> Monty has to give you information or there is no game, only pure chance.

Correct.

However, if you don't appreciate the information (and the controversy here and elsewhere shows that few contestants could have), and switch on a whim, and boost your odds accidentally, that part *is* pure chance, at least as I understand chance. Or, equally, you could stay on a whim: a "whim" implies no consequential thought or deliberation.

And, win or lose, you would never know whether your decision had boosted your odds or not. All you would have is a goat or an automobile or Monty's proffered cash (which complicates your strategy to come away with something valuable, based on the principle of the bird in the hand).