Just run the Monty Hall experiment and get it over with

May 5, 2009 | By | 15 Replies More

Back in the 60’s, Monty Hall was the host of a TV game show called “Let’s Make a Deal.”  I watched it often when I was a boy.  One of the specific games on the show involved offering a contestant to pick one out of three wrapped prizes.  Two of the prizes were valuable, but one was a worthless gag.  After the contestant chose one gift, Monty invariably removed one of the two wrapped gifts that the contestant did not take.  He then asked the contestant if he/she wanted to trade the box he/she originally chose in return for the other remaining gift.  Should the contestant stay put or should he/she switch?  My gut feeling says that there is nothing to gain by switching, but there are many experts who disagree with me.

Frankly, I’m tired of hearing about the Monty Hall problem.   Many mathematically-inclined experts insist that you should ALWAYS switch after Monte takes away one of the three hidden prizes. There’s all kinds of high end mathematics involved in many of these analysis (see here for instance).

The dispute gets really high-pitched sometimes, which is usually a clue that experts are claiming to be certain when they really don’t have any right to be.

What I’m wondering is this:  why don’t some social scientists simply gather empirical data in a lab?  Assign someone to be Monty and let college students play the roles of contestants.  Set the experimental parameters precisely (this needs to be done carefully because there is some question as to what, exactly Monty knows and does) and run the test over and over, until you’ve got LOTS of data.  Have some students always make the switch.  Have others never make the switch.  Then tally the results and tell the mathematicians that you have the real answer.

So that’s my thought: allow real-world trials tell the theoreticians the answer.  Then let’s move on, please.

Or has someone actually run the Monty Hall skit over and over in a lab yet and added up the results?  I haven’t seen it yet, if this has been done.

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Category: scientific method, Statistics

About the Author ()

Erich Vieth is an attorney focusing on consumer law litigation and appellate practice. He is also a working musician and a writer, having founded Dangerous Intersection in 2006. Erich lives in the Shaw Neighborhood of St. Louis, Missouri, where he lives half-time with his two extraordinary daughters.

Comments (15)

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  1. Jens says:

    Hi Eric. I actually did (write a program) and put the source on github (http://github.com/Jens-n/Monty-Hall-Problem/tree/master).

  2. It is a trivially easy experiment to do. Try it at home with a friend. (Make sure to do what you can to avoid "Monty" inadvertently giving away the position of the prize.)

    When I first encountered the problem, the mathematical explanations left me as unsatisfied as you are. So I exercised my (limited) computer programming skills to set up a simulation of the game. A million runs later, the stats were absolutely undeniable: you're twice as likely to get the prize if you switch. Period.

    I don't know if anyone has ever published an empirical test of the Monty Hall problem. It's such an easy thing to try that it probably isn't worth the bother of peer-reviewed publishing. If you'd like an official-looking writeup, though, just ask and I'll whip one together for you, including all of the computer code I used so you can replicate my results for yourself.

  3. TJ says:

    This is kind of a weird question. The results of science, or mathematics, are not always intuitive. That's why this puzzle comes up again and again.. the results seem counter intuitive, but the math is solid, and no one with a good understanding of probability would contest them. Discussions on the internet on this topic generate more heat than light, as they say, similar to the way debates about creationism vs evolution do, because people rely on their intuitions (or beliefs) rather than the math (or science).

    Assuming Monty Hall knows where the car (the usual example of the very valuable prize) is, and he must because he never accidentally reveals it, then you really should switch.

    As for the real world experiment, someone wrote a simulator to do what you suggest:

    http://github.com/Jens-n/Monty-Hall-Problem/tree/

    Under the assumption that Monty knows where the car is, the player who stays on the original door won 3286 out of 10000 times, the player who switched won 6618 out of 10000 times.. which is about what you'd expect.

    Part of the reasons our intuitions lead us astray is the small number of doors. Suppose there were 100 doors, 99 with goats behind them, 1 with a car. You pick door number 1. Monty then reveals the goats behind doors 2-99. You now can stay with door #1 or switch to door #100. Do you think it is more likely that you got it right the first time, or that it's behind the last door left after Monty eliminates 98 wrong answers?

    The results may be weird, but they are much less weird than the results of quantum physics.

