A new mathematical model is proposed for the action of counters such as the Geiger-Mueller or the scintillation counters. It is assumed that after each registration the counter is inoperative for a time interval of random length. The distribution of lengths of the inoperative periods is so defined that the Type I and Type II models familiar in the literature on counters are special cases. More important, it also allows an action that is a compromise between those two types. Assuming that the sequence being counted is a Poisson process with mean rate of occurrence $mT, m > 0$, in an arbitrary interval of length $T$, the process generated by the counter is discussed and rules are established for obtaining confidence intervals for the parameter $m$ from various types of counting experiments.