Various physical structures exhibit a fundamentally probabilistic nature over diverse scales in space and time, to the point that the demarcation line between quantum and classic laws gets blurred. Here, we characterize the probability of intermittency in the laminar-turbulence transition of a partially mode-locked fiber laser system, whose degree of coherence is deteriorated by multiple mode mixing. Two competing processes, namely the proliferation and the decay of an optical turbulent puff, determine a critical behavior for the onset of turbulence in such a nonlinear dissipative system. A new kind of polarization rogue waves is introduced at the point of transition to polarization turbulence. The probabilistic description of the puff-mediated laminar-turbulence polarization transition provides an additional degree of freedom for our understanding of the complex physics of lasers.