  In this paper we study two random analogues of the box-like self-affine
attractors introduced by Fraser, itself an extension of Sierpi\'nski carpets.
We determine the almost sure box-counting dimension for the homogeneous random
case ($1$-variable random), and give a sufficient condition for the almost sure
box dimension to be the expectation of the box dimensions of the deterministic
attractors. Furthermore we find the almost sure box-counting dimension of the
random recursive model ($\infty$-variable), which includes affine fractal
percolation.
