This article has the following graph in it:
I can understand how to achieve the cumulative probability of finishing the game in $N$ moves. I implemented it by finding the probability matrix $P$ and initializing a vector $u$ with all zeros except the first element. Then by successively applying $u = uP$ for let's say a hundred moves and plotting $u_{100}$ from each move arriving at this plot:
How can I come up with the first plot, namely the chance of winning a game in $N$ moves?
One way is to play 10,000 games and keep track of the results. Or, program a computer to play 1 million games. This technique is called Monte Carlo simulation. While the results may be aproximate, if you run enough scenarios it looks pretty good.
Annother approach is a Markov chain. You build a matrix, that tells you that for the current location of your token, what is the probability on your next roll being in each other location.
$A^n$ tells the probability that your token will be an any other location after n rolls.
and of the game being over after $n$ rolls.