In this two-part talk, we will study the behavior of union of translates of a given set by randomly chosen points from another set. A natural question is whether minimum is achieved by, say, the Euclidean ball. This turns out to be true, from which we can get another probabilistic interpretation of the Brunn-Minkowksi inequality. The proof is based on some combinatorial arguments and applying Steiner symmetrization in a new way. This talk is based on a paper by Ballister, Bollobas, Leader, and Tiba (https://arxiv.org/abs/2309.00103).