Hmm, anyone able to help with how to treat the draw?

I know that the probability of Homer winning (ignoring the new rule) is;

define W as winning, and N as not winning. Then and [tex]P(N)=2/3[/math.

So for homer to win;

etc.

So the probability for homer to win is just the sum of these;

which is the sum of a geometric process, so

So now I just have no idea how to incorporate the draw into the game. I know that at the end of each players turn, there will be 3 states (1) they won, (2) they rolled a 1 (3) they rolled a 2, 3, 4. In state (1) the game ends, in state (2) The other player will roll, and if a 1 the game ends. and (3) the game continues if rolls a 1, and ends if roll a 6.