  We consider $N$ events that are defined on a common probability space. Those
events shell have a common probability function that is symmetric with respect
to interchanging the events. We ask for the probability distribution of the
number of events that occur. If the probability of a single event is
proportional to $1/N$ the resulting count probability is Poisson distributed in
the limit of $N\rightarrow \infty$ for independent events. In this paper we
calculate the characteristic function of the limiting count probability
distribution for events that are correlated up to an arbitrary but finite
order.
