  Given a self-similar iterated function system $\Phi=\{ \phi_i(x)=\rho_i O_i
x+t_i \}_{i=1}^m$ acting on $\mathbb{R}^d$, we can generate a parameterised
family of iterated function systems by replacing each $t_i$ with a random
vector in $\mathbb{R}^d$. In this paper we study whether a Lebesgue typical
member of this family will satisfy the strong separation condition. Our main
results show that if the similarity dimension of $\Phi$ is sufficiently small,
then a Lebesgue typical member of this family will satisfy the strong
separation condition.
