Monty Hall Problem

[pgwenthold]

Why would Monty open the door with the car behind it and give you the choice to switch from your obvious goat to the other obvious goat? It's ridiculous.

Because Monty doesn't know where the car is?

Nothing says that Monty has to give you the option to switch if he opens the door and it is a car. In fact, it works just fine on a game show: "Let's see what's behind door number 3? Oh, that's the car; sorry, you lose!"

"You've lost. Would you like to switch to the other, equivalently losing position or just stay where you are? It doesn't matter in the least, so I'm not sure why I'm giving you this opportunity..."

Where is the requirement that he has to allow you to switch when you play the game?

The Monty Hall Problem has the contestant choosing one of three doors, where one door conceals a win and each of the other two conceals a loss; the host opening another of the doors, to reveal a loss; and the contestant being given the chance to either stay with his original choice or change his choice to the remaining door. The problem is assumed to be identical if repeated, because there is no information given to suggest otherwise

Unfortunately, there is no information to suggest either. That is why we keep saying, "There is not enough information."

YOU assume the problem to be identical if repeated. However, there is no basis for that assumption. As written in the OP, it only describes a single instance.

 

[CurtC]

Holy hell. What do you mean that there's nothing in the original problem statement to justify that assumption? That is the original problem!

Look, every time you play the game, Monty will open a door with a goat behind it. Every time. He will never, ever reveal a car. That's the nature of the problem. Deal with it.

You're not seeing the point. Let's take this mathematically equivalent scenario: you come upon a game of three-card monte - there are three cards on the table, one is an ace. He shows you where the ace is, then scrambles the cards too quickly for you to follow, then stops (let's also say that there is no cheating going on here, it's simply that he can scramble the cards faster than you can follow). You pick a card. The street hustler pauses, then says "hey, this other card over here is NOT the ace, now do you want to switch your guess to the remaining card?"

The question - should you switch? Any second grader can see that in this situation, if you switch, you lose. The only reason he offered the switch is because you picked the ace on your first guess.

Now, can you see any difference in this game and the Monty Hall problem? They're completely equivalent, except for assigning different motivations to the hosts.

 

[CurtC]

There you go again, setting conditions that are not in the original scenario.

Nobody stipulated that he was picking his door at random.

We all realize that. In fact, that's the point of this further discussion, that it was not specified in the OP how the host would behave, therefore there's not enough information. But we're then taking a set of interesting possibilities for his methods and working the numbers for those cases. These are not answers to the OP, but tangential discussions.

 

[CurtC]

What are you talking about? You cannot introduce abritrary '?' scenarios contradicting the experiment. Why don't you add a surprise case where a chimpanzee is revealed from time to time? Or a naked blonde? This is excluded in the very same way a car is.

Let's imagine that we take 300 people to play this game, each gets to play it only once so he has no knowledge of how Monty operates. Now let's group them into different sets depending on how they choose. Further, let's say that in this game, Monty doesn't know where the car is, but will always reveal one of the other two doors, and if he reveals a goat, he offers the chance to switch.

Set 1 - picks the car on the first guess, Monty then reveals a goat and offers the choice of switching. This group has 100 people.

Set 2 - picks a goat first, then Monty (by chance) reveals a goat and offers the choice of switching. This group has 100 people in it.

Set 3 - picks a goat first, then Monty (by chance) reveals the car. The game is over, the contestant loses. This group has 100 people in it.

Now the question is described as in the OP. This tells you that we're dealing with someone who is in either set 1 or set 2. The people in set 3 don't count, because they never get to this predicament.

Now, out of those 200 people, the 100 in set 1 would lose by switching, the 100 in set 2 would win by switching. This is a 50/50 proposition, AND IT'S VALID ONLY IN THE CASE WHEN THE HOST PICKS RANDOMLY WITHOUT KNOWING WHERE THE CAR IS.

(Again, I acknowledge that if the host knows where the car is and will show every contestant a goat every time, then he can win 2/3 of the time by switching. But both scenarios are consistent with the OP, and without making an assumption about how Monty operates, you can't say that the 2/3 answer is always correct.)

 

[Mobyseven]

This may be the intent, but it usually isn't stated as such.

