Motivated by problems that arise in computing degrees of belief, rye c
onsider the problem of computing asymptotic conditional probabilities
for first-order sentences. Given first-order sentences phi and theta,
we consider the structures with domain {1,...,N} that satisfy theta, a
nd compute the fraction of them in which phi is true. We then consider
what happens to this fraction as N gets large. This extends the work
on 0-1 laws that considers the limiting probability of first-order sen
tences, by considering asymptotic conditional probabilities. As shown
by Liogon'kii [24], if there is a non-unary predicate symbol in the vo
cabulary, asymptotic conditional probabilities do not always exist. We
extend this result to show that asymptotic conditional probabilities
do not always exist for any reasonable notion of limit. Liogon'kii als
o showed that the problem of deciding whether the limit exists is unde
cidable. Wt analyze the complexity of three problems with respect to t
his limit: deciding whether it is well-defined, whether it exists, and
whether it lies in some nontrivial interval. Matching upper and lower
bounds are given for all three problems, showing them to he highly un
decidable.