Here is the problem.
In a game(shown below) with the marker in the center on each turn the marker moves to a neighbouring region with all positions equally likely and the player collects the cash there(the cash is not replaced after that). The game is over once the marker returns to the center. On average how much cash does the player collect.
Here is what I have tried:
$\mathrm{Average} = \sum_{i=1}^4 P(i)i$, where $i$ is the number of dollars won.
$P(1) = 1/3$ and $P(2) = 1/3$ (after taking care of the case where the player cycles between empty positions)
But $P(3)$ and $P(4)$ are getting more difficult with all empty positions.
Is there any other way to solve this?