Laguerre polynomial approximation method is applied to study the period-doubling bifurcation and chaos behavior in double-well Duffing system with a random parameter subjected to harmonic excitation. Firstly the stochastic system is reduced to its equivalent deterministic counterpart, through which the response of the stochastic system can be obtained by numerical methods. Then bifurcation and chaos in the stochastic double-well Duffing system is explored. Numerical simulations show that the nonlinear dynamical behavior is similar to their counterpart in deterministic nonlinear system such as period-doubling bifurcation, but in some local areas they are shown to have their specific features.