Al Franken seems (for now, at least) to have won the Minnesota Senatorial election by 225 out of a total of about 3 million ballots cast: a margin of 0.0001, or 0.01%.
This margin of error is tiny; it's of the same order as the difference in length of your car between a day that's freezing and one that's in the 80's. (See here for steel's coefficient of thermal expansion if you want to check my math.)
This is so small that the result is a toss-up for all practical purposes. Presumably, however, society cannot accept that election results are random; we have to pretend that certainty can be had.
The margins of error of the voting process are sometimes larger than the margin of victory of the winner; this was certainly the case in Minnesota. Philip Howard of the University of Washington found seven such cases in the 2004 elections ("In the Margins: Political Victory in the Context of Technology Error, Residual Votes, and Incident Reports in 2004," 1/6/2005, PDF). He used three ways of thinking about error in an election: technology error, residual votes, and incident reports. For example, Howard cites a 2000 Caltech/MIT which found that the error rates for a large variety of vote counting processes were all 1% or more. (Recall that the margin of victory in Minnesota was one-one hundredth of this: 0.01%) He concludes: "In each case, the electoral outcome was legitimated by elections officials, not the electorate, because in very close races the voting process cannot reveal electoral intent."
In Minnesota, with all the recounts, many of those errors were removed. But there are many kinds of randomness in an election beyond the measurement: someone absent-mindedly ticking the wrong box, someone else deciding at random not to vote on a given day, or people who mistake one candidate for another. In the end, we just don't know the answer, and a coin toss (whether overt or hidden) is a fine way to decide the result. If it was a bad choice, the electorate can throw the bum out next time.