The classic Calvinist answer to the issue of free will is that humans are free within the limits of their nature, which Scripture says is utterly depraved and at absolute enmity against God. To put it more simply: no, Calvinists do not believe in free will.
My German Shepherds like Diamond brand dog food over most other brands of dog food. They like ‘people food’ even better. My dogs cannot, however, go out into the world, seek for themselves some sort of gainful employment, earn a wage, and then drop by McDonald’s for a Big Mac, or stop by the feed store for a bag of Diamond dog food. If my dogs were turned out to fend for themselves, they would likely fare very poorly (most stray dogs live at most only a year or two on their own, often much less). Their choices in what they eat are limited by their nature.
It is said that Adam was created able to sin or not to sin; his descendants, by virtue of Adam’s fall, were rendered unable not to sin; in Christ, the Elect will be made forever able not to sin. It is important to realise that God holds all humankind culpable for sin, and that because of sin we have no rights at all before God. He is absolutely within His rights to consign all the human race to perdition, and He has no obligation whatever to save any member of the human race. Moreover, humans in our natural state do not want to be saved, anyhow: we hate God, we want to replace Him with some sort of lesser ‘god’ of our own choosing (ultimately, ourselves), and we make ourselves enemies against God. God acts in grace to change the hearts of some–for His own good will and purposes, and not because of any merit or virtue in any of those whom He chooses–and in grace causes them to love Him, saves them, and sanctifies them.
**Syele Grumbles about accusations of Pelagianism.
Syele: I wrote with some degree of care to make it clear that Arminianism is NOT Pelagianism and has a great deal more in common with Calvinism than many of it’s amateur proponents fully grasp. By the same token, Calvinists are not antinomians, though some have used the illustration that an Elect person remains elect even if the last act of their lives were to be to commit some sort of horrific atrocity. The illustration may help in some contexts to drive home the point of Eternal Security, but it misses the larger point that the Elect would be extraordinarily unlikely to commit such an atrocity precisely because the Spirit of God working in them would have changed their nature so that their conscience would constrain them from grossly wicked behavior of that sort.
My point in mentioning Pelagianism is that many in the Arminian or Molinist camps often revert to Pelagian arguments to prove their point. Many Trinitarians like to illustrate the Godhead by speaking of water as having a liquid, vapor, and frozen form–missing the point that their illustration actually proves the heresy known as modalism rather than demonstrating the Trinity. Hope that clarifies my point.