A model is developed for the detection time of fires in nuclear power plants, which differentiates between competing modes of detection and between different initial fire severities. Our state‐of‐knowledge uncertainties in the values of the model parameters are assessed from industry experience using Bayesian methods. Because the available data are sparse, we propose means to interpret imprecise forms of evidence to the develop quantitative information, which can be used in a statistical analysis; the intent is to maximize our use of all available information. Sensitivity analyses are performed to indicate the importance of structural and distributional assumptions made in the study. The methods used to treat imprecise evidence can be applied to a wide variety of problems. The specific equations developed in this analysis are useful in general situations, where the random quantity of interest is the minimum of a set of random variables (e.g., in “competing risks” models). The computational results indicate that the competing modes formulation can lead to distributions different from those obtained via analytically simpler models, which treat each mode independently of the