A few years ago I went through a database of twentieth-century congressional elections and found that, in the period 1900-1992, there were 20,597 contested elections, of which 6 were decided by fewer than 10 votes and 49 decided by fewer than 100 votes. Which suggests that we might expect to see an exact tie about once every 400 years. Maybe yesterday’s election will be it! I’d estimate the probability as something like one in a couple hundred; probably Nate could come up with something a bit more precise.
P.S. See here for a discussion of the relevance of this to the decision of whether to vote. An objection sometimes arises about this sort of calculation that one vote never makes a difference, because if the election were decided by one vote, there would be a recount anyway. On page 674 of this article, we discuss why this argument is wrong, even for real elections with disputed votes, recounts, and so forth. This can be shown by setting up a more elaborate model that allows for a gray area in vote counting and then demonstrating that the simpler model of decisive votes is a reasonable approximation.
P.P.S. I noticed there were some questions about my calculation, so very quickly: If the margin is within 100 votes, then there are 201 possibilities: Dem wins by 100, Dem wins by 99, . . ., Rep wins by 99, Rep wins by 100. Of these 201 possibilities, only one is a tie. Thus, Pr(tie) approx= (1/201)*Pr(margin within 100 points). Historical data show that elections are within 100 votes approximately 50 times a century, thus (to extrapolate) approx 200 times every 400 years. Thus, based on this simple calculation, you’d expect an exact tie approximately once every 400 years.