Your "base case" is correct, no argument there and so is your secondary example. The error comes in by looking at specifics instead of the whole. The probability of success using a strategy is the sum of (the probability of success in a case multiplied by the probability of that case) for every case. Now, since the set of integers is infinite, the sum probability of having the first number you see be less than one googleplex if the number is chosen completely at random is 0 (not just "really small" like 1/2^googleplex, so small that it is zero in the sense that lim(x->inf)1googleplex/x IS zero). Therefore the chance that you will get numbers which will lead you to have a probability over 50% is 0.