Ask HN: The Case of the Mixed-Up Hats

There's a baseball team that consists of 20 kids. At the end of the season, they all go to the coach's house to have a barbecue. The coach says, "You can't wear your hats at the barbecue." So, all the kids take their hats and throw them in a big pile. They fill their little faces with hot dogs and corn on the cob. At the conclusion of the evening, they all reach in at random, grab a hat and put it on their heads. What's the probability that 19 out of 20 of these kids get their correct hat?