  4. JD says:

    Sounds good, let's test it! You be Monty Hall and I'll be the contestant. Every time I choose the correct door you give me 1$ and every time I choose the wrong door I'll give you $1. Of course, we'll need to run a few thousand trials just to be sure of our results…

  5. Tony Coyle says:

    Erich

    The issue central to understanding the Monty Hall challenge is this.

    At the beginning of the game, you have zero knowledge of the outcome. You would therefore choose randomly, with a probability of 1/3 that you chose the 'correct' door.

    Monty, then opens one of the two remaining doors. (demonstrating a door that is NOT a prize – and is NEVER a prize.

    The question is – should you switch?

    This is where the probabilistic fallacy occurs that most people fall prey to… namely – there are now two doors, there is a car behind one of them – so I have a 50/50 chance of winning the car no matter what I do.

    Wrong.

    The probabilities do not change – what changes is your knowledge of the landscape.

    The write up in Wikipedia (per your link) is extremely clear, and refutes the 'fallacy' clearly, both graphically and using probability theory.

    You can play the game as often as you want. The rules dictate the probabilities of outcome. If Monty always opens a door, he is always constrained to show a goat. That is why your probabilities improve to 2/3 if you switch (since you know that he can never show a car) He eliminates one of the 'thirds' for you. Your start point is still at 1/3. The remaining door has a 2/3 probability of success, no matter which way you cut it.

  6. Mark says:

    In general, people just don't understand probabilities. Hence the popularity of lotteries and gambling. It's one of the worst forms of ignorance — worst in the sense that it results in frequent and unnecessary pain and suffering.

  7. Erika Price says:

    Erich, I'm just going to reiterate the notion already conveyed in the previous comments: a formal experiment is unnecessary because a simulation will easily do the trick. In fact, the Monty Hall problem itself is not even a question fit for social science, as it involves a simple switch/don't switch binary of behavior, with only two binary possible outcomes. So if what you are questioning is the mathematical accuracy of the Monty Hall problem, you need only set up a simulation of numerous attempts at both switching and not-switching. The online simulator provided by David allows you do to this.

    I've long had a post on the back burner that is based on this article, which postulates that the evidence of cognitive dissonance in chimps may actually be an artifact of the Monty Hall problem. Basically, scientists concluded that Monkeys appeared to prefer a certain color because of an arbitrary preference that was later justified by becoming a lasting preference, a la cognitive dissonance. What the scientists neglected was that not all colors had an equal chance of being selected- just like 'switch' option in the Monty Hall Problem.

    I think the case is pretty much closed on the Monty Hall Problem, unless you have a qualm with which I'm unfamiliar. I'm interested in the social sciences for a reason, after all- I don't like to work out the math myself.

  8. Erika Price says:

    Actually, here is an even better simulation: http://math.ucsd.edu/~crypto/cgi-bin/MontyKnows/m

    This is great because it actually is a test of the Monty Hall Problem, because it shows the outcomes of every person online who has accessed the site. There you can see that those who switched have a success rate of over 60%, compared to one less than 30% for non-switchers. Consider this an unofficial, unpublished large-n experiment.

  9. Tony Coyle says:

    Mark – I agree with you about gambling. I don't gamble*. Knowing that the odds are truly stacked against me makes such an activity less than fun.

    Unfortunately, gambling is heavily marketed as harmless and enjoyable entertainment. If one was unkind, one might say that the gambling industry preys on the weak and the poor – but that would be so unkind and I'm sure that is not true.

    I suppose it is simply another of those strange American dichotomies.

    *caveat: I do play the occasional game of poker with friends — but that is more of an exercise in applied probability, than gambling. My friends might be gambling, but I'm doing math. I rarely lose (and if I do, not by much), but I'd never be tempted play in a casino — those guys are demonstrably much better at 'on the fly' math than me!

  10. Deena Mills says:

    Hi Erich

    First off, just to clarify, the way you've initially presented the Monty Hall Problem in your post is inaccurate.

    1) There is always one prize and two duds

    2) It is always a dud that is revealed

    This is very important because the nature of the problem is completely changed if there are two prizes and one dud or if one unknown prize/dud is taken away at random (i.e. not a revealed dud).

    Your post brought back memories of the first time I encountered this problem. It was in a first year university mathematics course. We were learning about statistics and the instructor brought up this problem near the end of class and asked us whether it was better to stick with our original choice or switch.

