  We examine properties of random numerical semigroups under a probabilistic
model inspired by the Erdos-Renyi model for random graphs. We provide a
threshold function for cofiniteness, and bound the expected embedding
dimension, genus, and Frobenius number of random semigroups. Our results
follow, surprisingly, from the construction of a very natural shellable
simplicial complex whose facets are in bijection with irreducible numerical
semigroups of a fixed Frobenius number and whose $h$-vector determines the
probability that a particular element lies in the semigroup.
