"Correct me if I'm wrong, but this sounds like the argument from improbability, i.e. that this couldn't have happened because it's just so astronomically improbable. This is a fallacious argument."
No. Math is math. Odds are odds, and random is random.
What the math shows is that you don't get long series of data sequenced randomly.
That's not to say that long series of data can't be sequenced, just that it can't happen *randomly*.
You can try this at home. Write a quick program to generate random 1's and 0's. Let it run until it outputs a working program longer than 64 machine commands. What you'll find is that you'll *never* get a useful program sequenced...but that doesn't mean that useful programs can't be written to your hard drive. It's not impossible. It's not improbable; it simply isn't done randomly.
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Let's assume, for the sake of argument, that there is only one useful program, and further that this useful program is N bits long. Then, if you flip a coin N times and record a 1 each time the coin comes up heads and a 0 each time it comes up tails, you generate a random sequence of N bits. There are 2^N possible such sequences. Since we assumed that only one of these is a useful program, this implies that this program can be generated randomly with a probability of 1/2^N. Admittedly, if you generate the sequence once, twice or even any reasonable number of times, you almost assuredly will not generate a useful program (assuming N is reasonably large). In principle, though, it is possible to randomly generate this sequence.