Incorrect. You're describing the wrong experiment. You're not using average corks, you're using a set of 20 corks in which one is guaranteed to be flawed. Because in your experiment, on the 20th cork there's a 1/1 chance of infecting the bottle (assuming you make it that far), while in reality there's about a 36% chance that a set of 20 bottles will be pristine even if each cork as a 1/20 chance of being tainted.
The correct way to run the experiment is:
(19/20) * (19/20) * (19/20) * ... etc., for as many times as you want to re-cork the bottle.
19/20 is the chance of a cork being clean. And every single (re-)corking must use a clean cork or your bottle will become tainted, so it's (19/20)^X where X is the number of times you want to (re-)cork the bottle. (Don't forget to include the original cork.)
(19/20)^20 is ~36% chance that the bottle is still clean after 19 recorkings (20 total counting the original cork).
(19/20)^2 is 90.25% chance that the bottle is still clean after 1 recorking (2 total corks). Or as Julian put it, a 9.75% chance that the wine as become damaged.