In probability theory, the van den Berg–Kesten (BK) inequality or van den Berg–Kesten–Reimer (BKR) inequality states that the probability for two random events to both happen, and at the same time one can find "disjoint certificates" to show that they both happen, is at most the product of their individual probabilities.

The special case for two monotone events (the notion as used in the FKG inequality) was first proved by van den Berg and Kesten[1] in 1985, who also conjectured that the inequality holds in general, not requiring monotonicity.

Reimer [fr; de][2] later proved this conjecture.

[3]: 159 [4]: 44  The inequality is applied to probability spaces with a product structure, such as in percolation problems.

be probability spaces, each of finitely many elements.

The inequality applies to spaces of the form

, equipped with the product measure, so that each element

is defined as the event consisting of configurations

can be verified on disjoint subsets of indices.

corresponds to tossing a fair coin

consists of the two possible outcomes, heads or tails, with equal probability.

that there exists 3 consecutive heads, and the event

would be the following event: there are 3 consecutive heads, and discarding those there are another 5 heads remaining.

[7] and their disjoint occurrence would imply at least 8 heads, so

[8] In (Bernoulli) bond percolation of a graph, the

Each edge is kept (or "open") with some probability

or otherwise removed (or "closed"), independent of other edges, and one studies questions about the connectivity of the remaining graph, for example the event

For events of such form, the disjoint occurrence

is the event where there exist two open paths not sharing any edges (corresponding to the subsets

in the definition), such that the first one providing the connection required by

[9]: 1322 [10] The inequality can be used to prove a version of the exponential decay phenomenon in the subcritical regime, namely that on the integer lattice graph

a suitably defined critical probability, the radius of the connected component containing the origin obeys a distribution with exponentially small tails:

may not be associative, because given a subset of indices

-ary BKR operation of events

where there are pairwise disjoint subset of indices

by repeated use of the original BK inequality.

[14]: 204–205  This inequality was one factor used to analyse the winner statistics from the Florida Lottery and identify what Mathematics Magazine referred to as "implausibly lucky"[14]: 210  individuals, confirmed later by enforcement investigation[15] that law violations were involved.

is allowed to be infinite, measure theoretic issues arise.

in the inequality is not defined),[13]: 437  but the following theorem still holds:[13]: 440 If

are Lebesgue measurable, then there is some Borel set