pick something out of a hat—select smth. completely at random:
- When asked how they chose Juneau to be the state capital, she said, “I don’t know, I think they picked it out of a hat.”
pull something out of a hat—produce smth. unexpected as if by magic:
- If you want to help Thomas, you’ve got to argue a lot of silly problems that he pulls out of a hat.
Note: The expression does not correlate in meaning with the phrase pull off a hat-trick—(usually of sports) have three successes in a row:
- This thoroughbred is ridden by Peslier, who has already pulled off a hat-trick of three consecutive Prix de l’Arc de Triomphe successes.
See also: have something under one’s hat / keep something under one’s hat.