ABSTRACTThe Boolean lattice consists of all subsets of partially ordered under the containment relation. Sperner's Theorem states that the largest antichain of the Boolean lattice is given by a middle layer: the collection of all sets of size , or also, if is odd, the collection of all sets of size . Given , choose each subset of with probability independently. We show that for every constant , the largest antichain among these subsets is also given by a middle layer, with probability tending to 1 as tends to infinity. This is best possible, and we also characterize the largest antichains for every constant . Our proof is based on some new variations of Sapozhenko's graph container method.