Probability Question: Choosing the Right Door
In this probability question, you have to choose one door with a prize. What do you do when the game show host eliminates an option?
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Choosing The Prize Behind The Right Door
(Published in Directory : Mathematics : Probability and Statistics)
There's an old probability question which makes a great discussion and experiment topic for a high school math class. The question centers around a game show, in which the game show host asks you to pick a prize behind one of three doors.
After he asks you to pick, the game show host doesn't open the door you picked. Instead, he opens a different door...one which he already knows is empty. Now, having eliminated one of the choices, he asks you, "Do you want to keep your guess? Or change it?"
Typically, people will answer, "There's a 1 in 2 chance that I'll get it right, so it doesn't make any difference if I change my guess or not, the probability is 1/2 either way!"
And the typical answer is not correct. Surprisingly, your odds are better if you change your guess. Why? Because the odds that you guessed incorrectly in the first place are 2 in 3, or 2/3. The game show's action of opening a door does not change that probability. Which means the odds are better if you switch doors. In fact, 2 times out of 3, you'll be better off switching.
If your students don't believe it, there are two ways you can help them see this.
Experiment
Have the class break up into pairs. In each pair, one person is the game show host. This student will pick a number (one, two, or three), but not tell the other student. The other student makes a guess, and the first student then removes one of the other numbers.
Have the first student always keep his guess the same, and count up how many times he wins. After they have done this many times, switch it around, and have the student always change his guess.
The students can now report back that they do win more frequently by changing their guesses.
Generalize
My favorite way of looking at this problem, which I found here: The Problem Site's Treasure Hunt, is to consider what would happen if there were a million doors. So the player makes his choice, and obviously his odds of having the right door are incredibly small. But now the game show host opens all but one of the remaining doors.
All of a sudden, students see the problem in a whole new light, when they realize that the odds strongly favor the one door the game show host didn't open.
About the Author
Name: Humphrey Nowlin
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Selected Member CommentsBelow you will find selected member comments about this article. To view all the user comments, please click here: Member Comments Page
| MrWallace wrote on Dec 3, 2007 |
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I couldn't resist the temptation to respond to Monty.
His first and third explanations imply the existence of a time traveling machine which allows us to alter the probability of a current event based on future actions. As far as I know, such a machine doesn't exist.
As for his second argument, he's making the rather absurd assumption that he can change the conditions of a problem without altering the outcome, which is TOTALLY bizarre!
And his last argument...I have no idea what he's even talking about - maybe someone else can make sense of it?
| | montyhall wrote on Dec 10, 2007 |
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People, people,
Let me start with the response by Mr Wallace to my first thought exercise: I am not postulating any sort of time machine, simply examining the illogic of the original case.
In my second point, I do not suggest changing the conditions, I simply accept the premise of the original question as posed on the website: The game player gets shown an empty door, there are two doors left, there is a 50-50 chance that the prize is behind one of the remaining doors.
Finally, the crux of the stupidity is revealed in Mr Wallace's third comment (no offence meant to Mr Wallace since others have fallen into the same trap). He indicates that he does not understand my third point. My third point is in fact based on the existing, well proven (not by me) actual mathematics of the problem. I did not present it first based on my assumption (proven correct) that people did not wish to understand the actual math involved. If people do not want to believe the actual mathementics involved then there is not any point in pursuing this further.
Permit me to go back to one more attempt to expose the illogic of the original answer. In the answer it is stated that the original 2/3 probability that you are wrong in your choice DOES NOT CHANGE when one empty door is revealed. Let us pursue this to the next step: Assume the host now opens another empty door and only one is left. According to the logic above, the probabilities do not change; therefore, even if there is only one door left and that door MUST have the prize. You people must (according to your own logic) still insist that there is only a 1/3 chance that the prize is actually behind that door. (I wonder where the prize goes two-thirds of the time?)
Oh my!!!
Monty Hall
| | Douglas Twitchell wrote on Dec 10, 2007 |
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While MrWallace's comments are a bit tongue-in-cheek, he is essentially correct.
His reference to a time machine comes from here: "Both of these bizarre answers are based on the incorrect logic that the probability of my original pick does not change when I am shown an empty door." Monty has stated that the probability of a past action changes based on a present event.
And MrWallace's observation about changing conditions of the problem is also correct. Monty, you have made the faulty assumption that the game show host's choice is not affected by my choice, so yes, you did change the conditions of the problem.
Regarding your final argument in the first post, to be honest, those of us who have actually done the simulation know your answer is wrong, which I suspect is what makes MrWallace not eager to try to figure out what you're talking about. I also was not interested in trying to decipher your explanation. Using phrases like "well known" and "well proven" does not exempt you from the responsibility of writing clear and accurate mathematical proofs if you want people to believe them.
Regarding your last argument in the second post, your problem is that you have assigned to Monty an action which can only be accomplished 2/3 of the time; if my initial guess is incorrect, he CANNOT open two empty doors.
Now, I have two requests to make. First, on both sides of this discussion, please maintain respectful courtesy and avoid sarcasm.
Second, Monty, I would request that before you post more discussion on this, you actually perform the simulation, and report to us your results.
Thank you,
Douglas Twitchell
AFE Administrator
-edited by Douglas Twitchell on Dec 10, 2007
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