The idea of “almost” solving a problem is not a new one and can be surprisingly powerful. I'll admit though that I'm thinking more along the lines of “probably” solving a problem which can be MUCH more efficient than solving a problem with certainty (even when the probabilities are quite high). See for example the probabilistic tests here:
I could look this up in my school notes, but a cool example is designing phone systems (or any networks). Phone traffic can be modeled using the Poisson distribution (http://en.wikipedia.org/wiki/Poisson_distribution). You design the system for the most probable traffic and deal with low-probability downtime unless you want to pay extra to deal with the rare high volume traffic.