Are You Smarter than a Monkey?
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Monty Hall Game |
Recently, the New York Times John Tierney reported on a serious instance of the Monty Hall fallacy.
"The economist, M. Keith Chen, has challenged research into cognitive dissonance, including the 1956 experiment that first identified a remarkable ability of people to rationalize their choices.Dr. Chen says that choice rationalization could still turn out to be a real phenomenon, but he maintains that there's a fatal flaw in the classic 1956 experiment and hundreds of similar ones.
He says researchers have fallen for a version of what mathematicians call the Monty Hall Problem, in honor of the host of the old television show, "Let's Make a Deal."
Somewhat coincidentally, Diane Levin at the Mediation Channel, also wrote a short article about the Monty Hall problem, and she reference her article from 7P productions' article which started off asking "If you made the wrong choice, are you smart enough to change your mind?"
In this article, I want to point out the value of simple decision models and the scientific method -working from easy solutions to hard problems.
Here is the generic description of the Monty Hall problem from John Tierney.
"Here's how Monty's deal works, in the math problem, anyway. (On the real show it was a bit messier.) He shows you three closed doors, with a car behind one and a goat behind each of the others. If you open the one with the car, you win it. You start by picking a door, but before it's opened Monty will always open another door to reveal a goat. Then he'll let you open either remaining door.Suppose you start by picking Door 1, and Monty opens Door 3 to reveal a goat. Now what should you do? Stick with Door 1 or switch to Door 2?"
One argument, which is relatively attractive, for not switching points out that the odds that you were right initially can not have changed, so that there is no point in switching. The other argument, for switching, is more subtle and basically requires you to count the possible ways in which switching works and does not work. There is a good diagram of this counting technique here.
I want to focus on an easier solution, and one that is not limited to discovering the flaws in reasoning about Monty Hall.
It is clear that the problem doesn't have an obvious or easy answer. So let's turn it into an easy problem. We suppose that Monty Hall is lazy and only picks Door 2, when the prize is in fact behind your first choice Door 1. He never opens Door 3, if your first choice was correct.
What should you do if Monty Hall opens Door 3 to show a goat? Well, the prize has to be behind Door 1 and Door 2. But we know that Lazy Monty Hall never opens Door 3 if the prize is behind Door 1. Therefore, the prize must be behind Door 2 and switch guarantees you will win. (What should you do in the Lazy Monty Hall game if Monty Hall opens Door 2 is harder, but I have detailed the calculations here -basically, it doesn't matter if you switch or not.)
Suppose Monty Hall was just a little bit less lazy, and in only 1 out of 100 times opened Door 3, if the prize was behind Door 1. Then, while not a guarantee, it still makes sense to switch. We can imagine making Monty Hall progressively less lazy until he is indifferent between showing you Door 2 and Door 3. And is now Coin-Flipping Monty Hall.
It is harder to show that the Monty Hall game is just like the Lazy Monty Hall game, where Lazy has been replaced by Coin-Flipping. Coin Flipping Monty Hall chooses to reveal Door 2 or 3, when your first choice was right, based on a coin flip.
Now we have a new puzzle because John Tierney said: "If you switch, you'll win whenever your original choice was wrong, which happens 2 out of 3 times." And he invites you to play simulated game to prove his point. This is the accepted "right" answer. But in the Lazy Monty Hall problem, switching if shown Door 3 wins all the time, and switching if shown Door 2 is irrelevant.
This new puzzle dissolves quickly. All of the simulators running Monty Hall games, such as this one and this one, are all models of the Coin Flipping Monty Hall game. Only in the Coin Flipping Monty Hall game is John Tierney's mathematical point correct. Most mathematical commentators assume that we are playing the Coin Flipping Monty Hall game.
One commentator, at the New York Times, summarized my feelings exactly:
"I am getting a bit tired of the media presenting versions of the Monty Hall problem without yet highlighting the basis for the issues that caused so much furor over the problem to begin with.Like so many mathematical conundrums, the issue that causes people difficulties with the Monty Hall problem is that the assumptions specified in working out the solution are not made precise."
Which is why we need simple models to spell out exactly the assumptions in our models of thinking about the world.
