  We study the geometric requirements on a threefold base for the corresponding
F-theory compactification to admit a weakly-coupled type IIB limit. We examine
both the standard Sen limit and a more restrictive limit, and determine
conditions sufficient for their non-existence for both toric bases and more
general algebraic bases. In a large ensemble of geometries generated by base
changing resolutions we derive an upper bound on the frequency with which a
weak-coupling limit may occur, and find that such limits are extremely rare.
Our results sharply quantify the widely held notion that the vast number of
weakly-coupled IIB vacua is but a tiny fraction of the landscape.
