Here's your fallacy (which goes in repeat mode with you):
You didn't take into account that the probability to not find "ab" at all in 256 attempts is 36%.
You didn't take into account that the probability to find "ab" more than once is not "very low", but at a good 28% (100 - 36 (for 0) - 36 (for 1)). Because you never added the probabilities for it to appear 3 times, 4 times, 5 times, and so on (up to 256 times). You only base some claims on the fact that it's very unlikely to appear "a second time", but it can also appear more than 2 times, not necessarily just one more time.
So while in principle you may be on to something (though nothing more than simply observing the normal behavior of a uniform random variable), your calculations are off quite a bit, to the point that the combined probabilities of failure once you go skipping over and over again, accumulate, and they accumulate in the fashion that you like really much: compounded.
Now I ask you.
After finding the 1st ab, I calculate to find the 2nd ab.
I calculate to find the 3rd ab based on the 1st and 2nd ab.
After finding the 1st, 2nd and 3rd ab, I find the 4th ab.
Now, are you saying that these are not probabilities but predictions?
Are you saying that probability never changes?
Or do you say that there is an independent probability but accept that there can be a conditional probability?