  We examine a family of discrete probability distributions that describes the
"spillage number" in the extended balls-in-bins model. The spillage number is
defined as the number of balls that occupy their bins minus the total number of
occupied bins. This probability distribution can be characterised as a normed
version of the expansion of the noncentral Stirling numbers of the second kind
in terms of the central Stirling numbers of the second kind. Alternatively it
can be derived in a natural way from the extended balls-in-bins model. We
derive the generating functions for this distribution and important moments of
the distribution. We also derive an algorithm for recursive computation of the
mass values for the distribution. Finally, we examine the asymptotic behaviour
of the spillage distribution and the performance of an approximation to the
distribution.
