Is there no end to circumstances that ask whether we should switch?
This one might take some thought.
Your rich uncle places a sum of money into an envelope, twice that amount into a second envelope, places them both into a black bag and invites you to draw one out of the bag. You open the envelope that you draw and find $1000. You now know the envelope in the bag contains either $500 or $2000. Your uncle gives you the option to take the other envelope in exchange for the one you drew.
Calculate the expected gain that comes from switching.