The Full Monty (Hall)
The Monty Hall problem is a terrific little example of how intuition, especially the intuition of those trained in statistics, can go horribly wrong. It appears from recent calculations of the economist, M. Keith Chen that some of the most famous experiments in psychology may have fallen victim to the same logical fallacy.
First, a reminder of the Monty Hall problem. You are on a game show and are offered one of three doors, one with a Mercedes Benz behind it, the other two with a goat. You choose one at random. Your probability of being correct is 1/3. No argument there. Then the game show host, Monty Hall, opens one of the other two doors and shows you a goat. You are now asked if you want to change your door? Should you change or not?
Almost everyone I have ever met, especially if they have done a stats course, says: “It doesn’t matter. You have a 50% chance of being correct either way”. This is incorrect. Your chance of winning is still 1/3! It cannot have changed just by seeing another door. Therefore, if you change your choice, the probability of you winning must be 2/3. You double your chance of winning by swapping. Monty showing you the goat gives you important information, which you can exploit. For more discussion of the Monty Hall problem go to Wiki (where else?). And here is Video (thanks to Ken Russell).
What has this got to do with psychology experiments? Well, psychologists perform lots of experiments on human and animal subjects where they have to make a choice, after being given some further information and stimulus. One famous experiment involves asking people to rank a list of wedding gifts. They are then given a choice between the two items, say A and B, they have listed as equally attractive. Some time later they are asked to rank the gifts again. It turns out that they rank the item that they chose consistently higher. For instance, if they chose B, then they now rank B as higher than A.
The theory explaining this is that you are dealing with a clash of conflicting thoughts – known as a cognitive dissonance. The conflict is set up by the fact that you first thought A was just as good as B. But then you didn’t choose it. So you banish the original notion and say it was worse all along. I am sure Shakespeare must have had a more poetic terminology for this human failing.
Another more recent experiment was performed on Capuchin monkeys, who were choosing between different coloured M&M’s. Strength of preference was measured by recording how quickly and how often different colours were chosen. After identifying three colours preferred about equally by a monkey — say, red, blue and green — the researchers gave the monkey a choice between two of them, say red and blue.
If the monkey chose, say, red over blue, it was next given a choice between blue and green. Nearly two-thirds of the time it rejected blue in favour of green.
Ring any bells? How about if the three M&M’s were not exactly equally ranked, only very close? This sounds very plausible. So when the monkey chose the red, he did so because it was preferred to blue. Monty Hall showing you the goat reveals to you more information about the relative odds. In the same way, the monkey’s choice of red over blue gives you more information which should change your assessment of his relative preferences. That changes the odds.
If you go through the permutations (see Dr. Chan’s paper here) it turns out that supposing the monkey favours red over blue by a tiny amount, the chance he originally preferred green over blue is 2/3. So the result is explained without assigning to the monkey the same tendency for post hoc rationalisation that we higher primates suffer from.
There is some agreement in the profession that Dr. Chan’s criticism of the Capuchin experiment reveals a real flaw but disagreement in the profession about how commonly this error has been made.
(Thanks to my colleague Kwanghui Lim for alerting me to this item)
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May 2nd, 2008 at 7:36 am
“It doesn’t matter. You have a 50% chance of being correct either way”. This is incorrect. Your chance of winning is still 1/3! It cannot have changed just by seeing another door. Therefore, if you change your choice, the probability of you winning must be 2/3. You double your chance of winning by swapping. Monty showing you the goat gives you important information, which you can exploit.
this is actually incorrect.
Or more exactly, it assumes that Monty only opens a door with a goat behind it and that he always does this. Other scenarios:
A. Monty opens a random door:
1/3 you chose the winning door, Monty randomly opens a remaining door with a goat behind it, you switch and lose;
1/3 you chose a losing door, Monty randomly opens the remaining door with a goat behind it, you switch and win;
1/3 you chose a losing door, Monty randomly opens the remaining door with the car behind it and you lose.
So you win 1/3 of the time if you switch (or 1/3 if you stick). Note, even conditional on Monty having opened a goat door, you win only 1/2 of the time.
B. Monty only opens the goat door when you’ve chosen the car in hopes of making you switch. You switch, you always lose. Unless you know he acts that way, in which case you would always stick and always win.
B`. Creating a scenario where Monty has to mix up his actions in step 2 to keep you guessing.
So there are three possible motivations for Monty’s behavior. Either he’s acting in a way beneficial to you, a way meant to deceive you or a random way. Why, in the absence of prior information, we should assume he’s acting in our benefit is beyond me. In the “benign” world where both you and he are acting randomly, you still only win 1/3 of the time and it doesn’t matter if you stick or switch.
Beyond that, even assuming he’s acting to our benefit, your explanation is not quite correct. Monty has given us no new information. We always knew that at least one of the doors we didn’t choose had a goat behind it. It’s the fact that it’s not new information that tells us to switch — our odds of having chosen the correct door to begin with are unchanged. It’s the illusion that this is new information that fools most people (including me when I first saw the problem).
