I want to find out the probability that a 1-dimensional asymmetric random walk, which steps to the right with probability $p > \frac{1}{2}$ and to the left with probability $1-p$, ever returns to the origin.
I tried to solve this by defining the probability of return to the origin after exactly $n$ steps, and taking the summation over these probabilities. This led me to
$$ \mathbb{P}(\text{"return to origin"}) = \sum_{n=0}^{\infty} \binom{2n}{n} p^{n} (1-p)^{n}$$
However, for any value for $p$ greater than $\frac{1}{2}$, the series converges to a number greater than 1. (Tested with WolframAlpha.) What am I doing wrong? How can I solve this problem?
Best Answer
What you're doing wrong is that you're double-counting those walks that return to the origin twice or more. Instead of $\binom{2n}{n}$, you may want to use the Catalan numbers to construct the number of paths that return to the origin for the first time after exactly $n$ steps.