Carrot at end of stick

It’s interesting to me how the old metaphor of a carrot at the end of a stick — an incentive to the horse to keep going — has morphed into “carrots and sticks.” It’s often a “mix of carrots and sticks.”

I think it’s the same carrot, the incentive carrot, but now we get to beat people with the stick, when before the stick was there to keep the carrot just out of reach.

Before the beating, do we pocket the carrot first, then tie it back to the stick later? Not sure how it works. Also, do people — er, horses, whatever — sometimes actually get the carrot now, instead of it being forever in front of them?