The Monty Hall problem

[humber]

bluster? That's pretty hostile. I purposefully chose silly things for fun.

Sarcasm...tsk...tsk.

It's just a math puzzle with some poorly chosen wording. No need to get personal.

I think your point was made that the puzzle assumes some commonly understood conditions, but is also full of holes.

 

[Rolfe]

Well, the wikipedia article is actually well written, with dozens of references to peer reviewed papers and the like. But, if you want to read the specifics on the vos Savant issue, here it is.

Many PhDs in mathematics, from obscure places like MIT )), understanding the assumptions correctly, disagreed with the argument that you should switch. It's a very non-intuitive problem, and got very heated debate. Of course, a subset of the debate was which assumptions did we make, and which are best to make, but my point in this thread is that is not the interesting part of the problem. It's interesting (to me, I grant) that even when all the assumptions are nailed down (wormholes nonwithstanding) very smart and educated people still disagree. Read the comments to her article - PhDs, grad students in mathmatics, all thought she was wrong, and continued to think so until they actually carried out an experiment based on what we all agree are the 'correct' assumptions for the 2/3 answer.

I'll selectively quote:

In other words, people made the correct assumptions, even ran the experiment, and still disagreed with what to us is the obvious solution.

Oh, thanks, I'll have a look. I read the wiki article on the problem some years ago, and my memory is that it wasn't very good at that time. I may be wrong about the Vos Savant affair, because I'm only going by what I've heard others say about it.

I remember first coming across the problem many years ago, before I had an internet connection as far as I remember (which puts it pre-1997). It did take me a couple of days to figure it out, but it all became clear as soon as I imagined 100 doors, 99 goats and one car. I realised you do need to assume that Monty is deliberately avoiding the car, but I agree, you're probably supposed to make that assumption. I can't see what's so complicated about it that people need to be publishing papers about it, I have to say.

I also recall mentioning the problem to a friend who was a university lecturer in computing and mathematics, and being told it was obvious there was no advantage, it was a 50-50 chance. I mentally excused him on the grounds that he had only answered off the top of his head, and I'd spent a couple of days thinking about it.

I'm very surprised that anyone with a reasonably-equipped brain, having had time to consider the problem, and even more so having had it explained to them, would argue with the answer that switching doubles your chances of success.

Ever since that first occasion, as I say back pre-internet for me so I didn't discuss it with anyone (I believe the problem had been featured in a newspaper and there had been an argument in the letters column - that's how I heard about it), every discussion I've had has featured a bunch of smart-alecks dissing the "correct" answer on the grounds that it involves making assumptions that aren't stated in the original problem. To the point where I tend to start pre-empting them.

And it is quite interesting to think of the other possibilities, in a masochistic sort of way.

Rolfe.

 

[roger]

As an aside, to explain my thinking on assumptions in puzzles, I think Marilyn got a different puzzle wrong. I wrote in, but never got a response.

I forget the exact formulation, but the problem was you had a set of coins, and you needed to put them in X envelopes such that condition Y was satisfied (the value contained in each envelope summed to some value - I think the condition was that every sum had to be unique, but I'm not sure). Her response was that it was impossible.

Well, to my way of thinking, if I have a puzzle "Do X", you figure out how to do X, you don't say "it's impossible". As it turned out, you could easily solve the problem by arranging the coins in the envelopes in a certain way, and then putting one envelope inside another envelope. In that way, one envelope allows you to sum the coins contained in it, and the coins contained in the enclosed envelope.

We could bicker endlessly about whether that is an "acceptable" interpretation of the problem. However, it was easily proven that w/o putting one envelope inside the other that the problem was not solvable, and it was also provable that my solution was unique (there weren't 10 different ways to put envelopes in other envelopes in a way to satisfy the problem). To me, that's a big part of puzzles - sussing out the assumptions. You don't get much traction on the "where do they bury the survivors" plane crash puzzle by arguing that perhaps in this county they do bury live people.

 

[roger]

Ever since that first occasion, as I say back pre-internet for me so I didn't discuss it with anyone (I believe the problem had been featured in a newspaper and there had been an argument in the letters column - that's how I heard about it), every discussion I've had has featured a bunch of smart-alecks dissing the "correct" answer on the grounds that it involves making assumptins that aren't stated in the original problem. To the point where I tend to start pre-empting them.