If it isn't stated as such, then the person isn't stating the Monty Hall problem. They are mistaken.

Interestingly, many posters are saying, "This is what the problem says" but we are the ones who go back and quote the OP and ask, "Where does it say that he always opens a door to reveal a goat?"

Monty presents 3 doors; behind one is a car, and behind the other two are goats. You pick a door. But before Monty opens that door, he reveals behind another door that there is a goat. He then gives you the opportunity to switch your choice. What is the probability that you will get the car by switching, or by staying?

Epic reading comprehension fail. There's nothing wrong with how the OP stated the problem, it is, indeed, entirely correct.

 

[Mobyseven]

You're not seeing the point. Let's take this mathematically equivalent scenario: you come upon a game of three-card monte - there are three cards on the table, one is an ace. He shows you where the ace is, then scrambles the cards too quickly for you to follow, then stops (let's also say that there is no cheating going on here, it's simply that he can scramble the cards faster than you can follow). You pick a card. The street hustler pauses, then says "hey, this other card over here is NOT the ace, now do you want to switch your guess to the remaining card?"

The question - should you switch? Any second grader can see that in this situation, if you switch, you lose. The only reason he offered the switch is because you picked the ace on your first guess.

Now, can you see any difference in this game and the Monty Hall problem? They're completely equivalent, except for assigning different motivations to the hosts.

Holy mother of Vishnu, how hard is this to understand? You're assigning motives to Monty that he just doesn't have. The scenario you just described, where Monty will only offer a switch if you've already chosen the car, is NOT the Monty Hall problem.

Look, people, this isn't so hard to grasp. The Monty Hall problem is not some horror movie about a game show host out to cheap unsuspecting players. It's a hypothetical scenario, specifically designed to show how unintuitive probability theory can be by presenting what seems to be a simple problem and showing that the 'gut' solution is incorrect.

If Monty is trying to cheap contestants then it's not the Monty Hall problem. If Monty is opening doors at random then it's not the Monty Hall problem. If Monty is a vampire, using goats to steal the souls of unsuspecting competitors, then it's not the Monty Hall problem.

 

[ddt]

Holy mother of Vishnu, how hard is this to understand? You're assigning motives to Monty that he just doesn't have. The scenario you just described, where Monty will only offer a switch if you've already chosen the car, is NOT the Monty Hall problem.

Look, people, this isn't so hard to grasp. The Monty Hall problem is not some horror movie about a game show host out to cheap unsuspecting players. It's a hypothetical scenario, specifically designed to show how unintuitive probability theory can be by presenting what seems to be a simple problem and showing that the 'gut' solution is incorrect.

If Monty is trying to cheap contestants then it's not the Monty Hall problem. If Monty is opening doors at random then it's not the Monty Hall problem. If Monty is a vampire, using goats to steal the souls of unsuspecting competitors, then it's not the Monty Hall problem.

That entirely depends upon what you call the "Monty Hall problem". Its usual formulation rests indeed on the rule that the host will always open another door which has a goat behind it.

However, in the real "Let's Make a Deal" show, this was not the case. From a NYT article on it:

Was Mr. Hall cheating? Not according to the rules of the show, because he did have the option of not offering the switch, and he usually did not offer it.

And although Mr. Hall might have been violating the spirit of Ms. vos Savant's problem, he was not violating its letter. Dr. Diaconis and Mr. Gardner both noticed the same loophole when they compared Ms. vos Savant's wording of the problem with the versions they had analyzed in their articles.

"The problem is not well-formed," Mr. Gardner said, "unless it makes clear that the host must always open an empty door and offer the switch. Otherwise, if the host is malevolent, he may open another door only when it's to his advantage to let the player switch, and the probability of being right by switching could be as low as zero." Mr. Gardner said the ambiguity could be eliminated if the host promised ahead of time to open another door and then offer a switch.

It is obvious that this influences the probabilities involved. It is also obvious that the game host not having a deterministic decision procedure, based only on the choice of the first door by the contestant, for offering the choice to switch or not, makes it impossible to calculate probabilities.

It is also obvious that it is impossible for the contestant to make a right decision if he does not know the decision procedure of the game host. A benevolent host may offer to switch only if you first picked a goat, in which case switching is always good; a malevolent host may offer to switch only if you first picked the car, in which case switching is always bad.

