In this paper, the effects of asymmetry of the potential on the mean first-passage times (MFPTs) in two opposite directions are investigated in an asymmetric b istable system driven by multiplicative noise and additive noise. We find that t he MFPTs in two opposite directions are no longer symmetric in an asymmetric bis table system. Furthermore, we calculate the MFPTs of an asymmetric Duffing model . Numerical results show that: (1) The MFPTs depend on the initial states in th e asymmetric bistable system, namely, T+(xs1→xs2) is not equal to T-(xs2→xs1). (2) The effec ts of noise intensity on the MFPTs T+(xs1→xs2) and T-( xs2→xs1) are different in the same kind of parameter plan e. There exists a peak on each of the curves of lnＴ- versus Ｄ,as we ll as the “resonant activation”, while the variation of lnＴ+ ver sus Ｄ is monotonic. (3) The influence of the asymmetry coefficient r on Ｔ± is entirely different. The curves of lnT- versus r have mon o-valleys but the variation of lnＴ+ versus r is monotonically risi ng.