We consider the problem of finding exact statistical characterizations of the Bernoulli trials network size estimator, a simple algorithm for distributedly counting the number of agents in an anonymous communication network for which the probability of committing estimation errors scales down exponentially with the amount of information exchanged by the agents. The estimator works by cascading a local, randomized, voting step (i.e., the i.i.d. generation of some Bernoulli trials) with an average consensus on these votes. We derive a tight upper bound on the probability that this strategy leads to an incorrect estimate, and refine the offline procedure for selecting the Bernoulli trials success rate.