TITLE: Random Walk of a drunk man
QUESTION [5 upvotes]: Problem Statement:
From where he stands, one step toward the cliff would send the drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 2/3, of a step toward the cliff 1/3. What is his chance of escaping the cliff?
My take:
Say the probability that he dies from where he stands right now is p. 
Then,
he could comfortably make one step left and end his life with probability 1/3
Or he could take one step away and two step towards and boom...take two steps away and three steps toward...so on and so forth
Resulting in p= 1/3 + 2/3 * (1/3)^2 + (2/3)^2 * (1/3)^3 +....
Summing this infinite sequence gives me probability of dying as 3/7 (around 43%). I was rather puzzled when i learnt that the correct probability is 1/2. Cant figure out what are the other 7% ways for my drunken man to die which I missed above?

REPLY [4 votes]: Here is a proof that does not rely on solving recurrence equations. 
Let $r$ denote the probability of hitting the cliff. Let $0,1,\dots$ denote the distance from the cliff. He starts at $1$. 
If in the first step goes to $2$, then will hit the cliff if and only if will ever go back to $1$, and then will ever go to $0$. But probability of going back from $2$ to $1$ is the same as hitting cliff starting from $1$, that is $r$. Summarizing: 
$$r= \frac 13 + \frac 23 r\times r,$$
or (writing $r=\frac 13 r + \frac23 r$): 
$$ \frac 13 (r-1) = \frac 23 (r^2 -r).$$ 
That is, if $r\ne 1$, we have 
$$ \frac 13 = \frac 23 r.$$ 
Or $r=\frac 12$.
It remains to show that $r\ne 1$. I'l allow myself to be sloppy and lazy, and will rely on the law of large numbers, which tells as that the position of this walk at time $n$ is of order $(\frac 23 - \frac 13)n$. In particular, the position tends to $+\infty$. If $r=1$, then by iterating, the probability of ever getting to $-1$ is also $1$, and the same for $-2$, etc. In particular, the path is unbounded from below. This contradicts the conclusion of the law of large numbers.