Mutation produces new variation in populations, and in each generation these variants are copied from parents to offspring. While almost all variants of genes are lost, they may remain in the population for many generations. We use branching process models to analyze counts of gene copies. In a population of constant size, on average, a gene copy produces one offspring copy at the next generation. An advantageous mutant will have a mean greater than 1, and a deleterious one will have a mean less than 1. It is thought that most mutations are slightly deleterious, and with high probability those variants become extinct rapidly. Nonetheless, the few deleterious mutants that are not yet extinct may achieve high numbers. Thus, we have a particular interest in those with a mean slightly less than 1. We use different probability models for the offspring distribution and consider the mutant’s survival about: the extinction probability over k generations, the expected copy count conditional on survival, and the probability of survival additional k generations conditional on surviving k already. We find that variances in addition to means of offspring distributions closely relate these statistics. By adjusting parameters of distributions, we let the mean and variance be approximately the same across distributions. Based on our simulations, when k is large and the mean and variance of offspring are the same, the mutant’s survival condition is uniform throughout. In other words, those statistics above can be estimated by the mean and variance exclusively, and the specific distribution does not affect much the conditional population dynamics. However, at the first few generations, these statistics are different for each distribution. Thus, if we know the mean and variance of a mutant, we can predict the long-term population behaviors conditional on survival without knowing the true distribution of the mutant.