Now, the wrong response to this argument is to simply say that we have a civic duty (or however you want to phrase it) to vote even though it's irrational. That's the wrong response.
So what is wrong with the argument? Because something has to be wrong with it, right? Because if not voting were the rational choice, that leads to a clear problem if we carry out the consequences: no one votes.
Where the argument goes wrong is in an assumption that most likely is never stated (i.e. it's hidden) or is only vaguely referred to without being at all fleshed out or explained. I don't know how it would or should be stated or fleshed out or explained, but that's because, first, I've never tried to work it out, and, second, it has to end up being incoherent, which is why it's where the problem with the argument lies. What I can tell you is that the assumption has to do with the nature of a vote, what it is, in the sense of its purpose, what it means in a certain kind of context.
The incoherence would arise so long as the nature of a vote is fleshed out in a way that makes it an individual thing, a singular object, a thing that is a thing itself. Why is that incoherent? Because the nature of a vote – and this relates to having to do with the nature of a democracy – can only make sense within a larger context: a vote can only be a vote when there are other votes existing in the same context that make it a vote. In other words, the very nature of a vote is that it is fundamentally part of a collective of votes; and without that context, there's no such thing as a vote. (Let me point out: it may perhaps be more correct to say a potential collective. Like I said, I haven't put much thought into this. I suppose that, whether it would make more sense to put it in terms of a potential collective or not would depend on some of the other details in fleshing out the whole theory on the nature of a vote and of voting.)
So the problem with the logical argument against voting is that it assumes a fundamentally individualistic conceptualization to the nature of a vote, such that, your vote is a single and individual thing all by itself, a vote-object distinct and individuated from all other objects that are also votes.
Once you reject that idea, and instead take the nature of a vote to be something fundamentally existentially relative to a collective context, the logical argument falls apart because it no longer makes any sense to ask whether your individual vote matters, since your vote cannot be a distinct, individual object that could be coherently conceptually considered in isolation from all other votes. So, to ask whether your vote matters is to ask whether the collective of votes to which your vote belongs matters. Once you ask it that way, the logic cannot lead you to that simple mathematical translation of the question being what the probability is that your individual vote will decide the outcome of the election (in your state). Instead, a mathematical and probabilistic translation is going to end up being a lot more complicated. But, I'm not sure you need the numbers to argue the case in favor of voting; I think a purely conceptual argument would suffice.
[So, I included J.S. Mill in my tags, because, surely, Mill has to figure into this discussion somehow. Right?)