Summary. Data comprising colony counts, or a binary variable representing fertile (or sterile) samples, as a dilution series of the containing medium are analysed by using extended Poisson process modelling. These models form a class of flexible probability distributions that are widely applicable to count and grouped binary data. Standard distributions such as Poisson and binomial, and those representing overdispersion and underdispersion relative to these distributions can be expressed within this class. For all the models in the class, likelihoods can be obtained. These models have not been widely used because of the perceived difficulty of performing the calculations and the lack of associated software. Exact calculation of the probabilities that are involved can be time consuming although accurate approximations that use considerably less computational time are available. Although dilution series data are the focus here, the models are applicable to any count or binary data. A benefit of the approach is the ability to draw likelihood-based inferences from the data.