If you choose the first option, $1 million will be immediately added to your bank account, however if you choose the second option, starting from today everyday your chances of winning $1 billion increase by 0.005%. If you accept your chances will start at 0% and keep increasing.
I think you are mixing up one-time probabilities and cumulative probability.
The probability of 0.005% increasing every day is the cumulative probability. So by year 5 the odds of having won the billion are 5x365x0.005%=9.125%, no additional formula. To get to 25% cumulative probability you’d have to wait some 15 years.
I think we’re both using the wrong word, as I can’t find an actual definition for “cumulative probability”. However my formula is what you’d use to calculate the probability of having won at least once over all the drawings.
While you are right that by year five the odds of winning are 9.125%, that is the odds of winning just on the fifth draw. However, you had a chance of winning each year prior, meaning that the odds of winning by year five are higher than 9.125%.
Its like with die, the odds of rolling a 3 is always 1/6, but the odds of rolling a 3 if you roll twice is ~30%.
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.