[math-fun] Asimov's 9s question

D.Asimov's question: Given an infinite string of random digits, what is the probability digits N to 2N are all 9s, for at least one value of N>0? I claim one can exactly solve this. The key is to set up the argument to count mutually-exclusive events. Consider a finitely-long string, L long. There are 10^L such. 10^(L-2) begin 99. 9*10^(L-4) begin x999 where x is not 9. 9*10^(L-5) begin xy9999 where y not 9. et cetera. So it seems to me all problems of this ilk can be solved by expressing as sums which then can be done in closed form.