Here’s the most amazing thing I read recently (in Mar-Apr edition of American Scientist–probably the best science writing around–which attributed this puzzle to David Blackwell).
Suppose that you write down two numbers, each on a slip of paper. You shuffle them up and put them, face-down, on a table. I am free to flip over the first one. Now I say whether or not the first number is larger than the second number. If I am right, you give me $100. If I am wrong, I give you $100.
First question: is this a fair bet for you? You should try for a minute to prove that this is a fair bet (and may “succeed”). Informally, you might think the first number doesn’t give me any information about the second since you could write down anything, so it seems like there is no strategy I can use to, on average, do better than breaking even.
Ahh our dumb intuition. Here’s how I can make money on this bet. This is the remarkable part. I flip over the first slip of paper. I then choose a number at random (say, from a normal distribution). If the random number is larger than the one I flipped over, I tell you that the hidden number is also larger than the one I flipped over. If the random number is smaller than the one I flipped over, I tell you that the hidden number is also smaller than the one I flipped over.
How could this possibly make me money on an even bet? Well suppose that you wrote down numbers X and Y. When my random number is less than both X and Y, then I will win 50% of the time since I will turn over the smaller number first half the time, and the larger number first half the time. In both of these cases, I will tell you the hidden number is the smaller of the two numbers, and so will be right half the time. Similarly, when the random number is larger than both X and Y, I will be right half the time.
But, when the random number is between X and Y, I will will all the time. To see this, suppose that X < Y. When I turn over X, my random number is larger than it, so I say that the unseen number, Y, is larger than the seen number X, and I am right. When I turn over Y first, my random number is smaller than it, so I say that the unseen number, X, is smaller than the seen number Y, and I am right.
So when I randomly choose a number between X and Y, I win all the time. Thus, if I sample from a distribution which assigns nonzero probability to every range, I will on average win more than half the time. Crazy, huh?