I am currently working on a high school project revolving around the 'Cliff Hanger Problem' taken from ”Fifty Challenging Problems in Probability with Solutions” by Frederick Mosteller.
The problem is 'From where he stands, one step toward the cliff would send the drunken man over the bridge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is $\frac{2}{3}$ of a step toward the cliff $\frac{1}{3}$ . What is his chance of escaping the cliff?'
Although the book provided a solution for the eventual probability of falling via the use of recursive equations, I decided to see if I can derive an expression to compute the probability of one falling down prior step N.
What I did was to calculate the number of paths that one can take such that he reaches $X_n=1$, where n=2m-1 for some m (this is because at even steps the person would not fall so I'm only considering the case with odd steps) without reaching $X_j=1$ for any $0\le j< 2m-1$.
The following expression is what I have calculated.
$P_n=1-\sum\limits_{i=0}^{m} p_{2i+1}\\ =1-\sum\limits_{i=0}^{m} \left({{2i+1}\choose{i+1}} - \sum\limits_{k=1}^{i}{{2k}\choose{k}}\right)\left(\left(\frac{1}{3}\right)^{i+1}\left(\frac{2}{3}\right)^{i}\right)$
(I actually made a mistake here as I considered 2m+1 rather than 2m-1, which left the summation sign with $\sum\limits_{k=1}^{i}{{2k}\choose{k}}$ undefined when i=0)
Where $p_2i+1=$ the probability of the path touching 1 at $n=2i+1$ without touching $1$ prior to the step.
The first binomial expression corresponds to choosing n+1 steps towards the cliff out of the 2n+1 steps. The second binomial expression is to subtract the paths that stemmed from previous 1's (in order to ensure that the path did not touch 1 prior to $n=2i+1$.)
However as I plotted this into excel I found that the probability does not converge to $\frac{1}{2}$ as n->infinity, which is the answer the book obtained through recursive relations.
I reviewed my argument but I don't know what did I do wrong (whether I've overcounted or undercounted).
Can anyone help?