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Let $S_0=0$ and let the independent and identically distributed random variables $S_n-S_{n-1}$ be $+1$ with probability $p>1/2$ and $-1$ with probability $q=1-p>0$. The strong law of large numbers gives $S_n\to +\infty$ almost surely, so the global minimum $W$ is finite and attained. For each nonnegative integer $m$, the unit-step property and translation of the starting point giveby the biased random walk hitting probability. Subtracting consecutive tails gives the displayed zero-based geometric distribution, with success parameter $1-q/p$. Inclusion of time zero is important: it ensures $-W\ge 0$. Reflecting the sign of the path yields the corresponding maximum of a downward-biased random walk.

The first-step analysis gives $h_0=1$ and, for $k\ge 1$,The minimal nonnegative solution for hitting probabilities theorem states that the probability of ever hitting a target is the pointwise smallest nonnegative solution of this transition recurrence with value one at the target. It follows by comparing any such solution with the probabilities of hitting within $n$ steps, then taking their increasing limit.

The characteristic equation $p{r}^2-r+q=0$ has distinct roots $1$ and $\beta =q/p\in (0,1)$. Hence every solution on the nonnegative integers with boundary value one has the formIf it is nonnegative for all $k$, taking $k\to \infty$ forces $A\ge 0$, and therefore $g_k-{\beta }^k=A(1-{\beta }^k)\ge 0$. The candidate $g_k={\beta }^k$ is itself nonnegative and satisfies the recurrence and boundary value. It is consequently the minimal solution, giving the biased random walk hitting probabilityOn the negative half-line the already established values $h_k=1$ satisfy the same recurrence away from zero. No unproved boundary condition at infinity has been imposed.