May 2nd, 2008 at 7:43 am
Sorry, a bit unclear. Monty has given us a smidgen of new info in the beneficial scenario. But really what he’s done is equivalent to giving us the chance to switch the one door we chose for the two other doors — which we would do even knowing that one of them has a goat.
May 2nd, 2008 at 9:14 am
There is an implication in the original problem that Monty always opens a door no matter what you choose, and that door always has a goat behind it. This rules out the scenarios that Walt raised.
The key behind this problem is as Walt says that we always knew there was one door we didn’t choose that was a goat, so the problem reduces to the question “Were you right with your initial guess, or not?” unless we can extract some additional information from Monty’s behaviour.
A complication arises however if we know of a pattern to the way that Monty chooses a door to open when you have chosen correctly, scenario 1 above. If Monty always opens the second door, for example, unless it contains the car, then we know that if he opens the third door there is probability 1 that the second door contains the car. This idea was summarised in an American Statistician paper.
I’m not convinced by Chris’ claim that studying statistics makes one especially susceptible to getting the Monty Hall question wrong. It is true that many statistically trained people don’t get this one right first up and need some convincing, but the same would be true of people who haven’t studied any statistics too. For example, consider the subtle point above about the true probability depending on Monty’s behaviour in scenario 1. This point was made by a bunch of statisticians in a statistics journal wasn’t it, not by academics with little statistical training. Of course in fine academic tradition they did have to think about the problem for a while, and only got around to publishing their results about 15 years after the original program stopped running!
May 2nd, 2008 at 10:01 am
David, I agree that those trained in statistics are ultimately better equipped to analyse the Monty Hall problem that those that are not. I am not bagging my own profession here! But I do think that if a quick answer is demanded then you and I are more likely to get it wrong than a lay person. The reason is because there are lots of other similar sounding problems where we know the result is a coin toss but lay people think there is a pattern. So our antennae our out for this kind of thing. .
Walt, Thanks for the interesting comments. You describe slightly different games to the standard Monty Hall game. I think the point of the Monty Hall example is that it is anti-intuitive (for some), not that it really models the Monty Hall game. Anyway, I contend that when Monty opens the door he does give us new information - a statistician would call it data. The new information is the particular door that has a goat. He also gives us a chance to act on this new information. The chance to switch doors would be worth nothing without the new information.
May 2nd, 2008 at 2:31 pm
The Monty Hall problem, and the discussion above, emphasize the importance of precise description of the ‘design’. Really, this should be done algorithmically, to remove any doubt. One should also code up a run of the experiment, with suitably configured electronic monkeys (the sort that sit inside all computers) making the choices, a good check that the experimenter has not stuffed up. One can try out possible strategies that the monkeys might follow.
All this translates well, with problems like this that are easy to describe to a secondary school classroom; those whom the probability calculation lose should be convinced by the simulation. It is a good way to train the students’ intuition.
There’s a related problem,called the Serbelloni problem on the website http://www.gnxp.com/MT2/archives/004048.html
“… it nearly wrecked a conference on theoretical biology at the Villa Serbelloni in the summer of 1966″. It is described in “Maynard Smith’s Mathematical Ideas in Biology (1968)”.
Translating it into the game show context, the contestant has no opportunity to switch. However, he/she is kept waiting till after an interval to know whether they’ve won the Mercedes Benz. In the meantime, he/she cajoles the host to name one of the two doors that does not have a car behind it. The host decides this provides no useful information, and informs them that door 2 does not have the Mercedes Benz behind it. Are the chances of winning the Mercedes Benz now 1/2, or still 1/3?
May 2nd, 2008 at 4:37 pm
… I would say that the probability is still 1/3.
May 2nd, 2008 at 7:07 pm
Personally I have no idea why anyone would prefer a car to a goat, but perhaps that is by the by.
But I do whole-heartedly agree with John Maindonald that one should CODE IT UP. And then simulate. (there used to be a language around called SIMULA or somesuch that would make this easy, but one can do it in any transparent procedural language).
English is NOT a good way of explicating the (sometimes hidden) assumptions and encoding their impacts. Hence we have a plethora of Marilyn Vos Savant type problems about which people passionately disagree, purely because the assumptions and the processes are not explicated. Convert the problem to code, in which case the assumptions are pretty explicit, and the ambiguities and disagreements disappear.
So, imho, statisticians and just about everyone else, should abandon verbal/semantic reasoning and CODE IT UP. (but not in Excel, please)
I wrote on some somewhat connected issues in http://dsanalytics.com/dsblog/why-simulation-is-better-than-statistics_80
and there are some links there of interest, notably
http://www.johnkay.com/in_action/458
June 12th, 2008 at 4:37 pm
John & John,
You are not alone in thinking that simulating the problem gives insight. They have been discussing Monty Hall at allstat this week. One of their readers, Fen Scott, wrote about the discussion thus:
…from yesterday’s discussion relating to the Goat Problem (or Monty Hall Problem). Having only come across this problem yesterday, I could not see past the “Intuitive” answer until an email I received from one of the list members (thanks Richard G) recommended a Monty Carlo study. A quick Excel spreadsheet later then convinced me of the answer, and also helped me to get my head around the theory behind this.