An interesting question in this is still: do you acquire enough information of watching, say, one season of "Let's Make a Deal" to deduce the decision procedure of the game host?

At least thanks to those who resurrected the thread that I learnt this tidbit of information about the show that problem was based on. I haven't seen a single instance of the original show. From Dutch game shows with the same set-up I got the impression that the host always indeed opened another door with a goat behind it. After all, the price of the car is a fraction of the cost of the production of the show and you want people in the majority of cases to go home with a price.

 

[CurtC]

Holy mother of Vishnu, how hard is this to understand? You're assigning motives to Monty that he just doesn't have. The scenario you just described, where Monty will only offer a switch if you've already chosen the car, is NOT the Monty Hall problem.

I agree, it's not part of the Monty Hall problem. However, the OP itself was not the Monty Hall problem, because it did not specify that Monty is constrained to always showing a non-prize door. At best, it was ambiguous - does the present tense wording mean that's what he always does, or a conversational way of stating what he did this one time? It sounds to me like it's describing only what he did this one time.

My point here is that when I've seen people attempt to present the Monty Hall problem, the statement very rarely includes the constraint on Monty, therefore you don't have enough information. The OP here was an example of that. Now we're having fun imagining different scenarios that are consistent with the OP and give different answers - OK?

If you want to talk only about the classic Monty Hall problem, then be sure to clearly specify how Monty must operate.

 

[Rolfe]

This may be the intent, but it usually isn't stated as such.

Interestingly, many posters are saying, "This is what the problem says" but we are the ones who go back and quote the OP and ask, "Where does it say that he always opens a door to reveal a goat?"

Given just the problem, as written in the OP, there is not enough information to answer the question. If you claim it is the "intent" of the problem, then you are basing your answer on more than just the problem, as written in the OP.

The question of what he "always" does arises because it is necessary to imagine multiple runs of the problem to identify the probabilities. However, in order to set up your multiple runs, you have to know what the ground rules are.

I could (if I was techie enough) set up a computer simulation so that a goat is always revealed at step 2, and as we know, this will most certainly demonstrate that switching doors will double your chance of getting the car.

I could also set it up so that either of the two unchosen doors are opened at random. And as we know, this will most certainly demonstrate that switching doors has no effect on your chances of getting the car.

(I could also set it up so that any of the three doors is opened at random. Again, this would demonstrate that switching doors has no effect.)

These are the only three possible "rules" I can think of that can reasonably be applied to the game. If the game is deliberately rigged so that the choice to switch is only offered when the contestant has already chosen the car, the puzzle is meaningless. And Monty is a vile cheat.

Holy mother of Vishnu, how hard is this to understand? You're assigning motives to Monty that he just doesn't have. The scenario you just described, where Monty will only offer a switch if you've already chosen the car, is NOT the Monty Hall problem.

Look, people, this isn't so hard to grasp. The Monty Hall problem is not some horror movie about a game show host out to cheat unsuspecting players. It's a hypothetical scenario, specifically designed to show how unintuitive probability theory can be by presenting what seems to be a simple problem and showing that the 'gut' solution is incorrect.

If Monty is trying to cheat contestants then it's not the Monty Hall problem....

Well, exactly. (I had to go back and edit this, because I don't agree that you can specify that the doors are not being opened at random in the Monty Hall problem - or you can, but if you do, then most of the surreal amusement goes out of it, see my next post below.)

And we can also exclude scenarios that say, well, Monty may choose a different scenario each time. This is because we are actually being asked about a single example of the game, in isolation. The repetitions are only necessary to demonstrate the odds. Thus by definition we have to set up the simulation with the rule in place which governs the example of the game which we are actually discussing.

Presumably the puzzle was invented by someone. It would be nice to be able to find that person and ask what the intent was. However, it's been around so long that the chances of finding the original inventor must be approaching zero. So, we have what we have. Just a bald description of the scenario.

