Do we choose a probability p the machine picks A, or does the machine start with a probability p, which we adjust to p+q chance it picks A?

Although I am no expert, I think your quantum computing comments are incorrect. To explore branches, retaining all histories, you need a "nondeterministic" computer that branches freely. This gives an exponential (2^n) speedup over a classical computer. Quantum computers apparently give only a polynomial one. For more detail, check out Scott Aaronson's blog "Schtetl-Optimized": http://scottaaronson.com/blog/

On the note of self-testing vs. controlled experiment, has anyone here tried the polyphasic ("uberman") sleep cycle? Does anyone know of any controlled experiments, either self-administered or larger-scale, which I could look at? I was interested in trying it a few years ago, but dropped in in about 24 hours (before I could have really even been said to try it) due to microsleep in waking hours.