Ahh! That's where our miscommunication laid. I thought you were bringing up smart-alecky objections, and you thought I was. Whereas we are both on the same page.

 

[JoelKatz]

I'm not really familiar with what all this stuff is about Vos Savant, but I think this is what it relates to - that she explained the answer adhering to the obvious assumptions, and then got roasted as being wrong, or stupid, or something, by people who decided that these assumptions were not inherent in the problem as presented..

Vos Savant got it 90% right. She received a very strange mix of letters. A lot of them were from people militantly insisting that she was an idiot and that switching can't possibly improve your chances -- some of these make very entertaining reading. Some people did point out that she failed to point out the assumptions that make her answer correct. I think that's entirely fair (if done politely) because the whole point of the problem is to understand what circumstances affect the answer. Not all of the people, even among those who were 100% correct, were polite.

This is a very complex problem with many layers of understanding that can be gained from it and many potential traps one can fall into. It would be a serious mistake to say 'I get it, you should switch" and dismiss it. There's more to learn. For one thing, working out different and better ways to explain its nuances to people really does help sharpen your own understanding.

 

[humber]

As an aside, to explain my thinking on assumptions in puzzles, I think Marilyn got a different puzzle wrong. I wrote in, but never got a response.

I forget the exact formulation, but the problem was you had a set of coins, and you needed to put them in X envelopes such that condition Y was satisfied (the value contained in each envelope summed to some value). Her response was that it was impossible.

Well, to my way of thinking, if I have a puzzle "Do X", you figure out how to do X, you don't say "it's impossible". As it turned out, you could easily solve the problem by arranging the coins in the envelopes in a certain way, and then putting one envelope inside another envelope. In that way, one envelope allows you to sum the coins contained in it, and the coins contained in the enclosed envelope.

We could bicker endlessly about whether that is an "acceptable" interpretation of the problem. However, it was easily proven that w/o putting one envelope inside the other that the problem was not solvable, and it was also provable that my solution was unique (there weren't 10 different ways to put envelopes in other envelopes in a way to satisfy the problem). To me, that's a big part of puzzles - sussing out the assumptions. You don't get much traction on the "where do they bury the survivors" plane crash puzzle by arguing that perhaps in this county they do bury live people.

Perhaps I should have said you were being ironic with the wormholes and other examples. They led to me looking up E-Prime. That is funny.
The Monty Hall puzzle, and those like them, do result in trying to second-guess the writer on what the expected answer is.

There are some examples of Vos Savant's errors in cache;
http://webcache.googleusercontent.c...cd=1&hl=nl&ct=clnk&gl=nl&source=www.google.nl

 

[humber]

I also recall mentioning the problem to a friend who was a university lecturer in computing and mathematics, and being told it was obvious there was no advantage, it was a 50-50 chance. I mentally excused him on the grounds that he had only answered off the top of his head, and I'd spent a couple of days thinking about it.

Paul Erdos is said to be one who would not accept the result.

"The army PhD who wrote in may have been correct that if all those PhDs were wrong, it would be a sign of trouble. But Marilyn was correct. When told of this, Paul Erdos, one of the leading mathematicians of the 20th century, said, “That’s impossible.” Then, when presented with a formal mathematical proof of the correct answer, he still didn’t believe it and grew angry. Only after a colleague arranged for a computer simulation in which Erdos watched hundreds of trials that came out 2-to-1 in favor of switching did Erdos concede that he was wrong."

Does anyone have proof that this is true?

 

[Rolfe]

I don't say that you should, but that is the intent of the writer of the question. For what other reason would the problem be posed as a game show? It is generally accepted that Monty works to avoid giving the prize away, and may even mislead the contestant. His intent is subsumed by the intent of the show's producers to provide entertainment.

In my observation, this isn't "generally accepted" at all. If anything is generally accepted, it is that we are supposed to assume that the situation always plays out as described in the example given - that the contestant chooses a door, then Monty opens another to reveal a goat, and the contestant is offered the opportunity to switch to the remaining closed door.

If we're playing mental games about "what can Monty do to minimise the chances of anyone winning the car", or "what is the most exciting way to make this play out so that the ratings will be good" then it's a psychology problem, not a mathematical one.

