As a matter of fact, the success of Wright brothers over other contenders most likely has a lot to do with the fact that they were meticulous experimenters. Basically every detail in the Flyer was "evolved" through a series of controlled experiments. For example, they systematically tested wing shapes first with fitting them to a bicycle and biking vigorously, and later built a wind tunnel where they tested well over 200 different miniature wings - a major improvement on earlier experiments such as that of Otto Lilienthal, who tested only a few wing shapes in a much cruder setup. These systematic explorations fed to the construction of several "Glider" models, and experience with these unpowered craft in turn fed back to the experiments.

I think they were about as certain as they could be that the Flyer I would work. A letter written by Wilbur Wright after first failed take-off on December 14 1903 states

"...but for a trifling error due to lack of experience with this machine and this method of starting, the machine would undoubtedly have flown beautifully." (Kelly's biography of Wright Brothers, 2002 edition)

Three days later, following repairs, it did.

At least according to some historians, the success of Wright brothers over e.g. Samuel Langley (who was funded by the U.S. government) can be attributed to this scientific approach where different aspects of the airplane were tested systematically and in controlled conditions. Generally, other aviation pioneers tended to change a lot of details at once in their designs after each failure; Wright brothers adjusted one little detail at a time and tested what effect that change might bring. The former method made learning much more difficult as it was hard to say which change in design actually had an effect. A good argument along these lines is put forward by G. Bradshaw in the following two papers:

Bradshaw, G. (1992). The airplane and the logic of invention. In R. Giere (Ed.), Cognitive Models of Science (pp. 239–250). Minneapolis: University of Minnesota Press.

Bradshaw, G., & Lienert, M. (1991). The invention of an airplane. In Proceedings of the Thirteenth Conference of the Cognitive Science Society (pp. 605–610).