  We introduce a probability distribution on $\mathcal{P}([0,1]^d)$, the space
of all Borel probability measures on $[0,1]^d$. Under this distribution, almost
all measures are shown to have infinite upper quasi-Assouad dimension and zero
lower quasi-Assouad dimension (hence the upper and lower Assouad dimensions are
almost surely infinite or zero). We also indicate how the results extend to
other Assouad-like dimensions.
