The Good Host Vs. The Carnival Shill

A recurring theme in this series is the way in which seemingly separate random events can become intertwined such that the probabilities are changed in unexpected ways. The classic modern example of this is the Monty Hall paradox, which we have discussed before. However, this type of phenomena is not new. Certainly, carnival hucksters have understood the principle for generations because it is the basis of several types of standard betting rip-offs.

To show better how this changing of probabilities happens, I will present several examples of the next few days. In some of these, the probabilities of occurrence will be changed by the conditions as in the Monty Hall paradox. In others, it might seem as though the probabilities have been changed, but they have not. So in these examples, your intuition might be correct or not. However, you cannot assume that the example contains a surprising paradox. It might be the moral equivalent of a medical placebo.

To get us started, today’s example has a more noble heritage than the carnival tricks. I believe it originates with Lewis Carroll’s pillow puzzles, which he probably wrote in his bed to take his mind off of other things before going to sleep. Like many other charming puzzles, I found it both in the original and in Martin Gardener’s books. What follows is my own re-telling.

The puzzle is elegantly simple. Suppose you are at a dinner party, and your host shows you a cloth bag of fine black velvet, and lets you feel it. Nearby is a table with a go board and two vases with go stones in it. They are not the slate and shell kind. They are plastic. Both colors weigh and feel the same. The only difference is that half are white and half black. As you feel the bag, you detect what is obviously a go stone in the bag, but you cannot tell whether it is a white or black stone. Your host reaches into the go stone vases and withdraws a single white stone. He puts it in the bag and shakes it thoroughly.

At his invitation, you reach into the bag and withdraw a single stone. It is white. You cannot tell from the stones’ vases if a stone is missing from either, and you did not glimpse inside the bag to see the color of the remaining stone.

Your host now suggests a simple wager. Whoever wins pays for dinner. The host bets the remaining stone is white, and, being a good host, suggests he is giving you the advantage since you have already pulled a white stone out. Therefore the odds favor the second stone being black.

Is your host a good host or an apprentice carnival shill?

Answer to follow…

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.