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Tuesday, October 27, 2009

The Monty Hall Problem


PRETEND YOU ARE ON A GAME SHOW WITH MONTY HALL and he offers you the following scenario as described in an article from the Journal of Experimental Psychology.

You face three doors and behind one door is a car, while the other two hide goats. Your goal is to pick the door that hides the car. Here are the rules. First, the car and the goats were placed randomly behind the doors. Second, after you choose a door, the door remains closed for now. Third, Monty knows what is behind each door. Fourth, he has to open one of the two remaining doors. Fifth, the door he opens must have a goat behind it. Sixth, if both remaining doors have goats behind them, he chooses one randomly.


After Monty opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or switch to the last remaining door. Pretend you chose Door 1 and Monty opens Door 3 containing a goat. With only Doors 1 and 2 remaining-one of which contains a car-he asks you, "Do you want to switch to Door 2?"

From a probability standpoint, are you more likely to win the car by staying with your original choice of Door 1, switching to Door 2, or does it make any difference at all if you stay or switch? Before reading further, think of your answer then return to the next paragraph.

As you contemplated your answer, you may have reasoned that since one of the two remaining doors contains the car, you have a 50/50 chance of winning, so there is no need to switch. That may sound reasonable, but it is not correct. Presented with this three-door scenario, you should always switch, in fact, by switching, you have a 2/3 probability of picking the car.

Here's the explanation, according to Michael Shermer writing in the February 2009 issue of Scientific American.

At the beginning of the game you have a 1/3 chance of picking the car and a 2/3 chance of picking a goat. Switching doors is bad only if you initially chose the car, which happens only 1/3 of the time. Switching doors is good if you initially chose a goat, which happens 2/3 of the time. Thus, the probability of winning by switching is 2/3, or double the odds of not switching.

Over countless studies using this "Monty Hall" problem, the vast majority of participants think that staying and switching are equally good alternatives. So, if you are in that camp, you have lots of company.

For investors, the fact that the majority of people who take the "Monty Hall Challenge" get it wrong suggests that there may be times when the majority of investors are "wrong," too. At crucial turning points in the stock market, when there is evidence to support two opposite directions for the major averages, the majority of investors may "misread" the data (as in the Monty Hall Challenge) and draw a conclusion that subsequently turns out to be incorrect. While we will not always be "smarter" than the crowd, we do realize that, like the Monty Hall problem, the crowd is not always right. And because of our open mind, our willingness to think differently, we are constantly scanning for opportunities or turning points that may be overlooked by the crowd.

Sidebar: Are you still shaking your head about the answer to the "Monty Hall" problem? Here's another way to look at it from mathforum.org.

What if there were 1,000 doors? You would initially have a 1/1,000 chance of picking the correct door. If Monty opens 998 doors, all of them with goats behind them, the door that you chose first will still have a 1/1,000 chance of being the one that conceals the car, but the other remaining door will have a 999/1,000 probability of being the door that is concealing the car. Here switching sounds like a pretty good idea.

4 comments:

  1. I was just thinking this is prety good stuff!

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  2. It's only good if you pick the correct door...unless of course, you like goats!

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  3. Here's the psychological answer:

    This is experimental psychology not math analytics. The bottom line is that faced with a 50/50 decision, the majority would say that it doesn't matter if you switch doors. And from a psychological perspective, that is wrong. Why? Because in a majority of instances, the majority is always wrong. Thus, by switching doors, you have a much better chance of choosing the correct one. "Deep Thoughts"!

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  4. Robert,

    The reason the majority get it wrong is that the majority will stare at the two remaining doors while they decide - almost in an attempt to see through the doors. The correct place to stare is into Monty's eyes as he knows what lies behind each door. Boxers, karate gurus, swordsmen, poker players - they all watch their opponent's eyes, not his hands or cards (or closed doors).

    That's my theory and I'm sticking to it, even if it's nonsense.

    Oh, wait. Monty's got two eyes. Damn. That takes it back 50/50.

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