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In this paper, we investigate the distribution of $k$-free numbers in a classof $\alpha$-random walks on the integer lattice $\mathbb{Z}$. In these walks,the walker starts from a non-negative integer $r$ and moves to the right by $a$units with probability $\alpha$, or by $b$ units with probability $1-\alpha$.For $k\geq 3$, we obtain the asymptotic proportion of $k$-free numbers in apath of such $\alpha$-random walks in almost surely sense. This provides ageneralization of a classical result on the distribution of $k$-free numbers inarithmetic progressions.

On the distribution of $k$-free numbers on the view point of random walks