Understanding the Monty Hall Problem

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801
  1. ‘There is no point in Monty Hall revealing the goat.’

Various posters make this assertion ad nauseam, though it is quite incorrect. While the ‘swap to two doors’ explanation illustrates the role of the sum of probabilities in the MHP solution that it is indeed to the contestant’s advantage to switch, the MHP is clear that the contestant has to choose one door. Without finding out the location of the goat they are unable to achieve better than 1/3. Monty’s method of providing that information is by revealing a goat; any other method of doing so could be used to similar effect, but it must be done - to vary the method would be pointless hair-splitting.

802

I’m referring to the description of the game at the top of this page.

The host will always reveal a goat after the contestant chooses their initial door. Then the contestant can keep their first choice or switch. They cannot switch to the door the host has opened. The contestant knows (or should know) in advance that the host will always reveal a goat, and they also know that once the host opens a door they cannot choose that door, so it doesn’t matter which door the host opens.

This is logically equivalent to choosing a door, then being offered the choice to trade what’s behind that door for whatever is behind the other two. Since the host’s reveal also removes that door from the contestant’s choices it makes no difference if the contestant sees a goat and gets to switch to the remaining door, or if they don’t see the goat and get to switch to the other two doors.

I know this has been pointed out before, not to change the rules, but to show with simple logic that switching is the better strategy. If you take out all the theatrics of the host and the goats it’s obvious.

803

The only thing that is obvious is the remarkable similarity between the reasoning, turns of phrase and writing style (yes all three!) of “Greg Jorgensen” and “Richard Buxton”. The probability that a person could be so mistaken as to make the same spurious arguments, and express them in such a similar way, as Buxton seems remote to say the least.

804
  1. ‘If you take out all the theatrics of the host and the goats …’

… there is no advantage to switching.

The first step in the solution to the MHP is to understand that the sum of probabilities gives 2/3 chance of the car being behind a different door after the contestant has made their first choice. Buxton, Jorgensen, Saltmarsh and Nobody appear to grasp this confidently. However, the next step is to understand the mechanics of how to choose ONE of these doors, since the contestant is only offered the chance to switch to ONE other door. It is essential that the contestant knows that the location of a goat (or a car, but that would make a boring TV show) is, or will be, revealed for them to gain any advantage by switching.

To argue that this need not be done by Monty revealing a goat is spurious. It has to be done by some means or other, and Monty’s goat is as good a way as any. It could be done by informing the contestant they can open both doors and choose the better (if any) of the two, or by pinning an honest notice to one saying ‘Goat’, or by Monty saying “If one of them’s a car I’ll leave it parked out the back with the keys in for you to collect”. Any way up, it is essential for the location of a goat to be revealed by some means. While this could be done another way, it is absolutely wrong to claim Monty’s goat reveal is irrelevant. The only value of such a claim could be as a smokescreen to obfuscate and cover up one’s inability to calculate that Monty’s reveal changes the remaining door’s chance of car from 1/3 to 2/3.

The answer to the following version of the MHP is “No”:
‘Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, says to you, “Do you want to pick a different door?” Is it to your advantage to switch your choice?’

805

That’s the same thing I was trying to convey. If there’s no reveal, then you still have 3 mystery doors and you’d be repeating the same mistake if you changed your mind regardless how many times…

Perhaps its just not coming out clear because everyone has a different way of understanding and explaining.

806
  1. ‘You want to switch instantly…this is a given.’
  2. ‘Moral of the story being that switching was always going to be smarter …’

Neither of those statements is true unless the goat reveal is executed.

I assert that the goat reveal results in the chance of car of the remaining door increasing from 1/3 to 2/3. That single door - not both doors, not two doors, just that one individual door. 2/3.

807

If there are 2 goats and 1 car, thus meaning youre more likely to pick a goat on the first try than a car, then why switching ASAP is already indicated.

Monty enables you to actually have something to switch to.

Honestly, humans have far too many different ways of describing things and other people dont understand because they use different words and concepts to convey them. I guess it is all irrelevant as long as we have our own way of understanding. I am just glad I was able to think about this problem long enough to understand the mechanism by which it works. Whether someone else understands it in pure mathematics or another means I guess is just as well.

808

@Jonathan I’m not using a pseudonym, I only came across this page yesterday. I haven’t posted before under a different name as you imply.

What’s remarkable, but not surprising, about reading the comments on this interesting problem is the number of people who insult, talk down to, and ridicule other people’s comments, for no reason or gain other than to make themselves feel smarter. Enjoy that glow.

