  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.
