Abstract: Prospective phase II trials usually result in failures in phase III trials. For randomized controlled phase II and phase III trials which are conducted with patients randomized to one of two treatments where the variances of the normally distributed responses are assumed to be known, we analytically obtain the estimated and theoretical assurances for the three cases (no, additive, and multiplicative bias adjustments). Under some minor assumptions, we show that the estimated assurances for the three cases are increasing functions of the per group number of patients and the observed treatment effect of the phase II trial, respectively; and for Case 3, the estimated assurance is an increasing function of the retention factor. When the true treatment effect of phase III is assumed to be a known constant, we show that the theoretical assurances for the three cases are constants which are equal to the designed power or one minus the type II error. Moreover, we show that the estimated assurances are always less than the theoretical assurance. We also obtain the analytical formulas of the probabilities of launching a phase III study for the three cases. Moreover, for Case 3, we show that the probability of launching a phase III study is an increasing function of the retention factor. According to our theoretical investigations, we find that the true treatment effect of phase III has no effect in the simulations. Finally, the simulations are conducted to illustrate the theoretical investigations.