We are concerned with random ordinary differential equations (RODEs). Our main question of interest is how uncertainties in system parameters propagate through the possibly highly nonlinear dynamical system and affect the system's bifurcation behavior. We come up with a methodology to determine the probability of the occurrence of different types of bifurcations (sub- vs super-critical) along a given bifurcation curve based on the probability distribution of the input parameters. In a first step, we reduce the system's behavior to the dynamics on its center manifold. We thereby still capture the major qualitative behavior of the RODEs. In a second step, we analyze the reduced RODEs and quantify the probability of the occurrence of different types of bifurcations based on the (nonlinear) functional appearance of uncertain parameters. To realize this major step, we present three approaches: an analytical one, where the probability can be calculated explicitly based on Mellin transformation and inversion, a semi-analytical one consisting of a combination of the analytical approach with a moment-based numerical estimation procedure, and a particular sampling-based approach using unscented transformation. We complement our new methodology with various numerical examples.