The fact remains that it is clearly possible to set up the simulation in two ways (I'll exclude the third one above, because I don't think it adds anything to the discussion). Both sets of rules produce our proposed scenario - one door chosen, another opened, and that revealing a goat. We can't tell which set of rules is in force from the initial premise. Nevertheless, the set of rules in force has a huge effect on the outcome. One set - switching doubles your chances of winning; the other - no advantage to switching.

I think the reason the puzzle seems surreal is that it's very difficult to imagine the what Monty is thinking can influence the odds. However, it becomes easier if you imagine it as playing a computer simulation - I understand there are such simulations available on the net.

Are you playing a game where the programmer has set it up with "always a goat"? SWITCH!

Are you playing a game where the programmer has set it up with "random unchosen door is opened"? Don't bother.

And this still applies even if you only play once.

Does this make it any clearer? Now, instead of Monty's human and no doubt capricious mind, we simply have two computer games. It's just that you don't know which one you're playing.

Stop. Think about it. Now please explain to me where I'm wrong.

This now gets me back to my original point, which is that excluding a rigged game, you should switch. Because even if you're on the latter programme, switching won't decrease your chances. But there is a less-than zero possibility that you are on the former programme, when swithcing is beneficial. Therefore you should rationally decide to switch, to allow for the possibility that the former is the game in town.

I thiink this is what I like about it. The surreal conclusion that Monty's very intent affects the odds (made a bit less surreal if we substitute Monty with a pre-programmed computer game). And then, the extra leap that says, stop arguing. If you don't know which scenario you're dealing with, and in one the switch is beneficial while in the other it's neutral, then just switch anyway - you can't lose by it, and by switching you are in a position to exploit the possibility of the beneficial scenario.

Rolfe.

 

[pgwenthold]

Holy mother of Vishnu, how hard is this to understand? You're assigning motives to Monty that he just doesn't have.

1) No one is assigning any motives to Monty but you
2) We are saying that, given the wording of the problem, we _don't know_ his motives, and in order to come up with an answer requires that we make assumptions that are not inherent in the original problem as it is stated here.

The OP of this thread is readily available to you, and has even been quoted twice in the last two days. Therefore, it should not be hard for you to show us how the problem shows that he does not have these motives.


Notice, you are the one claiming he "doesn't have" motives. We are claiming that we don't know that he doesn't have motives.

 

[Rolfe]

I agree, it's not part of the Monty Hall problem. However, the OP itself was not the Monty Hall problem, because it did not specify that Monty is constrained to always showing a non-prize door. [....] If you want to talk only about the classic Monty Hall problem, then be sure to clearly specify how Monty must operate.

Well, maybe. However, I've been aware of this problem for a long time, and it has never been presented in that way to me. I first became aware of it in the early to mid 1990s, when it was apparently the subject of much correspondence in the Guardian newspaper. While I only had second-hand exposure to that correspondence, it's quite clear from the nature of the arguments that it was similar to the OP. Only one example described, in the present tense. No information given about whether this was or was not part of a longer run of games, or about how any other runs of the game turned out, or about what underlying rules should be assumed. Indeed, the name "Monty Hall" wasn't even mentioned that I know about. It was "a game show host in an impoverished South American republic".

I think the point is that if you specify the ground rules exactly, it takes just a little bit longer than GreyICE's boxes and coins puzzle to figure out, and one you've done that, it's done. Move on. You completely understand the problem, and its solution. Might have taken you a couple of hours, if you're slow.

However, by not specifying the rules, the problem takes off into a completely different sphere. First you think you have the answer one way, then you realise the answer may be different if you assume a different set of rules, then you sit back, boggled, contemplating the possibility that what the host is thinking can affect the odds attached to your choice.

That's why the problem keeps resurfacing, and why this zombie thread is still going even after six years.

Rolfe.

 

[sphenisc]

Lol, Rolfe, thanks for making me feel better.

The coin-box puzzle is interesting, the answer is 2/3rds (like switching in Monte Hall) despite the fact that at face value it looks like 1/2

(this is because despite your 50/50 chance of having the 2 gold coin/1 silver 1 gold box, you only have a 50/50 chance of grabbing a gold coin from the 1 silver 1 gold box).

It's easier because the boxes don't need motivation.

The final stage to that trick is set it up, and offer to bet $10 that the coin will be the same color - which is why it's the start of a con game too.

