We consider permutations avoiding a pattern of length three under the family of Mallows distributions. In particular, for any pattern $τ\in S_3-\{321\}$, we obtain rather precise results on the asymptotic probability as $n\to\infty$ that a permutation $σ\in S_n$ under the Mallows distribution with parameter $q\in(0,1)$ avoids the pattern. By a duality between the parameters $q$ and $\frac1q$, we also obtain rather precise results on the above probability for $q>1$ and any pattern $τ\in S_3-\{123\}$.