An estimate for the probability of dependent events

In this note we prove an estimate for the probability that none of several events will occur provided that some of those events are dependent. This estimate (essentially due to Filaseta, Ford, Konyagin, Pomerance and Yu) can be applied to coverings of ZZ by systems of congruences, coverings of ZdZd by lattices and similar problems. Although this result is similar to the Lovász local lemma, it is independent of it. We will also prove a corollary in the style of the local lemma and show that in some situations our lower bound is stronger than that given by the Lovász lemma. As an illustration, we shall make some computations with an example considered earlier by Chen.