Rolling Dices

[panz]

here is how the problem goes:
Charlie and Dave are betting over a sequence of rolls of a single die. Charlie will win the first time consecutive rolls produce the same number. Dave will win the first time there are three consecutive rolls with all different numbers.
Example:
Charlie wins on these sequences
3-3
6-4-6-6
3-4-4
Dave wins on these sequences
4-6-3
6-4-6-4-2

The question asks: What is the probability that Charlie wins this game.

Basically the conclusion I have came up with is - the only way for the game to continue is for the two numbers to keep repeating, as in - 4-6-4-6-4-6-4-6, and whenver that sequence breaks, one person will win.
So, the probability I came up with (the probability that the game continues or Charlie wins) is (1/6)+ (1/6)(5/6) + (5/6)(1/6)^2 + (5/6)(1/6)^3 + (5/6)(1/6)^4...etc, and the limit of this sequence converges to 1/3. So the probability that Charlie wins is 1/3.

Is this right?

Thanks