TITLE: 2 dimensional symmetric random walk
QUESTION [6 upvotes]: Let $\left\{\left(X_{n}, Y_{n}\right)\right\}_{n=0}^{\infty}$ be a 2 -dimensional symmetric random walk. Namely, this is a Markov chain where $\left(X_{n+1}, Y_{n+1}\right)$ takes one of the following 4 values with equal probability:
$$\left(X_{n}+1, Y_{n}\right),\left(X_{n}-1,Y_{n}\right),\left(X_{n}, Y_{n}+1\right),\left(X_{n}, Y_{n}-1\right)$$
Suppose that $X_{0}=Y_{0}=0$.
Define $T:=\inf \left\{n \geq 0: \max \left(\left|X_{n}\right|,\left|Y_{n}\right|\right)=3\right\} .$
I want to find the value of $\mathbb{E}[T]$ and $\mathbb{P}\left(X_{T}=3, Y_{T}=0\right)$.
Thanks for your help!

REPLY [3 votes]: The answer above is entirely correct. Am just here with some simple code to give you confidence that $\mathbb{E}[T] = \frac{135}{13} $