Elections aren’t strictly probabilistic, so there’s no good answer to this question without adding enough assumptions to make the answer meaningless for general elections.
Also, for any election larger than a few thousand people, there’s enough noise that we don’t necessarily have confidence the results are totally valid down to the single vote, anyway (recounts often tally up differently than the official results).
As for the argument that one vote doesn’t matter, it certainly doesn’t matter a lot when compared against the aggregate, but it clearly matters, because you don’t have a reasonable aggregate unless sufficient people vote, i.e. if everybody bought into this argument, suddenly your single vote would clearly matter a great deal.
One thing you can say is that your single vote doesn’t matter enough to affect the aggregate, as you are reasonably assured of a reasonable turnout and especially close results. That’s usually true but not always true. For example, the 2000 Presidential elections could reasonably be said to hinge on Florida. Even then, a single vote wouldn’t have swung the tide, but, in aggregate, were there sufficient Nader voters who may have voted differently (or not at all) had they realized how close the result would have been? There’s certainly room to argue that point, since the result hinged on a few thousand votes.
And at that point, you have a pile problem. A single vote isn’t a sufficient aggregate. 10 votes probably isn’t, either. A million votes certainly is. 10000 votes is probably big enough, too. What about 50? Or 100? Or 1000? The problem is magnified when we have different levels in voting. Presidential elections depend on the individual states, and states have different numbers of electors. So, in a sense, a voter in Alaska has more influence than one in California. But California, in aggregate, has more influence than Alaska.
