I read that a go around is 1-3 out of every 1000 and I wondered how rare it is for a large commercial plane to actually hit the runway before deciding to go back up again.
An approach abort that results in touchdown is called a "balked landing" and generally happens when you've started the landing transition, reducing power and starting the flare, where you are in a "low energy state" (speed decelerating below Vref and engines going to idle) at the time you make the go-around decision. It's considered a high(er) risk maneuver like high speed rejects on takeoff, because at that point you are likely going to touch down, however briefly, whether you want to or not.
Normal missed approach go-arounds happen mostly because of weather where the runway environment wasn't visible at minimums, and maybe occasionally from a runway incursion that is noticed early, or something like a report of a flock of birds in the way, which would be most of those 1-3 per 1000 landings. They happen at at least a hundred or couple hundred feet and usually don't involve actually touching down.
A balked landing will generally only be for something like a runway incursion at the last second, leaving you with a situation where landing and rolling out is obviously more dangerous than going around because you are probably going to hit something on the runway, and they are pretty rare events. Balked landings are usually special maneuver covered in the initial type course and may be included in recurrent training.