Monty Hall Problem

by Brian on July 20, 2004

Via “Justin Leiber”:, here’s “a playable version”: of the “Monty Hall Problem”: It’s simultaneously a lesson in decision theory and in the perils of small sample sizes – my first two plays I lost the car by switching.



Martey 07.20.04 at 2:32 pm

Your link makes the decision for us! You should change it to


Motoko Kusanagi 07.20.04 at 2:54 pm

Here’s a variation, by Lewis Carrol

“A bag contains a counter, known to be either white or black. A white counter is put in, the bag is shaken, and a counter is drawn out, which proves to be white. What is now the chance of drawing a white counter?”


Brian Weatherson 07.20.04 at 3:32 pm

Thanks Martey. It’s fixed now.


Keith M Ellis 07.20.04 at 3:38 pm

Oh, yeah, you would post about the MHP only two months after I took down my nearly ten-year-old MHP site (which was probably the most famous and oft-referenced MHP site). Dammit. I might’ve merited a mention instead of being merely a lowly commenter! :)

Motoko, I’m not sure why you think that’s a variation on the MHP.

Over the years, I’ve exchanged many, many emails from folks who’ve been sure that it doesn’t matter if one switches or stays. This, after my page included journal citations, a warning that the intuitive solution isn’t correct, and an exhortation for the skeptical to code a simulation or reenact a series of trials with a friend. But, nope, quite a few are sure that it’s 50-50 and I must be clearly some nut who fails to understand elementary probability.

The MHP has held long fascination for me, and discussing it with such people has been rewarding. I’ve tried a varity of different pedagogical approaches to explaining the problem and along the way developed a keener sense of how people misunderstand it. And that last bit is why it fascinates me. The misunderstanding of the MHP is, in my opinion, exemplary of a very common but interesting variety of human miscomprehension.

I moved to my own personal domain from my old ISP, and that’s the practical reason the page came down. Why I left it down has everything to do with the sort of planning/hand-wringing/OCD type stuff that comes from an “opportunity” to recreate and improve upon an old but relatively important project. The result, of course, is that I’ve done nothing at all. I’ve thought a great deal about various simulations, either server-side and cumulative, like this one, or some java stuff. But it won’t convince everyone. One person with whom I exchanged several emails kept insisting that any such simulation didn’t prove anything because it would rely upon pseudo-random numbers. That’s representative of how a little but not enough knowledge (or understanding) is quite dangerous and this is often the case with the MHP.


eudoxis 07.20.04 at 4:03 pm

A similar process, which I haven’t seen mentioned elsewhere, applies to the situation with Iraq. Once the Bush administration has made a serious push for war, and has been denied a green light by other world players (a door with a goat has been opened), the effective impact of inspections alone has changed.


Morat 07.20.04 at 4:08 pm

I once spent three weeks arguing the Monty Hall problem with a friend. He was a smart guy, but he simply refused to accept the math.

I’d show him the math (it’s fairly easy, if you’ve got basic probability down) and he’d swear it had to be wrong. I coded up a simulation and ran it through ten thousand trials or more, and he swore the results had to be wrong and that I’d made an error somewhere (I shared the source code, but nada).

I pointed to a million web sites, explained it over and over, fricking offered to drive the [i]six hours[/i] to his home with and play a modified version until he was satisified.

I even used the Extended (or Insane) Monty Hall Varient (Say there are a million doors, behind one of which is a car. You choose one. Monty throws open all but one of the other doors, showing them to be empty. Do you stay or switch?) which I didn’t think anyone could argue with….

Nothing. Zero. Zilch. Nada. It was more frustrating than arguing with Young Earthers.


Keith M Ellis 07.20.04 at 4:16 pm

Yes, it’s very interesting, isn’t it? The (then, I suppose) science writer for one of the Houston daily newspapers was a skeptical correspondent once. Nope, he was certain that switching couldn’t possibly make a difference. I asked him to consult some mathematicians at one or more of the Houston universities. “It’s not important enough to spend that much time on it”, he said. He had been thinking about writing about it.


harry 07.20.04 at 5:51 pm

There’s a lovely explanation of it in *The Curious Incident of the Dog Who Died*. I suspect that the problem will become much mroe well-known as result of it being a best-seller (and a gripping one too) so Keith, get your site back up. And morat, give the book to your friend.


Leonard Richardson 07.20.04 at 6:52 pm

I once wrote a simulator that can play thousands of rounds of the game to empirically demonstrate the probability split:


Sharon 07.20.04 at 7:34 pm

It’s a sign of my statistical illiteracy that it took me hours to understand the maths of this one. Crank, creak… aha! So I got there in the end. (Although I’m still not sure that I could turn round and explain it to anyone else.) And there are the statistics to prove the theory… Must send this to the big-brained friend who patiently tried to explain it to me in the first place!


Keith M Ellis 07.20.04 at 7:45 pm

Here is my MHP page, transposed to my new domain. Please keep in mind that I wrote this ten years ago, and have hardly changed anything since then. (This feeling that it’s now inadequate is why I’ve wanted to just redo the whole thing from scratch. You know how it is.) And, um, this is from an archive from 2001 I scrounged up—I think there are some uncorrected typos in there.


Tom Runnacles 07.20.04 at 9:20 pm

I had to write a damn simulation of the problem to prove to myself that switching made a difference.

I think I can follow the mathematical argument, but it still seems a bit like Witchcraft to me. Ho hum.

A sidenote: I hadn’t realised that Marilyn Vos Savant was a real person – as a Brit, I’d only come across her as a walk-on character in a Dilbert cartoon.


WillieStyle 07.20.04 at 11:29 pm

Neat little problem. It didn’t take very long for I and my co-workers to be convinced that switching worked better.

As a side note, does Marylin Vos Savant do anything other than right that column for Parade?
It seems like such a waste of IQ.


Keith M Ellis 07.20.04 at 11:54 pm

It’s hard to believe she’s a real person, but a variety of sources insist she is.


bondra 07.21.04 at 2:08 am

For what it’s worth, Keith, I (a decidedly non-mathematical sort) found your old site’s explanation of the problem to be a good bit clearer than the one initially linked — particularly the notion that the remaining, unchosen box “inherits” the probability associated with the box that Monty opens. I actually do think I understand it.


Motoko Kusanagi 07.21.04 at 10:24 am

Keith: I don’t know what I was thinking when I wrote “variation”…


Michael Greinecker 07.21.04 at 6:46 pm

There’s a German book, Das Ziegenproblem by Gero von Randow which is all about the problem, giving many intuitions, proofs and teaches some very basic probability and decision theory along he way.

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