In my college genetics lab, we ran a computer simulation on allele frequency in a population of purebred vs. "mutts." Basically, you're almost as likely to end up with a sick dog either way, but with a purebred, it's easier to know the likelihood of running into a certain genetic problem, since there are usually 3-5 "common" genetic problems in any pure breed. With a mixed breed, you have almost as high of a likelihood of finding a problem, but it's anyone's guess what that problem might be--and the problems are usually less severe. But that's why we have vets and the internet
I don't really think you have an advantage either way, the mutt at the pound could have a better temperament and overall health than the purebred dog with a champion pedigree, or it could be the other way around. Every dog is unique.