$n$ balls on a circle are colored white or black according to $n$ mutually independent binomial trials. It is shown here that, when their expectations converge with $n$, (a) counts of runs of various lengths are asymptotically independent Poisson; (b) counts of certain configurations other than runs yield asymptotic correlated Poisson distributions; (c) counts of configurations with structure independent of $n$ can be partitioned into equivalence classes, with asymptotic equivalence (equality with probability one) and asymptotic independence respectively within and among classes. It is also shown that (d) there cannot, essentially, exist configurations whose counts, asymptotically, are marginally, but not multivariate, Poisson.