In this paper we study self-trapping of Bose-Einstein condensates in a symmetric double-well potential with uniform noise or Gaussian noise existing respectively. We find that both uniform noise and Gaussian noise destroy the critical behavior of self-trapping with the interaction increasing and create a transition zone between Josephson oscillation and self-trapping, between which exists a critical point originally. Furthermore, the stronger the noise becomes, the wider the transition zone is. Meanwhile, we find that the phase space falls into complete confusion when we increase noise intensity to a certain extent with interaction fixed, and the trajectories reappear after we increase the interaction without increasing the noise intensity. In the full-quantum situation, when noise exists there is a transition zone instead of a critical value, and stronger noise creates wider transition zone, which is the same as in mean field treatment. What is different is that in full-quantum situation noise creates self-trapping, and the stronger the noise, the more obvious the self-trapping becomes.
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