Bayesian predictive power is the expectation of the probability to meet the primary endpoint of a clinical trial, or any statistical test, at the final analysis, where expectation is computed with respect to a prior distribution over the true underlying effect. Bayesian predictive power is thus a way of quantifying the success probability for the trial sponsor while the trial is still running. The existing framework typically assumes that once the trial is not stopped at an interim analysis, Bayesian predictive power is updated with the resulting interim estimate. However, in blinded Phase III trials, typically an independent committee looks at the data and no effect estimate is revealed to the sponsor after passing the interim analysis. Instead, the sponsor only knows that the effect estimate was between predefined futility and efficacy boundaries. In the first part, we show how Bayesian predictive power for a time-to-event endpoint can be updated based on such knowledge only. In the literature it is often suggested to use a Normal prior. In the second part, we show that, unless the prior sample size is large compared to the Phase III trial, using a Normal prior leads to potentially undesired features of Bayesian predictive power that seem to be generally overlooked in applications.