I think that perhaps applies to Parade's reason for publishing the puzzle, though not necessarily Vos Savant's intent

You think it's meant to be a psychology problem, and it has been misinterpreted as a mathematical one? Well, it's a point of view I suppose.

He can do that or not, if the contestant picks the car, because there is always one goat. So it could be said his intent is to show the contestant a goat.

This is the obvious assumption, and it's the assumption that makes for an interesting mathematical puzzle, indeed. Psychological conundrums are something else.

Yes, but I don't think actual games are reduced to one prize and two identical alternatives. One may be cash, or a holiday for example. Perhaps the car has a higher monetary value, but a holiday may be preferred if that allows a week away from the spouse.
If it were to become known that the prize were more likely to be behind the first curtain, I suppose the show would react to that too.

But none of that is stated in the usual presentation of the problem. If it's necessary for the person strying to solve it to know who Monty Hall is, and how his game show played out week after week, then I think it's getting a bit esoteric these days.

As hgc said, he always reveals a goat.
But, I agree, remove the trimmings and it's just a simple example of conditional probability.

Yes, and I think that was all it was really intended to be. When the problem was originally formulated, it's unlikely anyone really thought about all the what-ifs involving Monty trying to trick the contestants.

But that machine must follow an algorithm that mimics Monty's action of always revealing a goat.

Which is exactly wat I said.

There is a website dedicated to her errors, http://wiki.wiskit.com/marilyn,
"Marilyn is Wrong", but is not currently accessible.

Now I thought Roger was saying she was right, but lots of people didn't get it and argued. Now you're saying she was wrong? I suppose I have to go and read up about it now, but which is it?

Rolfe.

 

[Rolfe]

Ahh! That's where our miscommunication laid. I thought you were bringing up smart-alecky objections, and you thought I was. Whereas we are both on the same page.

Pretty much.

The simple answer that switching doubles your chances of winning is to me so self-evident now that it's fairly boring. The interesting bit is trying to see what other conditions you can put on the puzzle that will also provide interesting answers that don't include "it depends".

Rolfe.

 

[Rolfe]

There are some examples of Vos Savant's errors in cache;
http://webcache.googleusercontent.c...cd=1&hl=nl&ct=clnk&gl=nl&source=www.google.nl

OK, I see, she was right but a bunch of smart-alecks decided to have a go because they didn't agree that the obvious assumptions should be assumed.

Rolfe.

 

[Rolfe]

Having looked at the web page Humber linked to, I noticed one reply that seemed to put the thing in its widest perspective.

Tim Mann mann@pa.dec.com wrote to point out that Marilyn's analysis does not require that the game show host offer every contestant the opportunity to switch. All that is required is that the host's decision about whether to offer the contestant an opportunity to switch is independent of whether the contestant's initial choice is correct. For example, the host could make the decision whether to offer the switch before the contestant chooses a door, or the host could make the decision based on the flip of a coin, the day of the week, or some other variable independent of the contestant's initial choice.

At first glance that seems right to me. So long as the host's decision isn't based on whether the initial choice was correct, then there is never a disadvantage to switching, and there may be an advantage.

Have I missed something?

Rolfe.

 

[SOdhner]

I was having a little trouble wrapping my head around this when I first heard about it, until I approached it like this: "What are the odds I picked a goat? 2/3. So my current door is PROBABLY a goat. Later opening of doors doesn't change that."

 

[humber]

In my observation, this isn't "generally accepted" at all. If anything is generally accepted, it is that we are supposed to assume that the situation always plays out as described in the example given - that the contestant chooses a door, then Monty opens another to reveal a goat, and the contestant is offered the opportunity to switch to the remaining closed door.

Yes. And that action will only be fulfilled if a goat is chosen. Otherwise it terminates when the car is revealed. As general rule, a TV trope if you will, game show hosts do not show the prize to the contestant.

If we're playing mental games about "what can Monty do to minimise the chances of anyone winning the car", or "what is the most exciting way to make this play out so that the ratings will be good" then it's a psychology problem, not a mathematical one.

It is a psychology problem. If it were simply mathematical, then it would be expressed that way. There is a reason why this problem gains so much attention, and one is that it contradicts commonly held expectations around chance.

