For a sequence (c_n) of complex numbers we consider the quadratic polynomials f_(cn) (z):= z~2 + c_n and the sequence (F_n) of iterates F_n := f_(c_n) o ... o f_(c1). The Fatou set F_((c_n)) is by definition the set of all z implied by C-circumflex such that (F_n) is normal in some neighbourhood of z, while the complement of F_((c_n)) is called the Julia set J_((c_n)). The aim of this paper is to study the connectedness of the Julia set J_(c_n)) provided that the sequence (c_n) is bounded and randomly chosen. For example, we prove a necessary and sufficient condition for the connectedness of J_((c_n)) which implies that J_((c_n)) is connected if |c_n | <= 1/4, while it is almost surely disconnected if |c_n| <= #delta# for some #delta# > 1/4.
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