I touched on this under the heading 'Modelling Monopoly as a Markov Process' (pasted below). Looking back at what I wrote, however, it definitely needs some elaboration.
First, a brief note on turns and dice rolls: the majority of turns (5/6, to be exact) will end in a single roll of the dice. However, if doubles are rolled, the dice must be rolled again within the same turn. Players rolling 3 doubles in a row (P = 1/216 in any given turn) go straight to jail. This has been taken into account in the data.
The mean number of rolls/turn is approximately 1.2. There is no relevant reason to count in turns rather than rolls, and the latter will help simplify the analysis.
First, I draw the distinction between a 'turn' and a 'roll'. A turn can include multiple rolls (up to 3) as a result of rolling doubles.
I then assert that there isn’t any “relevant reason to count in turns rather than rolls”. What I mean by this is that either unit will lead to the same landing probabilities for each square.
One potential objection to my view is that rolling three doubles in a row sends a player to jail. I tried to account for this by increasing the landing probability for jail by 1/216, which is the probability that any given roll is the third double in a row (1/6 * 1/6 * 1/6). I then multiplied each other landing probability by 215/216, such that the probabilities of landing on each space total to 1.
This isn’t strictly accurate — the probability of rolling a third double given that a player is on a particular space on the board isn’t uniformly distributed; however, after the first couple dice rolls, it is a very close approximation.