You think it's meant to be a psychology problem, and it has been misinterpreted as a mathematical one? Well, it's a point of view I suppose.

I think that this one, and so many others, are like that. How many members of the general public are otherwise interested in problems of probability?
Why did so many "react angrily" to the solution?

This is the obvious assumption, and it's the assumption that makes for an interesting mathematical puzzle, indeed. Psychological conundrums are something else.

Is it normal to consider intent in general problems of probability?
You did, and so did I. I thought about it too, and then from the point of information and game theory. No need for math at all.

But none of that is stated in the usual presentation of the problem. If it's necessary for the person strying to solve it to know who Monty Hall is, and how his game show played out week after week, then I think it's getting a bit esoteric these days.

Would, say, tribesmen living in the deepest jungle, have a clue what the problem was about? They may need to have all that explained.

Yes, and I think that was all it was really intended to be. When the problem was originally formulated, it's unlikely anyone really thought about all the what-ifs involving Monty trying to trick the contestants.

It is is taken from the use of "Monty knows where the car is"
In this example, he can't actually improve on his chances of not giving the prize away, if he sticks only to opening doors. He reveals a car, or the goat. His best option for not giving the prize away while avoiding terminating the game, is not to open any doors. If the contestant gets the car if he reveals it, then he is worse off again.
Not opening doors leaves the contestant with only the 1/3 information that a car is there. His action of showing a goat adds information to the benefit of the contestant.

Now I thought Roger was saying she was right, but lots of people didn't get it and argued. Now you're saying she was wrong? I suppose I have to go and read up about it now, but which is it?
Rolfe.

She is right, but trivially so. If her answer to this question says she doesn't understand the problem she set.

"Q: You are on a game show with three doors. A car is behind one; goats are behind the others. You pick door No. 1. Suddenly, a worried look flashes across the host’s usually smiling face. He forgot which door hides the car! So he says a little prayer and opens No. 3. Much to his relief, a goat is revealed. He asks, “Do you want door No. 2?” Is it to your advantage to switch?
—W.R. Neuman, Ann Arbor, Mich."

Marylin: Nope. If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch

It is the showing of the goat that gives the contestant the information that allows 2/3 to be inferred, not that Monty knows it or not.

 

[GlennB]

A parallel for card players (principle of restricted choice)

In a game of bridge, declarer holds these spades in his hand (South) and dummy (North), needing to bring in 5 spade tricks without loss (the ???'s are other suits and don't concern us here):

A 10 8 6 4
???
???
???


K 9 7 5
???
???
???
​

Declarer (south in this diagram) has the lead. The usual play when missing 4 cards that include a potential trick for the opponents (here the QJ32) is to play off the A-K aiming for a 2-2 break.

Declarer leads the spade 5 from the south hand, and left hand opponent plays the spade J. Should South still win with N's A and then play towards the K, aiming for a 2-2 break?

 

[humber]

Having looked at the web page Humber linked to, I noticed one reply that seemed to put the thing in its widest perspective.




At first glance that seems right to me. So long as the host's decision isn't based on whether the initial choice was correct, then there is never a disadvantage to switching, and there may be an advantage.

Have I missed something?

Rolfe.

I don't think so (?) Probability does not really enter the question unless purposefully introduced. I found this:
http://www.math.leidenuniv.nl/~gill/essential_MHP.pdf

 

[Rolfe]

I don't think so (?) Probability does not really enter the question unless purposefully introduced. I found this:
http://www.math.leidenuniv.nl/~gill/essential_MHP.pdf

That's quite interesting, though I think a little precious.

We're back, really, to the split between those who believe the problem was never intended to be more than a simple puzzle assuming that every iteration is the same - Monty opens a door to reveal a goat and offers the contestant the opportunity to switch his choice - and those who want to be "smart alecks" and introduce alternative scenarios where Monty might do any number of different things.

I note that author names Craig Whitaker as the person who posed the original question. It would be interesting to know what his intention was. Did he merely expect the answer yes, switching doubles your chances of winning, or was he anticipating the complex answers that depend on which side Monty had got out of bed that morning? Vos Savant's original response, as described here and elsewhere, would suggest that all the complications are later convolutions added by the smart-aleck brigade, rather than intended in the original proposition.

