That's a nigh impossible question to answer. There is a theory which suggests that a perfect understanding of a person's brain and all its neuronal connections--a schematic of the brain's circuitry, if you will--coupled with perfect knowledge of all inputs being fed to the brain would allow one to perfectly predict that person's action in the given situation. This is the same as understanding a program's code and all inputs into the program and being able to predict the outputs (which is possible). In this sense, the brain (which gives rise to the mind) is being viewed as a marvelously complex computer, and is therefore of a predetermined nature (and thus lacking of free will) based upon its configuration. Of course, the brain is much more plastic than a computer, and has the ability to (somewhat) alter itself as time passes, and it is indeed marvelously complex, therefore we have no way at this time to put this theory to any real test.
Of course, there are other theories, many of a spiritual nature, which suggest that something special (like the presence of a soul) is going on and that this other thing gives rise to something beyond the mechanical workings of the brain, and gives us free will. This is even more difficult (impossible even) to test, which (at this time) brings this debate down to a matter of personal belief.