The probability to win in a single round is $\frac 13$.
The odds of winning in a single round are $1:2$. That is 1 possibility to win versus 2 equally likely possibilities to not-win.
It's the same for drawing.
If we keep going after a draw until someone wins, then the probability to win becomes $\frac 12$, and the odds to win becomes $1:1$.
The probability for drawing becomes $0$, and the odds for drawing becomes $0:1$.