UPDATE FOR JUNE 27, 2011: This one won’t go away. Just in this week, check out Let’s Make a Probablistic Deal, which includes a podcast.
Below is a post from 2005 that discusses this very subject. Note that the Monty Hall app we created has since been taken offline, as it was done in ColdFusion and I didn’t bother to keep up with the program’s server requirements.
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Maybe you’ve heard of it. It’s been called the Monty Hall Scenario, the Monty Hall Problem, the Monty Hall Riddle, and the Monty Hall Puzzle. Persistent arguments over the years have transformed this initially harmless and entertaining logic exercise into a phenomenon known the world over as the Monty Hall Dilemma, the Monty Hall Dispute, the Monty Hall Debate, and the Monty Hall Controversy. Or my personal favorite, the Monty Hall War.
For those not familiar with it yet, it goes like this:
A TV host shows you three numbered doors. Behind one door is a brand new car. Behind the other two doors are goats. You get to pick a door, winning whatever is behind it. Regardless of the door you choose, the host, who knows where the car is, then opens one of the other two doors to reveal a goat, and invites you to switch your door for the remaining closed door if you so wish. Does switching increase your chances of winning the car?
For our international friends and those not thoroughly familiar with American pop-culture, this riddle is modeled after a classic game show from the ’60s and ’70s called “Let’s Make A Deal” with host Monty Hall.
OK, now on to the good stuff.
The correct answer to the question is yes. Trading your original door for the remaining closed door effectively doubles your odds of winning the car. Your original door has 1-2 odds (a 1 in 3 chance, or 33% probability), while the remaining door offers 2-1 odds (a 2 in 3 chance, or 67% probability). Many people initially believe the odds to be even, with both your original door and the remaining door each offering a 50% chance of winning the car, but this is inaccurate. When the correct answer is given to them, most people are astounded, and quite amused at the twist. But the truly stimulating conversations come not from the converts, but the skeptics, who simply refuse to believe the truth.
I came across this gem on The Science Channel during a show called “Understanding Odds”. I found it interesting, and worth passing along to friends who enjoy logic problems. However, what I found even more interesting was the emotional and heated debates I ran into when sharing this riddle with others. It’s fascinating how the human mind can be so locked into its own little box as to miss something that should be obvious to anybody with reasonable thinking skills. From a psychological point of view, I found it amazing how a member of the human race can become so emotional and defensive when it comes to proving something it truly believes in. This “thinking with the heart and not the brain” is a common first step towards ignorance that we see everywhere in society today.
While researching the history of this puzzle on the WWW, I learned that this debate has been going on for a while now between very respectable and competent thinking people. Mathematicians, physicists, doctors, and computer programmers alike have cursed, yelled, and even reverted to personal attacks when trying to prove that the odds are even between the two final doors. In their mind, switching doors changes nothing. In their mind, each door has a 50% chance of having the car. But sadly, they’re wrong. Even worse, their argument digs their ego into a hole it can’t get out of, getting deeper by the minute until the proverbial crow is swallowed, and that’s where things start getting emotional. They turn into the intellectual equivalent of a my dog immediately before her bath; she thinks I’ve trapped her and tries to resist, when in reality I’m only trying to help.
Self-professed statisticians have a particularly hard time accepting the truth of this Monty Hall Scenario, which is surprising, since one would assume they of all people would understand it. Their background in the science of statistics causes them to assume they’ve got the answer, based on what they’ve learned in statistics. It’s as if they’re prisoners to their own theoretical training, making a pride-driven error in equating the ability to calculate with the ability to reason. While the two are certainly related on several levels, they’re actually two very different things. Without a logical foundation, calculation is an aimless exercise. In other words, an answer is pointless if the question isn’t fully understood.
Fortunately there are plenty of resources on the WWW which offer perfectly adequate explanations to this problem, alleviating me of any rehashing chores. Here is a link to one of the more eloquent and well-rounded papers on the subject, with a few explanations including some geeky conditional probability mathematical equations using Bayes’ Theorem:
For more resources than you probably have time for, go Google it.
For laughs, we’ve also programmed a simulation in ColdFusion, which conveniently makes the semi-abstract concept quick and easy to comprehend right there in your web browser. Statistics show that two dozen tries or so is usually enough for percentages to stabilize and achieve an observable pattern. After about 24 tries, using the same switching strategy every time, the winning percentage will level off and the advantage should become clear to the user. For the more stubborn skeptics, we encourage you to play the game about 24 times while holding your original door every time, then clean the slate, and compare your winning percentage to that of 24 tries with a switching strategy.
For the technically minded, below are the basic rules the program uses to in order to simulate the Monty Hall Scenario:
A. The computer generates a random number. Numbers are limited to 1, 2, and 3. The computer defines this random number as the ‘winningNumber’.
B. The user is presented with three images of closed doors numbered 1, 2, and 3. Under the hood, the computer has already assigned each of the three images as a unique number. These numbers were limited to 1, 2, and 3, with each unique number corresponding to its associated closed door image.
C. The user selects one of the three closed door images. The computer stores the value (1, 2, or 3) of the image that the user selected, and defines this value as ‘firstSelection’.
D. At this point, the computer has (in memory) the ‘winningNumber’ (stored from Step A) and the user’s ‘firstSelection’ (stored from Step C). These two values may or may not be the same.
E. The computer now randomly selects a number (1, 2, or 3) that is NOT the ‘winningNumber’ and is NOT the ‘firstSelection’, and reveals it to the user. It will always be a goat.
F. The computer then presents the user with the image of the goat (the revealed door) and the two remaining closed door images. Note that the computer still has, in memory, the initial random winning number (‘winningNumber’) AND the number that the user initially selected (‘firstSelection’). Note that these two values may or may not be the same.
G. The remaining two doors will always be a losing number AND the ‘winningNumber’, where one and only one of these remaining values is equal to the value of ‘firstSelection’. *Again, note here that ‘winningNumber’ and ‘firstSelection’ may or may not be the same value.
H. The user is now presented with the opportunity to change her initial selected value ‘firstSelection’ to the remaining value OR keep it the way it is. The technical possibilites of ‘firstSelection’ being ‘winningNumber’ have remain unchanged since Step A (a 1/3 chance), while the chances of the remaining value being ‘winningNumber’ are 2/3.
I. After the user makes her second selection, the computer stores the value in memory and defines this value as ‘finalAnswer’.
J. The computer compares ‘winningNumber’ with ‘finalAnswer’. If the values are equal, the user wins. If not, the user loses.
*Please note that we are not suggesting that having a goat around the house would necessarily be a bad thing.
“If you program it, they might listen.” –Anonymous