The Poisson probability distribution is used to model the number of cycles in the generalized Collatz problem. First, interrelated cycles are defined and used as a criterion in counting the cycles for a given q value. Initially, archived data in the mathematical literature (giving the known 3n + q cycles) is analyzed. For large samples, the Poisson probability distribution gives a poor fit of the data (there are too many cycles for the large x values). Associated cycles are defined and used as an additional criterion in counting cycles; this improves the data fit substantially. Some theory and empirical results are given in an attempt to explain the origin of this distribution of cycle counts. Degrees of freedom in probability distributions involving the difference between the number of odd and even elements in a cycle are shown to be a partial explanation for the distribution of cycle counts. (L, K) trees (generalized associated associated cycles) are defined and used to account for the smallest difference between the number of odd and even elements in the cycles for a given q value. The article consists entirely of analysis of empirical results; no proofs are given.