I think what irritates me is the assertions that the problem IS this that or the other. It can be taken more than one way. I don't think the assertion that it IS a psychological problem has any more inherent validity than the assertion that it was only intended to be an exercise in conditional probability, and the obvious assumptions are intended to be assumed.

Rolfe.

 

[Rolfe]

Yes. And that action will only be fulfilled if a goat is chosen. Otherwise it terminates when the car is revealed. As general rule, a TV trope if you will, game show hosts do not show the prize to the contestant.

Why not? Open second door, reveal car, oops you lost. Perfectly decent entertainment, especially if the contestant is a smart aleck.

It is a psychology problem. If it were simply mathematical, then it would be expressed that way. There is a reason why this problem gains so much attention, and one is that it contradicts commonly held expectations around chance.

It is perfectly possible for someone intending to present a mathematical problem to frame it poorly, so that more options are available than they in fact intended. The problem seems to get plenty attention even in its simple form of "Monty always does what is described, in every iteration of the game".

I think that this one, and so many others, are like that. How many members of the general public are otherwise interested in problems of probability?
Why did so many "react angrily" to the solution?

Depends how you define the general population. How many members of the "general population" give a monkey's about the Monty Hall problem? And in my experience, people react angrily because the solution appears counter-intuitive, and they are annoyed at being shown to be wrong.

She is right, but trivially so. If her answer to this question says she doesn't understand the problem she set.

I think that's taking it too far. She didn't realise she hadn't defined the problem rigorously enough. That's rather different. You may think that in doing so she opened the door to a different, more interesting, problem.

Is it normal to consider intent in general problems of probability?
You did, and so did I. I thought about it too, and then from the point of information and game theory. No need for math at all.

"Q: You are on a game show with three doors. A car is behind one; goats are behind the others. You pick door No. 1. Suddenly, a worried look flashes across the host’s usually smiling face. He forgot which door hides the car! So he says a little prayer and opens No. 3. Much to his relief, a goat is revealed. He asks, “Do you want door No. 2?” Is it to your advantage to switch?
—W.R. Neuman, Ann Arbor, Mich."

Marylin: Nope. If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch

It is the showing of the goat that gives the contestant the information that allows 2/3 to be inferred, not that Monty knows it or not.

Now that's actually an interesting way of looking at it, and perhaps a different way of expressing a point I was getting at above.

Maybe that's why I keep opening threads with "Monty Hall" in the title - because there's a possibility of getting a handle on an extra wrinkle each time.

Rolfe.

 

[69dodge]

So we agree you should switch if Monty knows and reveals a goat and the advantage is 2/3 to you. We also agree that a random choice by Mr. Hall reduces that to 1/2. So here's a variation that might have legs.

Monty doesn't want to give away where the car is so he purposefully doesn't look ahead of time. You pick and he secretly flips a coin. He then peeks behind the door is he is about to reveal.

If he sees a car, he says, "You picked a goat, so I'm going to reset and give you another chance." The three items are remixed (so neither you or Monty know what's what) and the procedure is repeated. This continues until Monty's peek shows him a goat and he opens that door.

Under these circumstances, should you switch or not switch or does it matter?

Interesting question. The answer is that it doesn't matter. The probability is split 50:50 between the door you picked and the other unopened door.

(This was not my first guess, but then I worked it out with pencil and paper.)

ETA: Simulation confirms the 50:50 answer.

If your odds are reduced with this procedure instead of the regular one, isn't it counter-intuitive that him showing you your choice was wrong sometimes actually hurts your chances?

I don't understand this question.

 

[Vorpal]

If a goat is revealed, then it makes no difference if that is done by intent or not.

Yes, it does.

The added information provided by the goat, and suggesting swapping as the best option, does not come from Monty, but knowing that a car is there.

No, it doesn't. That's been explained in several people in several ways already, so I guess 69dodge's prediction is likely correct.

If Monty reveals the car, then the game is over...

Yup. And both staying and switching win exactly half of the cases when the game is not over.

It's the timing of the goat information that is interesting. If the goat is revealed prior to the contestant's choice, then the chance is 50/50, so there is no motive to swap, but if revealed after the contestant's choice, then it becomes advantageous to swap.

Nope. Count it and see.

