Here the problem :
In the morning, Alice goes from her home to her office and returns home in the evening. She has a scarf : on each journey, if it is cold and her scarf is in place (in her office if Alice is in her office or at home if she is at home), Alice takes her scarf. If it is not cold, she leaves her scarf in place. Let $\{X(n),n∈ \mathbb{Z}\}$ be the process such that $X$ is $1$ if Alice has the scarf available where she is (i.e the scarf and Alice are both at the office or both at Alice's house), and $0$ otherwise. Each half-day, it is cold with probability $p$ and warm with probability $1-p$ independently from one half-day to the next. It is assumed that the process $X$ is at the stationnary state.
What is the probability that Alice will find herself without her scarf at her disposal ?
I've modeled the problem as a MC with 2 states :
Using this, I found that $\pi_0^{\star} = \dfrac{1 - p}{2 - p}$.
However, I don't know if my answer is correct, can someone tell me if this is correct ?
Any help is welcome, thanks