The fluctuation in the number of collisions suffered by particles as they slow down in a moderating medium is studied via a probability balance equation. The equation describes the collision history of foreign particles slowing down in a host medium and also accounts for the recoil particles produced in the collision. The equations are solved by the introduction of a generating function from which the space and time dependent probability distributions are obtained. That is, the probability that a particle will suffer just N collisions to reach energy E at a time t after injection. In the space dependent case it is the probability that a particle suffers just N collisions to travel a given path length before coming to rest.
Explicit expressions for the means and variances are obtained by solving a difference equation. From this solution it has been possible to obtain exact expressions for hard spheres and for a variety of models based on the inverse power law approximation. A number of new results are presented and some old ones rederived in a more efficient and general manner. The results are of value in the understanding of radiation damage cascades and in neutron slowing down in moderating materials.