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rhollerith_dot_com comments on More art, less stink: Taking the PU out of PUA - Less Wrong
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The (fatal) flaw in your argument is that you multiplied probabilities without checking your model of reality for any obvious reasons to believe that the probabilities might be significantly dependent on each other.
In other words, if all we know about a man is that he is trying to mate, is the probability that he will succeed with woman #900 given that he struck out with #1 through #899 really the same as the probability that he will succeed with woman #1?
The general point still holds. P(at least one success) can be very large even if P(nth attempt succeeds) is small, for all n.
The independence assumption is implicit in my calling them Bernoulli trials, but you are correct that this may not be valid. Still, the general point stands. Good catch!