The method I described gives nonzero probability to every positive integer. The method you describe a) will never terminate and thus b) gives zero probability t

The method I described gives nonzero probability to every positive integer. The method you describe a) will never terminate and thus b) gives zero probability to every positive integer (since there's no stopping condition, even if you ever get to a particular integer you're always going to generate another digit and move on past it). The essential issue is that probabilities have to sum to 1, but you can't just give the same probability to an infinite number of things because there's no number for which p*infinity = 1. So the only way to get an infinite sum to equal 1 is if the probabilities are unequal, and in particular the probabilities have to go to zero in the limit (e.g. 1/2 + 1/4 + 1/8 + 1/16 + ... = 1).