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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 :

enter image description here

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

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  • $\begingroup$ Not sure I understand the question. Does "find herself without her scarf at her disposal" mean "it is cold out and her scarf is at the other location"? Something else? Just to say, if my interpretation is correct (highly doubtful) then the answer is wrong since it might exceed $p$ (if $p=0$ your formula yirlds $.5$ for instance). $\endgroup$
    – lulu
    Jan 14 at 17:34
  • $\begingroup$ Maybe it means "the scarf is at the other place" with no reference to the temperature? I guess that fits your formula better....if $p=1$ then your formula yields $0$ which is correct (if it is always cold, she always keeps her scarf at hand). Likewise $.5$ makes sense when $p=0$. So is that what you meant? $\endgroup$
    – lulu
    Jan 14 at 17:36
  • $\begingroup$ Yes, imagine that in the morning the temperature is sufficiently hot for Alice to not bring her scarf, then if it’s cold on the evening she doesn’t have her scarf at her disposal. $\endgroup$
    – Bozu
    Jan 14 at 18:03
  • $\begingroup$ Well, that doesn't answer my question since you specifically refer to the temperature in your example. My best guess (possibly wrong) is that you are just referring to the state in which the scarf is at the other location, with no reference to the current temperature at all. Is that right or not? $\endgroup$
    – lulu
    Jan 14 at 18:07
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    $\begingroup$ Anyway the OP modelling with the M.C. looks correct and nicely done $\endgroup$
    – Thomas
    Jan 14 at 18:22

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