A more common random fractal construction involves statistical
selfsimilarity: instead of specifying exact scalings, at each iteration
the scaling of each piece is selected randomly from a set range. 
Suppose the scaling factors are selected randomly from the
range [1/4, 1/2]. A square with diagonal corners (a, b)
and (a+L, b+L) is replaced by four squares arranged as shown here. Each
of the scaling factors r_{1}, r_{2}, r_{3}, and r_{4}
is selected randomly from [1/4, 1/2]. 

For example, here are three realizations of the first iterate of this random process;
below each is the sixth iterate. 

Click
here
for an animation of the first six steps of the left example. 
Statistical selfsimilarity refers to the fact that subpieces of each piece have
the same distribution of sizes. 

With probability 1, the dimension of this set is given by the
randomized Moran equation: 
E(r_{1}^{d} + r_{2}^{d} +
r_{3}^{d} + r_{4}^{d}) = 1. 
For example, construct a random Cantor set by at each stage either 
replacing each interval with two intervals scaled by 1/3, with probability 1/4, or 
replacing each interval with two intervals scaled by 1/4, with probability 3/4 

Then the randomized Moran equation becomes 
1  = E(r_{1}^{d} + r_{2}^{d}) 
 = (1/4)⋅((1/3)^{d} + (1/3)^{d}) + (3/4)⋅((1/4)^{d} + (1/4)^{d}) 
 = (1/2)⋅(1/3)^{d} + (3/2)⋅(1/4)^{d} 

Solving numerically, we find d ≈ 0.529053 