  We describe the typical homological properties of monomial ideals defined by
random generating sets. We show that, under mild assumptions, random monomial
ideals (RMI's) will almost always have resolutions of maximal length; that is,
the projective dimension will almost always be $n$, where $n$ is the number of
variables in the polynomial ring. We give a rigorous proof that
Cohen-Macaulayness is a "rare" property. We characterize when an RMI is
generic/strongly generic, and when it "is Scarf"---in other words, when the
algebraic Scarf complex of $M\subset S=k[x_1,\ldots,x_n]$ gives a minimal free
resolution of $S/M$. As a result we see that, outside of a very specific ratio
of model parameters, RMI's are Scarf only when they are generic. We end with a
discussion of the average magnitude of Betti numbers.
