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# The harmonic mean formula for probabilities of unions: applications to sparse random graphs

@article{Aldous1989TheHM, title={The harmonic mean formula for probabilities of unions: applications to sparse random graphs}, author={David Aldous}, journal={Discrete Mathematics}, year={1989}, volume={76}, pages={167-176} }

- Published 1989 in Discrete Mathematics
DOI:10.1016/0012-365X(89)90316-6

Under a symmetry condition, the probability of a union of events can be expressed in terms of the harmonic mean number of events which occur, conditioned on a specific event occurring. This leads to improvements of Boole’s inequality in settings where standard inclusion-exclusion techniques fail. Applications to asymptotic independence numbers and partition numbers of sparse random graphs are given.