In this paper, we study randomized consensus processing over general random graphs. At time step k, each node will follow the standard consensus algorithm, or stick to current state by a simple Bernoulli trial with success probability pk. Connectivity-independent and arc-independent graphs are defined, respectively, to capture the fundamental independence of random graph processes with respect to a consensus convergence. Sufficient and/or necessary conditions are presented on the success probability sequence for the network to reach a global a.s. consensus under various conditions of the communication graphs. Particularly, for arc-independent graphs with simple self-confidence condition, we show that Σk pk is a sharp threshold corresponding to a consensus 0 1 law, i.e., the consensus probability is 0 for almost all initial conditions if Σk pk converges, and jumps to 1 for all initial conditions if Σk pk diverges.