Dear Brian Hayes,
thank you for this helpful review as well as for your highly interesting regular column Comp Sci.
A lot has been written about the Monty Hall affair, however I haven't yet heard of a convincing solution that doesn’t rely on statistical reasoning but logical ones. Here is my attempt:
First it is to be realized that—according to the puzzle's concept—there are restrictions concerning the door(s) Monty Hall is allowed to open:
Neither does he open the door that hides the prize nor the one chosen by the subject.
With this in mind, three situations are to be considered:
1) If the subject initially chooses the door that hides the prize, then of course switching would not be advisable.
2), 3) If the subject initially chooses one of the two other doors, then switching the doors would be advisable. (These are the two situations in which Monty Hall has no choice of which door he opens.)
Consequently, in two of three situations switching is successful.
With best regards
posted by Helmut Gluender
July 2, 2008 @ 1:30 PM
Brian Hayes makes use of the "Lets Make a Deal" puzzle to make the point that computer simulation is a good way to sort out controversies such as the one Marilyn vos Savant created by asserting there is an advantage to switching one's initial choice of three doors, one of which has a prize behind it, once one door has been opened that does not reveal the prize. Actually, he inadvertently makes a quite different point which is that Monte Carlo simulations are only as good as the structure of the model used to run them. He gets a surprise free outcome by structuring the probabilities a la vos Savant as 1/3 for the original choice and 2/3 for switching to the other remaining door. Run that case 100,000 times, and yes, you'll get the vos Savant answer that it makes sense to switch choices from the original door. That doesn't make it the correct answer.
The initial statement of the problem as three doors with a prize randomly placed behind them does have a 1/3 probability of being the right choice prior to the opening of a door that doesn't contain a prize. Once door is opened without the prize, the problem changes to two doors each of which is equally likely to have the prize behind it. There is no advantage to switching doors. If a second door is opened without a prize behind it, one doesn't need a computer simulation to know the probability of it being behind the final door.
Monte Carlo simulations are most useful when no explicit analytical structure will give a precise answer or there is doubt about data accuracy. Usually there are many random variables that have different frequency distributions that are developed empirically. Some of the variables are correlated with one another, others are entirely independent. In these situations the model generates a distribution of outcomes which can actually be of more predictive value than a single exact number based on static assumptions.
Chapel Hill, NC 27514
posted by Robert Sampsell
July 15, 2008 @ 3:01 PM
June 6, 2008
To: Brian Hayes
c/o: American Scientist
I read with a great deal of interest your column in the American Scientist, July-August 2008, entitled Programs and Probabilities, in which you discuss the Monty Hall affair. This was discussed in the New York Times by John Tierney on April 8, 2008 and at that time I wondered why such a simple problem could generate so much controversy.
Your Monte Carlo results suggest that switching doors doubles your chances of winning but does not really explain why that is happening. The probabilistic explanation offered by "the other side" in your article is correct but does not explain in simple terms why that result occurs.
What if one wanted to explain what is happening to a bright high school sophomore who has never studied probability? There is a direct approach which is similar to what happens in our thinking in a chess game: if I do this then this happens but if I do this other thing then that could happen. And the problem here is much simpler than any real chess game.
Here's the direct approach:
Let's assume without any loss of generality that my initial choice is always door 1. There are three different cases to consider: the prize itself is behind door 1 or it is behind door 2 or it is behind door 3.
Let's now assume the prize is behind door 1. Monty Hall opens either door 2 or door 3. In either case I have two choices: I can stay with my original choice, door 1, or I can switch to the door that is still closed (to door 2 if Monty opens door 3 or to door 3 if Monty opens door 2). If I don't switch I win the prize and if I switch I lose.
Now assume the prize is behind door 2. Monty opens door 3 and shows me that it is empty. If I don't switch (stay with door 1) I lose but if I switch to door 2 I win.