If the probability is 2/3s it's going to be a different colour then this seems an excellent way of losing money fast!!!

 

[CurtC]

The OP of this thread is readily available to you, and has even been quoted twice in the last two days. Therefore, it should not be hard for you to show us how the problem shows that he does not have these motives.

To be completely fair, I didn't notice it before, but the wording of the OP could possibly be interpreted as saying what Monty always has to do. I didn't read it that way, and I think most people wouldn't either, but the use of the present tense could be ambiguous: "But before Monty opens that door, he reveals behind another door that there is a goat. He then gives you the opportunity to switch your choice."

Does the present tense there indicate what he always does? At best, it's ambiguous, but certainly it's not explicit in saying what rules Monty follows.

However, by not specifying the rules, the problem takes off into a completely different sphere. First you think you have the answer one way, then you realise the answer may be different if you assume a different set of rules, then you sit back, boggled, contemplating the possibility that what the host is thinking can affect the odds attached to your choice.

Exactly. I have come across this puzzle many many times, and I don't think I've ever seen it worded to constrain Monty's behavior. This lack of constraint adds an interesting twist to the puzzle which seems to irritate some people.

 

[ddt]

To be completely fair, I didn't notice it before, but the wording of the OP could possibly be interpreted as saying what Monty always has to do. I didn't read it that way, and I think most people wouldn't either, but the use of the present tense could be ambiguous: "But before Monty opens that door, he reveals behind another door that there is a goat. He then gives you the opportunity to switch your choice."

Does the present tense there indicate what he always does? At best, it's ambiguous, but certainly it's not explicit in saying what rules Monty follows.

I, for one, interpret the OP as fixing the rules of Monty's behaviour. If Monty had a choice whether he'd open another door or not, and even one with the car behind it, I had expected a modality in the form of an auxiliary verb (can, may) or a subjunctive . As the OP is stated, in a present tense indicative, it means how he always behaves.

Disclaimer: English is a second language for me.

Exactly. I have come across this puzzle many many times, and I don't think I've ever seen it worded to constrain Monty's behavior. This lack of constraint adds an interesting twist to the puzzle which seems to irritate some people.

Actually, all the times I've heard the puzzle it was always with the automatic behaviour of the host to open a second door with a goat...

 

[Mobyseven]

Stop. Think about it. Now please explain to me where I'm wrong.

You aren't. We're in agreement.

I agree, it's not part of the Monty Hall problem. However, the OP itself was not the Monty Hall problem, because it did not specify that Monty is constrained to always showing a non-prize door. At best, it was ambiguous - does the present tense wording mean that's what he always does, or a conversational way of stating what he did this one time? It sounds to me like it's describing only what he did this one time.

My point here is that when I've seen people attempt to present the Monty Hall problem, the statement very rarely includes the constraint on Monty, therefore you don't have enough information. The OP here was an example of that. Now we're having fun imagining different scenarios that are consistent with the OP and give different answers - OK?

If you want to talk only about the classic Monty Hall problem, then be sure to clearly specify how Monty must operate.

1) No one is assigning any motives to Monty but you

Bugger off. Seriously, this is just starting to piss me off. I haven't assigned any motive to Monty. I have shown what his function in the scenario is. I haven't come anywhere remotely near assigning him a motive, unlike others in the thread.

You say I've assigned him a motive, I say put up or shutup. Quote the part where I assigned him a motive, or concede that I bloody well didn't.

2) We are saying that, given the wording of the problem, we _don't know_ his motives, and in order to come up with an answer requires that we make assumptions that are not inherent in the original problem as it is stated here.

Wrong. I quoted the problem as it was presented, and bolded the relevant part. There's no issue with it outside of your inability to accept what is written in plain English.

The OP of this thread is readily available to you, and has even been quoted twice in the last two days. Therefore, it should not be hard for you to show us how the problem shows that he does not have these motives.

I know the OP is readily available. I quoted it, and bolded the relevant part. Moreover, you're talking out of your arse when you ask me to prove that the problem as stated shows that he does not have motives A through Z. I can't prove that, and while you think you're just being oh-so-clever in 'trapping' me, you're missing the whole bloody point: His motives don't matter one bleedin' scrap. His function in the scenario is clearly defined, and regardless of what imaginary motive you want to toss around the outcome of the scenario won't change. If you decide to assign a motive to Monty that changes his function in the scenario, go right ahead - but if you do that, you're no longer discussing the Monty Hall problem.

