Apart from Bayesian approaches, the average run length (ARL) to false alarm has always been seen as the natural performance criterion for quantifying the propensity of a detection scheme to make false alarms, and no researchers seem to have questioned this on grounds that it does not always apply. In this article, we show that in the change-point problem with mixture prechange models, detection schemes with finite detection delays can have infinite ARLs to false alarm. We also discuss the implication of our results on the change-point problem with either exchangeable prechange models or hidden Markov models. Alternative minimax formulations with different false alarm criteria are proposed.