In common with other non-linear models, the optimal design for a limiting dilution assay (LDA) depends on the value of the unknown parameter, θ, in the model. Consequently optimal designs cannot be specified unless some assumptions are made about the possible values of θ. If a prior distribution can be specified then a Bayesian approach can be adopted. A proper specification of the Bayesian approach requires the aim of the experiment to be described and quantified through an appropriate utility function. This paper addresses the problem of finding optimal designs for LDAs when the aim is to determine whether θ is above or below a specified threshold, θ0.