with probability of occurrence for each . Suppose the bivariate probabilities are given by for every pair of indices . Kounias derived the following upper bound: which subtracts the maximum weight of a star spanning tree on a complete graph with nodes (where the edge weights are given by ) from the sum of the marginal probabilities . Hunter-Worsley tightened this upper bound by optimizing over as follows: where is the set of all spanning trees on the graph. These bounds are not the tightest possible with general bivariates even when feasibility is guaranteed as shown in Boros et.al. However, when the variables are pairwise independent (), Ramachandra-Natarajan showed that the Kounias-Hunter-Worsley bound is tight by proving that the maximum probability of the union of events admits a closed-form expression given as: where the probabilities are sorted in increasing order as . It is interesting to note that the tight bound in depends only on the sum of the smallest probabilities and the largest probability . Thus, while ordering of the probabilities plays a role in the derivation of the bound, the ordering among the smallest probabilities is inconsequential since only their sum is used. Comparison with the Boole–Fréchet union bound It is useful to compare the smallest bounds on the probability of the union with arbitrary dependence and pairwise independence respectively. The tightest Boole–Fréchet upper union bound (assuming only univariate information) is given as: As shown in Ramachandra-Natarajan, it can be easily verified that the ratio of the two tight bounds in and is upper bounded by where the maximum value of is attained when , where the probabilities are sorted in increasing order as . In other words, in the best-case scenario, the pairwise independence bound in provides an improvement of over the univariate bound in . Generalization More generally, we can talk about k-wise independence, for any k ≥ 2. The idea is similar: a set of random variables is k-wise independent if every subset of size k of those variables is independent. k-wise independence has been used in theoretical computer science, where it was used to prove a theorem about the problem MAXEkSAT. k-wise independence is used in the proof that k-independent hashing functions are secure unforgeable message authentication codes. See also Pairwise Pairwise disjoint References Theory of probability distributions Independence (probability theory)