Finally assume the prize is behind door 3. Monty opens door 2 and shows me that it is empty. If I don't switch (stay with door 1) I lose but if I switch to door 3 I win.
Thus if I don't switch I win only when the prize is behind the door of my original choice (door 1) but if I switch I win in two cases, when the prize is behind door 2 or is behind door 3.
The switching strategy does twice as well as the "don't switch" strategy.
Of course this is closely related to the probabilistic approach but if the problem is approached in this direct way I don't think there would have been any controversy.
One more thought: sticking with the original choice neglects the additional information given to us by Monty when he opens a door. The additional information given to us by Monty when he opens door 3 is that he didn't open door 2. The additional information given to us by Monty when he opens door 2 is that he didn't open door 3. The switching strategy takes advantage of this additional information and may help the reader understand that there is no paradox involved here.
Los Angeles, CA
posted by Ethan Aronoff
July 15, 2008 @ 3:41 PM
In response to Brian Hayes’ postscript “Monty Hall Redux” (Sept-Oct 2008):
I had the wrong Monty Hall answer at first until I read about the computer simulation and decided to rethink the problem, which then prompted me to develop a new understanding. When confronted with an anomaly like conflicting results from a computer model, we all respond differently (locating the error in self vs. other/other's model) because even people who share a belief in logic as a means of solving problems have widely varying degrees of faith in their own powers of reason. Dissent is healthy but digging in one’s heels in the face of compelling evidence hinders progress, and the line between the two isn’t always clear. (Forget Monty Hall, how do you persuade others that humility is a virtue?)
Rather than continuing a fruitless debate or acknowledging dissent and moving on without resolution, a third option is to try translating the problem into different terms. In this case we can use the language of human behavior, recognizing that Monty Hall is actively responding to the player’s initial choice. To make it even clearer we can imagine that Monty Hall wants the player to lose; he’s greedy and wants to keep the prize.
Consider the only two possibilities: the player’s initial door selection either has or does not have the prize behind it. As a player, you don’t know if you’ve selected the door with the prize. But Monty Hall does “know” and acts accordingly. (Even if this knowledge isn’t explicitly stated, it’s implicit in Monty Hall’s behavior of opening only unchosen doors, coupled with his knowledge of where the prize is.)
Case 1 - If your chosen door doesn’t have the prize behind it (probability of 2/3), then Monty Hall has no choice. He must open the lone door that was neither selected by you nor has the prize, and hope that you don’t switch. Because in this case, switching will always give you the prize.
Case 2 - If your chosen door does have the prize (probability of 1/3), then Monty Hall has a choice; he may open either of the doors you didn’t select. But of course it doesn’t matter which door he opens. He’s just hoping you’ll switch, because if you switch you’ll never get the prize.
Since you don’t know whether the door you chose has the prize behind it (determining whether you’re operating in Case 1 or Case 2), you should switch since it’s twice as likely that you’re in Case 1. Twice as likely that Monty Hall had no choice when he opened the door he did. The “always” and “never” consequences to switching make it mathematically simple - switching gives you the prize 2/3 of the time.
posted by Barbara Blossom
August 9, 2008 @ 3:42 PM
Well, for the Monty Hall problem the argument that convinced me right away was this: Suppose there were 1000 doors and one prize. You chose one door, and Monty opens 998 other doors. In this situation, everybody would switch to the door Monty didn't open, although this setup is as symmetric as the 3-door case. From here it is much easier to convince people that the advantage of switching persists down
to 3 doors.
This argument follows the general strategy of arguing in
extremes. Another strategy (favored by physicists) is to simplify things
as much as possible (but not more!), sometimes even to the absurd.
The general question raised in "Monty Hall redux" is very interesting. Even in a world of
mathematical proofs and established experimental results, convincing people is an issue of its own. Sometimes even false arguments for a right statement are more convincing than all correct arguments.
posted by Stephan Mertens
August 31, 2008 @ 10:50 AM