  We consider permutations avoiding a pattern of length three under the family
of Mallows distributions. In particular, for any pattern $\tau\in S_3-\{321\}$,
we obtain rather precise results on the asymptotic probability as $n\to\infty$
that a permutation $\sigma\in S_n$ 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 $\tau\in S_3-\{123\}$.
