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In this work, we develop a computational framework to efficiently quantify the temporal exceeding probability of ship responses in a random wave field, i.e., the fraction of time that ship responses exceed a given threshold. In particular, we consider large thresholds so that the response exceedance needs to be treated as rare events. Our computational framework builds on the parameterization of wave field into wave groups and efficient sampling in the group parameter space through Bayesian experimental design (BED). Previous works following this framework exclusively studied extreme statistics of group-maximum response, i.e., the maximum response within a wave group, which is however not straightforward to interpret in practice (e.g., its value depends on the subjective definition of the wave group). In order to adapt the framework to the more robust measure by the temporal exceeding probability, novel developments need to be introduced to the two components of BED (on surrogate model and acquisition function), which are detailed in the paper. We validate our framework for a case of ship roll motion calculated by a nonlinear roll equation, in terms of the agreement of our results to the true solution and the independence of our results to the criterion of defining groups. Finally, we demonstrate the coupling of the framework with CFD models to handle more realistic and general ship response problems.