I would use this definition, that is in the OP, the probability given is the one of having won the draw by that time: the first year when the draw happen the probability is 0, the second year the probability of having won is 9.125%, while the probability at the third year is 18.25%. This is the sum of the probability of having won either of the two draws (you can’t win more than once).

If you want to interpret the probability as a one time probability, then I agree with you.