This talk is focused on the comparison between the probabilities that the sums and differences of independent, identically distributed randomelements take values in very general sets. Depending on the setting--abelian or nonabelian groups, or vector spaces, or Banach spaces--many of these inequalities are sharp in various cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities. (Joint work with Zhao Dong, Wenbo Li, and Mokshay Madiman).