Perhaps the best well-known example of asking patients to do something at random was performed by Benjamin Libet in 1983. Libet asked patients to wait until a spontaneous moment and push a button as they watched an animated clockhand circle. Surprisingly, what he found was that there were about 200 ms between cerebral activity indicating the patient was about to push the button and the perception of the impulse by which they pushed the button. Libet could predict when they were "randomly" going to push the button before even they knew it, calling into question the concept of free will.
In 2008, another experiment was done by another research group, this time asking patients to "randomly" choose between left and right hand while hooked up to an fMRI. Researchers were able to determine participants decisions up to 10 seconds before the participants themselves "impulsively" made their decision by analyzing the voxels of their fMRI readout.
So there is obviously a fair bit more of determinism involved in our choice making then we are able to perceive. Are brain only seems to cue us in when it's time to act.
Libet, B., Gleason, C. A., Wright, E. W., and Pearl, D. K. (1983). Time of conscious intention to act in relation to onset of cerebral activity (readiness-potential). The unconscious initiation of a freely voluntary act. Brain, 106:623-642.
 Soon, C., Brass, M., Heinze, H., Haynes, J. (2008). Unconscious determinants of free decisions in the human brain. Nature Neuroscience 11, 543 - 545.