Plant transient calculations for reactor accidents and operational transient events must be performed and evaluated despite uncertainties in input data, empirical correlations, modeling assumptions, and evaluation criteria. The use of best-estimate analysis in conjunction with systematic evaluation of uncertainties provides an alternative to bounding analyses that may introduce unnecessarily large conservatisms. A methodology has been developed that combines established techniques for statistical analysis to deal with a general class of problems having uncertainties both in the evaluation criterion and the modeling parameters. Response surface and Monte Carlo sampling methods are used to determine the probability density function of the key output variable. This function is in turn combined with a probabilistic form of the evaluation criterion to estimate the probability that the criterion may be violated. The methodology is applied to an evaluation of the likelihood that high-pressure safety valves would be challenged by increasing pressure transient events in a boiling water reactor. Plant transient results are obtained from RETRAN-02 calculations, and safety valve characteristics are obtained from test data.