likely much
larger. We find that the good bases are concentrated at specific "end points"
with special isolated values of $h^{1,1}$ that are bigger than 1,000. These end
point bases give Calabi-Yau fourfolds with specific Hodge numbers mirror to
elliptic fibrations over simple threefolds. The non-Higgsable gauge groups on
the end point bases are almost entirely made of products of $E_8$, $F_4$, $G_2$
and SU(2). Nonetheless, we find a large class of good bases with a single
non-Higgsable SU(3). Moreover, by randomly contracting the end point bases, we
find many resolvable bases with $h^{1,1}(B)\sim 50-200$ that cannot be
contracted to another smooth threefold base.
