Abstract. In the course of attempting to estimate the arrival rate of a single server queue using an active probing experiment, the authors found it necessary to derive the distribution of the number of arrivals between two probes under the conditions that the busy period of the queue lasts this long. In this paper we derive this distribution. The key building blocks in the derivation of the distribution are the classical ballot theorem and its generalized forms. Keywords: active probing; ballot theorem; busy period; M/D/1; M/G/1 1