And now, a short interlude for your amusement...​

Signor: This cheese tastes a bit off.
Signora: But what if it was beef?
Signor: ...but it's not beef. It's cheese.
Signora: But what if it was beef?
Signor: Then I guess it would taste...like beef?
Signora: Ah! So you're saying that this cheese tastes like beef!
Signor: *Applies head to wall vigorously*

fin​

Notice, you are the one claiming he "doesn't have" motives. We are claiming that we don't know that he doesn't have motives.

I'm the one pointing out that his motives are worth a rat's fart in a typhoon. It's his function that is key. You can go crazy like a freshman literature student imagining motives for the man, but unless you change his function the outcome is the same - and if you change his function then you aren't discussing the Monty Hall problem anymore.

 

[CurtC]

I'm the one pointing out that his motives are worth a rat's fart in a typhoon. It's his function that is key.

Whatever.

You call it function, he calls it motives, you're talking about the same thing. The proper Monty Hall puzzle specifies Monty's behavior as always being forced to show a non-prize door, and offer the switch. However, the puzzle as described in the OP of this thread does not specify that. At best, it's ambiguous. As I mentioned before, I have never seen the puzzle presented in the wild (as opposed to its Wikipedia page), where this constraint was explicit. And without this constraint, you have to make some assumptions about how Monty behaves (his function/motive) in order to come to a solution.

 

[Michael C]

I, for one, interpret the OP as fixing the rules of Monty's behaviour. If Monty had a choice whether he'd open another door or not, and even one with the car behind it, I had expected a modality in the form of an auxiliary verb (can, may) or a subjunctive . As the OP is stated, in a present tense indicative, it means how he always behaves.

No, that doesn't imply that this is how he always behaves. Consider these two problems:

1. I throw a coin. It comes up heads. Can I deduce that there is anything unusual about the coin?

2. I throw a coin. It always comes up heads. Can I deduce that there is anything unusual about the coin?

In any case, I am happy that the OP left the possibility of interpreting the problem in different ways, since it led to the discussion on whether the version where Monty always knows the location of the prize is different from the one where he chooses a door at random. The two versions are indeed different and have different solutions.

 

[Michael C]

The coin box is weird, conceptually.

You have three boxes, one with two gold coins, one with two silver coins, one with a gold coin and a silver coin.

If you take a random coin out of a random box, and it's gold, what chance is there that the other coin in the box is silver?.

The coin-box puzzle is interesting, the answer is 2/3rds (like switching in Monte Hall) despite the fact that at face value it looks like 1/2

I hope you meant to say "the answer is 1/3"

 

[Michael C]

I just found something interesting in Wikipedia. Marylin vos Savant, who may be considered to be one of the world's experts on the Monty Hall problem, wrote about the version where Monty chooses the door at random in her column in November 2006:

This difference can be demonstrated by contrasting the original problem with a variation that appeared in vos Savant's column in November 2006. In this version, Monty Hall forgets which door hides the car. He opens one of the doors at random and is relieved when a goat is revealed. Asked whether the contestant should switch, vos Savant correctly replied, "If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch" (vos Savant, 2006).

 

[jsfisher]

You and Monty are playing a game of "Guess which door has the car." The only problem is, you don't know the rules for the game. The only thing you know are the events described in the opening post. (You picked a door; Monty revealed a goat behind another door; you now have an option of changing your door choice.)

Suppose these are the actual rules:

1. You get to pick a door.
2. If the door you pick has a goat behind it, then the game is over with you winning a goat.
3. Otherwise (i.e. the door you picked has the car), Monty opens one of the remaining doors to reveal a goat.
4. You get an option to change your door choice.
5. The game ends with you getting whatever is behind your final choice.

I am not saying these have to be the rules; they are just offered as a for-instance. Are these rules in any way inconsistent with the conditions stated in the opening post?

If they are not consistent, what is the specific inconsistency. If they are consistent, would you switch door choices if given the opportunity?