One of the eerie things about physics is that sometimes theories of very different phenomena end up having basically the same mathematical structure. Statistical mechanical probabilities arise from entropy maximization, and because of this weird analogy, perhaps you can think about quantum mechanical probabilities arising from some sort of maximization or minimization process. Except in quantum mechanics, you have to deal with complex numbers… what does it mean to maximize or minimize a complex number?
Go check out John Baez on the remarkable analogies between statistical mechanics and quantum mechanics and the idea of quantropy.
The Finch and Pea 