In this paper, we consider random iterations of polynomial maps $z^2 +c_n$ where $c_n$ are complex-valued independent random variables following the uniform distribution on the closed disk with center $c$ and radius $r$. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter $c = -1$, almost every random Julia set is totally disconnected with much smaller radial parameters $r$ than expected. We also introduce several open questions worth discussing.