I am given the following problem:
Every time I teleport, the teleportation machine decides to transport me to the very same place I am with probability $α ∈ (0,1)$, or, if I don't repeat the same location, it randomly select one of the other $N−1$ places to which I can teleport and places me there instead. Find the limiting probability $$ \lim_{n→∞} P(\text{nth teleportation place is the same as the first place}) $$ I am also given the following hint: Define a suitable two state Markov Chain with states {initial place, other place}.
I approach the problem in the following way: Let $X_n = 0$ be the state in which we are at the initial place. Let $X_n = 1$ be the state in which we are at other place. Then the transition matrix is partially given by $$ P= \begin{bmatrix} \alpha & 1-\alpha \\ \beta & 1-\beta \end{bmatrix} $$ However, I can't seem to find $\beta$. I tried used Bayes' rule and conditional probability but to no avail. Any help would be greatly appreciated
If you are in another place, the probability you come back to the initial place is the probability you do not stay still $(1-\alpha)$ multiplied by the probability the initial place is chosen $\left(\frac{1}{n-1}\right)$, so
$$\beta = \frac{1-\alpha}{n-1}$$