  We consider a special one-parameter family of d-dimensional random,
homogeneous self-similar iterated function systems (IFSs) satisfying the finite
type condition. The object of our study is the positivity of Lebesgue measure
and the existence of interior points in these random sets and in particular the
existence of an interesting parameter interval where the attractor has positive
Lebesgue measure, but empty interior almost surely conditioned on the attractor
not being empty. We give a sharp bound on the critical probability for the case
of positivity Lebesgue measure using the theory of multitype branching
processes in random environments and in some special cases on the critical
probability for the existence of interior points. Using a recent result of Tom
Rush, we provide a family of such random sets where there exists a parameter
interval for which the corresponding attractor has a positive Lebesgue measure,
but empty interior almost surely conditioned on the attractor not being empty.
