A lot of board games involve rolling dice and moving around a cyclical board. Monolopy is the most common example. On the 16 position board below, the player’s piece was on the bottom row as depicted and the player rolled two dice that summed to 7. Then the player would move the piece to the top corner. If the dice are fair and the pieces only moves around the board according the dice roll, what is the steady state probability of being in the “starting” location (bottom left corner). You should be able to figure this out without solving the system. The starting location is 3 spots away.
I worked out on my own that the player needs to roll a 3, so rolling a 3 has the probability of 1/18. Is 1/18 my steady state? or is it something else?
The steady state is the set of positions such that if you moved things around randomly after they have a certain probability of being in particular positions, then the probability of being in a particular place does not change.
I.e. if you have some Markov matrix/operator, x=Px for the steady state, where x is a set of probabilities.
The steady state probability of a given position is the value of x when it is in that position. If you would like to try your homework problem, I can try to tell you if your answer looks good.