  We show that the naive mean-field approximation correctly predicts the
leading term of the logarithmic lower tail probabilities for the number of
copies of a given subgraph in $G(n,p)$ and of arithmetic progressions of a
given length in random subsets of the integers in the entire range of densities
where the mean-field approximation is viable.
  Our main technical result provides sufficient conditions on the maximum
degrees of a uniform hypergraph $\mathcal{H}$ that guarantee that the
logarithmic lower tail probabilities for the number of edges induced by a
binomial random subset of the vertices of $\mathcal{H}$ can be
well-approximated by considering only product distributions. This may be
interpreted as a weak, probabilistic version of the hypergraph container lemma
that is applicable to all sparser-than-average (and not only independent) sets.
