Stochastic dynamics in a turbulent swirling flow are reported in this paper via the probability density functions (PDFs) of responses with the generalized cell mapping (GCM) method. Based on the short-time Gaussian approximation (STGA) procedure, the influence generated by the time average and the amplitude of the fluctuation to the turbulent flow on the probabilistic responses are demonstrated. We observe that the shapes of the steady-state PDFs change from two peaks to the single peak with the change of system parameters, indicating that the rotation to shear ratio will change from two stable states into one stable state, while the torque difference of the propellers in the von-Karman turbulence experimental setup becomes large or changes in a wide range. That is to say, the stochastic P-bifurcation phenomena occur. The evolutionary mechanism of the transient response is revealed with the global portraits. Furthermore, the idea of block matrix is devoted to solving the storage problem due to the amount of image cells for the STGA procedure in high dimensional system. Monte Carlo (MC) simulations are in good agreement with the proposed strategy.