It's conditional probability, and it is the added information that makes it that way. One resorts to probability, when there is insufficient information.

You're not calculating your probabilities correctly.

Just write a simulation if you don't believe it. Or just examine and re-use the MATLAB one I wrote here, to confirm the probabilities I calculated here: in the case of a clueless Monty revealing a random door that's not the player's, what you do when it happens to be a goat is irrelevant to your chances. (Of course, just what happens when he shows a car makes a difference, but that's not the disputed case.)

ETA: That a car is there, is information that is common to Monty and the contestant. Monty also knows where the car is, but within the scope of the problem as stated by Vos Savant, that specific knowledge is of no use to him, unless he wants to show the car.

"[The host] forgot which door hides the car! So he says a little prayer and opens No. 3. Much to his relief, a goat is revealed."

Look, if you're saying that the problem is incompletely defined because we still don't know how Monty picked his door, I agree with you: in that case we can't say much of anything at all. But there are two interesting subcases: (1) Monty picks a random door, and (2) Monty picks a random door that's not the player's. I've considered both of these in the second post linked above. I guess one could make even more possible assumptions compatible with the statement in Marylin's column, but (2) is actually the most natural.

 

[humber]

That's quite interesting, though I think a little precious.

OK.

We're back, really, to the split between those who believe the problem was never intended to be more than a simple puzzle assuming that every iteration is the same - Monty opens a door to reveal a goat and offers the contestant the opportunity to switch his choice - and those who want to be "smart alecks" and introduce alternative scenarios where Monty might do any number of different things.

Not "Smart Alecs". The criticisms of many puzzles of this type, including Vos Savant's, are valid. Those puzzles work around the alternative cases that you may make, and mislead. Many, like the wave example, are factually wrong.

I note that author names Craig Whitaker as the person who posed the original question.
It would be interesting to know what his intention was. Did he merely expect the answer yes, switching doubles your chances of winning, or was he anticipating the complex answers that depend on which side Monty had got out of bed that morning? Vos Savant's original response, as described here and elsewhere, would suggest that all the complications are later convolutions added by the smart-aleck brigade, rather than intended in the original proposition.

There are more objections than simply sniping. I have a book where the author does a real hatchet job on her, and that may perhaps be spleen.

The problem was posed long ago:
"One of the earliest known appearances of the problem was in Joseph Bertrand's Calcul des probabilites (1889) where it was known as Bertrand's Box Paradox. It later reappeared in Martin Gardner's 1961 book, More Mathematical Puzzles and Diversions, as The Three Prisoner Problem and then resurfaced in 1975 - inspired by Monty Hall's U.S. gameshow "Let's Make a Deal" - in an article in The American Statistician by Steve Selvin entitled A Problem in Probability."
( The journal is behind a pay wall, but he comments here)
http://montyhallproblem.com/as.html

The problem, as presented by Ms vos Savant in 1990 (apart from the change in notation) ran as follows.
Suppose you're on a game show and you're given the choice of three doors. Behind one is a car, behind each of the others is a goat. You pick a door, say door A, and the host, who knows what's behind the other doors, opens another door, say B, which has a goat. He then says : "Do you want to switch to door C?" Is it to your advantage to take the switch?

Despite the fact that you, the contestant, have only doors A and C to choose from, and so it seems your probability of having the car is 1/2 each, Ms vos Savant answered "yes", switch to C.

It appears not to be a general question, but a specific procedure. The advantage in swapping can be found from strategy alone, and needs no probability at all. I think it is about information and reaction to unfamiliar situations. Much as Mark6 wrote.
http://www.internationalskeptics.com/forums/showpost.php?p=7063207&postcount=51

I think what irritates me is the assertions that the problem IS this that or the other. It can be taken more than one way. I don't think the assertion that it IS a psychological problem has any more inherent validity than the assertion that it was only intended to be an exercise in conditional probability, and the obvious assumptions are intended to be assumed.
Rolfe.

I did not say that other possibilities were not worthy of exploration.

But, I think that the puzzle, as presented for public consumption, has only one solution that is remotely interesting. As Roger wrote;

"You don't get much traction on the "where do they bury the survivors" plane crash puzzle by arguing that perhaps in this county they do bury live people."

Nor, the prize being revealed, nor all doors opened etc...