  In this paper, we study the pattern occurrence in $k$-ary words. We prove an
explicit upper bound on the number of $k$-ary words avoiding any given pattern
using a random walk argument. Additionally, we reproduce several already known
results and establish a simple connection among pattern occurrences in
permutations and $k$-ary words. A simple consequence of this connection is that
Wilf-equivalence of two patterns in words implies their Wilf-equivalence in
permutations.
