Alice takes two envelopes and puts some number of dollar bills in each one. These can be any positive whole numbers (we're putting realism aside), but they cannot be the same number.
Bob chooses one envelope and looks at its contents, but not those of the other envelope. Bob then decides which envelope to keep. Devise a strategy by which Bob has a greater than $1/2$ chance of keeping the envelope with more money. This must be true no matter what distinct numbers Alice chose.
(This is a classic that many of you are probably familiar with, but I didn't find it on the site. If you've seen it, I'd kindly request that you wait before answering to give others a chance.)