Supose that $(S_n)_{n\geq0}$ is a random walk on $\{0,1,2,\dots,N\}$ with up prbability of $p$ and down probability of $(p-1)$. Find $v_k$ the probability of absorption at $N$ if the walk starts at $S_0=k$ for $0 \leq k \leq N$
$v_k=\mathbb{P}($win$)$
$v_k=pv_{k+1}+(1-p)v_{k-1}$
and $v_0=0$ and $v_N=1$. The auxillary equation for this difference equation is
$p\lambda^2-\lambda+(1-p)=0$
$(p\lambda-(1-p))(\lambda-1)=0$
i.e. $\lambda=\frac{1-p}{p}$ or $1$. If $p=\frac{1}{2}$, then the homogeneous equation ($w_k$) is
$w_k=A+B$
I am unsure of where to go from here because $v_0=0=A+B$ and $v_N=1=A+B$, which seems to suggest that $A=B=0$, ie. $v_k=0$, which cannot be true.
I know how to do it if $p\neq(1-p)$. I am guessing that perhaps a different strategy is in order?
