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Given events A and B on a product space (S={prod }_{i = 1}^{n} S_{i}), the set (A Box  consists of all vectors x = (x1,…,xn) ∈ S for which there exist disjoint coordinate subsets K and L of {1,…,n} such that given the coordinates xi,i ∈ K one has that x ∈ A regardless of the values of x on the remaining coordinates, and likewise that x ∈ B given the coordinates xj,j ∈ L. For a finite product of discrete spaces endowed with a product measure, the BKR inequality $$ P(A Box B) le P(A)P(B) $$(1)was conjectured by van den Berg and Kesten (J Appl Probab 22:556–569, 1985) and proved by Reimer (Combin Probab Comput 9:27–32, 2000). In Goldstein and Rinott (J Theor Probab 20:275–293, 2007) inequality Eq. 1 was extended to general product probability spaces, replacing (A Box  by the set Open image in new window consisting of those outcomes x for which one can only assure with probability one that x ∈ A and x ∈ B based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality Eq. 1 results, due to the fact that Open image in new window . In particular, it may be the case that (A Box  is empty, while Open image in new window is not. We propose the further extension Open image in new window depending on probability thresholds s and t, where Open image in new window is the special case where both s and t take the value one. The outcomes Open image in new window are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.

A BKR Operation for Events Occurring for Disjoint Reasons with High Probability