    I thought – of course it doesn't matter, the chance of winning was 1/3 before a dud was revealed and now it would be 1/2. The instructor told us the answer to the problem – your chance of winning doubles if you switch – and breezed through an explanation. I gawped at my classmates sitting around me looking so gullible and trusting at the instructor, some sleepily nodding…and I thought "WTF!?!?!? What is this mathematical trickery?!?!? This instructor is so obviously wrong, why aren't any of you saying anything?? Have you all lost your minds?!?!?"

    I stuck up my hand and challenged the instructor. I asked her, with open incredulity, if I did this experiment 100 times – I would win 2/3 of the time if I always switched from the original choice and I would win 1/3 of the time if I always stuck with the original choice?

    The instructor said "yes" and the class ended with me sitting there looking baffled.

    I went home and told my hubby (Tim – your first commentor) about the lesson. I felt relieved that he didn't get it either. He suggested writing a program. Even after seeing the results of the program I still had a very hard time accepting them. The problem was sooooo counter-intuitive to me that I had to think about it for a long time and in many different ways before something finally clicked in my brain and I understood. If we hadn't done our own empirical test of this I don't think I would have ever been able to understand.

    Now here's an interesting tidbit. I'm going to ramble on about monkeys and Smarties now so you may want to stop reading here.

    I was listening to an old episode of the podcast the Skeptic's Guide to the Universe today (Episode #142) and the skeptical rogues were talking about the Monty Hall Problem and how it can sneak in and even dupe scientists who are trained in statistics. Basically, some economist wrote a paper demonstrating that 40 years of research in Choice Theory involving experiments with chimps and monkeys had fallen prey to the Monty Hall fallacy.

    The experiment is you offer a monkey a choice between a red or blue Smartie. Let's say monkey chooses red, so you then offer the monkey a choice between a blue or a green Smartie. 2/3 of the time the monkey will choose green.

    Choice Theory says that the reason for this is that in choosing red the first time the monkey was also rejecting blue and will therefore reject it again in order to validate the initial rejection. Hmmm, interesting theory.

    Mr Smartie Pants economist came along and said "ah, but wait…"

    Let's assume that each monkey has an order of colour preferences:

    for example first red, then green, then blue (rgb)

    there are six possible orders of preference overall (rbg, rgb, grb, gbr, brg, bgr) right?

    In the above example, the monkey's first choice (red over blue) narrows the list of possible preferences for that monkey from 6 to 3: rbg, rgb, or grb. Two of these have green above blue; one has green below blue. So, assuming all the possible preferences are equally likely, there is a 2/3 chance at this stage that the monkey will choose green over blue, and a 1/3 chance that it will choose blue over green.

    So this whole Choice Theory idea of the monkey rejecting the same colour a second time in order to validate the initial rejection is completely unsupported. The reason why 2/3 of the monkeys reject blue or rather twice as many monkeys reject blue is because there are twice as many orders of colour preference that allow for this option.

    Cool.

    – Deena

  11. Erich Vieth says:

    Thank you, everyone, for all of the good information and links. Maybe I'll now be able to get over my mind-block on this issue.

  12. Tony Coyle says:

    Erich

    Maybe a refresher in probability theory?

    And the best layman summary of 'why' the probabilities are as they are… from the NYT article linked earlier by Erika:

    This answer goes against our intuition that, with two unopened doors left, the odds are 50-50 that the car is behind one of them. But when you stick with Door 1, you’ll win only if your original choice was correct, which happens only 1 in 3 times on average. If you switch, you’ll win whenever your original choice was wrong, which happens 2 out of 3 times.

    Said that way – it is even intuitively right!

  13. Tony Coyle says:

    Erich – be thankful that you're never going to be as bad as Walter Wagner (he of the LHC lawsuit in Hawaii!)

    Thanks to this piece on the LHC on Comedy Central's Daily Show his idiocy is showcased. It's a beautiful moment.

  14. Erich Vieth says:

    Here's a youtube setting out the Monty Hall problem.

    I must confess that I didn't quite understand that parameters of the problem until after writing this post. Thanks to everyone writing in to set me straight.

    <object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/mhlc7peGlGg&hl=en&fs=1&"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/mhlc7peGlGg&hl=en&fs=1&&quot; type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344"></embed></object>

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