A dog named Alfred is sitting in the corner $A$ of a square $ABCD$. A bone is placed in the center $O$ of the square. We assume that Alfred moves randomly, following the segments created by $OABCD$. We call $P_n$ the probability of Alfred reaching the bone in the center $O$ for the first time after $n$ moves and $S_n$ the probability of Alfred reaching the center at least one time after $n$ moves.
The questions regarding this problem are:
(1) Compute $P_n$
(2) Find the link between $P_n$ and $S_n$ and compute $S_n$
(3) Calculate $\lim\limits_{n \to \infty} S_n$
Now I understand that $S_n = \sum_{k=1}^{n} P_k$, and I have tried to find a formula for $P_n$ using matrices but I don't see a pattern emerging and without the help of wolfram alpha it would take me decades to do the calculations (which are a $5 \times 5$ matrix to the power of $n$). Can someone help me figure out how to compute $P_n$ ?
There was a mistake with my reasoning, here is a fix :
If $X_n$ is the position at set $n$, we have : \begin{align*} P_n&=\mathbb P[X_n=O,X_1\neq O,\dots,X_{n-1}\neq O]\\ &=\mathbb P[X_n=O|X_{n-1}\neq O]\cdot\left(\prod_{k=2}^{n-1} \mathbb P[X_k\neq O | X_{k-1}\neq O]\right)\cdot \mathbb P[X_1\neq O]\\ &=\frac{1}{3}\cdot \left(\prod_{k=2}^{n-1} \frac{2}{3}\right) \cdot 1\\ &=\frac{2^{n-2}}{3^{n-1}} \end{align*}