There are two settings where we can state rough answers. The simplest setting is
In a political election that opinion polls suggest will be very close, the chance your vote is decisive is about 13/N, where N is the number of votes.
The actual number "13" here (and the "3" below) depends on some rather arbitrary assumptions below, and if you asked three different statisticians you would likely get three different numbers in place of "13" -- it's just a rough estimate.
Of course the estimate above is only relevant where there are opinion polls, typically for political elections with a large number N of voters. In a different setting, which I'll call a club election, there are say 50 -- 5000 voters. Here you would have only haphazard knowledge -- maybe you and 6 out of your 9 friends favor candidate A ( From which you can't make any statistical inference, because your friends are likely to have opinions similar to yours) -- but what's more relevant is whether or not there's a general sense that one candidate is a clear favorite.
In a club election where there seems to be no clear favorite, the chance your vote is decisive is about 3/N, where N is the number of votes.
The mathematics for these estimates is indicated below. An interesting article What is the probability your vote will make a difference? by Gelman et al gave a very detailed analysis, based on opinion polls as of 2 weeks before the 2008 U.S. Presidential Election, of the chance of a voter in each State having the deciding vote, taking into account the complexities of the Electoral College system. Their conclusion was that in each of 4 particular States, the chance was about 1 in 10 million. This paper also discusses the academic literature.
In the second example all we know is "no clear favorite". Interpreting that crudely as "a prior distribution for proportion favoring candidate A is uniform on [a,b]" gives a probability your vote is decisive as 1/N \times 1(b-a). Taking [a,b] = [1/3,2/3] gives the stated formula.
Comment. It is a complete blunder to analyze elections as if people voted at random.