Here's a puzzle.
Suppose I made you an offer. I have two check made out to you. One check is hidden in my left hand and one check is hidden in my right hand. One of the checks is for an arbitrary, positive, non-zero amount and the other check is for an amount either ten times larger than the other check or one-tenth the amount of the other check. The amounts have arbitrary precision (i.e., the amounts can contain fractions of a penny). It is random which hand contains the larger check.
Here's the deal.
You can pick the check in either hand. After you look at the check in the hand you picked (but you're not allowed to see the other check), you have to decide which check you want: the check that you've seen in the hand you've picked or the other check that you haven't seen. The game is then over and you walk away with your check.
Here's the puzzle.
Common sense would tell you that since the contents of my hands are random and unknown, it shouldn't matter what you do. Pick the check in either hand and be done with it, or pick either hand and then choose the check in the second hand - shouldn't matter.
Common sense would be wrong.
Let's say you've chosen a hand and find that the check is for $10. The other check is therefore either $100 or $1. The expected value is ($100 + $1) / 2 = $50.50 which is substantially better than the $10 you hold. So you should always go for the second hand after looking at the check in the first.
Which, of course, makes no sense since it's random.
What's the explanation?
The answer to this question is also the key to the secrets of the universe.