  In this paper, we study the problem of finding the probability that the
two-dimensional (biased) monotonic random walk crosses the line $y=\alpha x+d$,
where $\alpha,d \geq 0$. A $\beta$-biased monotonic random walk moves from
$(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(\beta + 1)$ and
$\beta/(\beta + 1)$, respectively. Among our results, we show that if $\beta
\geq \lceil \alpha \rceil$, then the $\beta$-biased monotonic random walk,
starting from the origin, crosses the line $y=\alpha x+d$ for all $d\geq 0$
with probability 1.
