This is a [cross-post](https://math.stackexchange.com/q/3299314/64809) from math.stackexchange.com. There has been no response there.


Given a Markov chain of finite states with constant transition probabilities, what is the method to compute the probability of a path finally landing in a particular state for the first time having traversed all other states? Are there any references? The following is a puzzle as an example.


$12$ people sit on a round table to play a variation of the telephone game. They are numbered from $1$ to $12$ in clockwise order, i.e. the people with adjacent numbers (including $1$ and $12$) sit next to each other. Person #1 chooses a secret word and starts the game by randomly selecting one of the two neighbors and whispering the word to that person. Upon hearing the word, each person continues the game by randomly selecting one of the neighbors and whispering the word. The game ends when everyone knows the secret word.


What is the probability that the last person whispering the word is numbered $6$?



Probability of traversing all other states and finally landing on one state
For the example, the considered event $E$ is the intersection of two events $A$ and $B$, namely that both 5 and 7 are reached before 6. The union of the two events has probability $1$. So the probability of $E$ is $P(A)+P(B)-1$. By this answer:


<https://math.stackexchange.com/questions/725996/reaching-a-level-before-another-for-a-random-walk>


$P(A)=7/11$ and $P(B)=5/11$ so the sought probability is $1/11$. The argument can be adapted to any position on the circle : they all have the same probability to be reached last.


