After the U.S. Civil War, the conduct of elections grew much more professionalized and this trend has continued over time. These days, an election win by more than 0.5% of the vote as determined as soon as complete preliminary and unofficial election results are received is unlikely to be challenged at all. Races are usually much closer than that in election contests where the final result differs from the preliminary and unofficial election results.
The chance of a race being this close are impossible to determine in the absence of data. The way electoral districts are drawn means that the partisan divide in the average district is quite far from 50-50, but candidates of both political parties tend to field candidates who are closer than average in their party to the median voter in the district.
My intuition from watching political news regularly, without doing the math, is that the contest rate is about 1% of multi-candidate races below the Presidency, and more like 10% in Presidential races.