  We use various combinatorial and probabilistic techniques to study growth
rates for the probability that a random permutation from the Mallows
distribution avoids consecutive patterns. The Mallows distribution behaves like
a $q$-analogue of the uniform distribution by weighting each permutation $\pi$
by $q^{inv(\pi)}$, where $inv(\pi)$ is the number of inversions in $\pi$ and
$q$ is a positive, real-valued parameter. We prove that the growth rate exists
for all patterns and all $q>0$, and we generalize Goulden and Jackson's cluster
method to keep track of the number of inversions in permutations avoiding a
given consecutive pattern. Using singularity analysis, we approximate the
growth rates for length-3 patterns, monotone patterns, and non-overlapping
patterns starting with 1, and we compare growth rates between different
patterns. We also use Stein's method to show that, under certain assumptions on
$q$, the length of $\sigma$, and $inv(\sigma)$, the number of occurrences of a
given pattern $\sigma$ is well approximated by the normal distribution.