No probabilities change during the game. From the outset one choice has 100% chance of winning a car, and two choices have 0% chance. The host knows where the car is, he is working with complete information. When the host reveals a goat that doesn’t somehow make the other door twice as likely to hide the car.

The contestant does not have full information: he or she knows that one door hides a car, and is allowed to choose one door with 1/3 chance of choosing the car. Then one of the other doors is shown to hide a goat – a fact that is known in advance from the rules of the game. At this point the contestant is allowed to abandon their intial choice and go with the remaining door if they want to.

The probability of the car hiding behind either any of the two unopened doors doesn’t change: it is still 100% likely to be behind one door and 0% likely to be behind another. The probability that the contestant chose the winning door on their first try remains 1/3, even though they have now seen a goat and one of the doors has effectively been taken out of the game.

The insight that the contestant may or may not have at this point is that the chances are still 1/3 that the door they chose at first hides the car, and therefore 2/3 that they didn’t, so switching is the best strategy.

I have not described a different game or changed the rules. All I have been writing about is that it’s logically equivalent for the host to allow the contestant to trade their first choice for the other two choices, without opening any doors. That would not be as entertaining on TV but it would give exactly the same outcome.

@Freddie et al, go back and read my comments and you’ll find that I never proposed a version of the game where the contestant was offered to trade their first choice for EITHER of the other two doors; that is obviously not the same game and switching would have no advantage.

If the interactive version of the game at the top of the page was changed so that after you made your first choice you could trade for to the other two, without seeing what was behind one of them, and it just told you whether you won or lost without revealing the goats, it would be the same game: sticking with your first choice would win 1/3 of the time, switching would win 2/3 of the time.

If anyone can explain without a barrage of insults and condescension exactly what relevant information the contestant learns by seeing the goat revealed, please do. By “relevant” I mean information that would affect the contestants decision to keep or switch. Seeing the goat is not the important thing; being offered the chance to abandon the initial choice for is the important thing, trading a 1/3 chance for a 2/3 chance.

809

“No probabilities change during the game.” Yes they do (from the player’s point of view which is all we’re interested in). At the outset each door had a 13 chance of hiding the car, when Monty opens a door with a goat behind that door now has a ZERO chance of hiding the car.

“When the host reveals a goat that doesn’t somehow make the other door twice as likely to hide the car.” Yes it does otherwise (since the opened door has a zero chance of hiding the car) it’d be 50/50 - which it isn’t. Your door has a 1/3 chance, the door Monty didn’t open has a 2/3 chance, twice as likely. And you even agree when you say “trading a 1/3 chance for a 2/3 chance.”

810

@PalmerEldritch Congrats on posting comment #800.

Possibly this is just a difference due to semantics.

The objective probabilities don’t change. From the outset one door has 100% chance of hiding the car, the other two have 0% chance. That remains true throughout the game.

At the beginning the contestant chooses a door with 1/3 chance of winning the car, and a 2/3 chance of not winning. After the host opens a door (always a door that has no objective chance of winning), the contestant’s odds are still 1/3 that they chose the correct door at first, and 2/3 that they didn’t. So no probabilities relevant to winning have changed.

The only question is whether the contestant realizes that they have a chance to trade their 1/3 chance for a 2/3 chance. Many people will stick to their initial hunch or intuition for psychological reasons, and that is reinforced by the wrong idea that seeing a goat behind one door has no changed the game to a 50/50 proposition.

This can be restated, as you have done, as if opening a losing door has given that door a 0/3 chance of winning and the other unchosen door a 2/3 chance of winning – that is logically equivalent (yields the same result), but it implies that the probabilities have somehow migrated from one door to another. What has actually happened is a theatrical removal of one choice to give the appearance that the contestant’s chances have changed from 1/3 to a coin toss, when actually their chances haven’t changed at all (still 1/3 chance they chose right).

The point I have been trying to make is that the contestant knows going into the game that one of the doors they don’t choose hides a goat, and that the host will show that door to them. Therefore no relevant information is gained when that happens, and after it happens the odds that the contestant chose the wrong door initially are still 2/3.

811

Many suspected Richard Buxton was posting under multiple IDs including “Saltmarsh” and “C.Bond”. And now “Greg Jorgensen”. Then Buxton slipped up and left no doubt that he and C.Bond were one and the same when under his Richard Buxton ID he corrected a post by “C.Bond” as if it were his own. Did others not notice the following?

712 C. Bond
"Acck!! Edit to comment 707, 2nd last paragraph.

Switching does not guarantee a win but increases your chances from 1/3 to 2/3, even when you know 1 of the 2/3 chances is a goat. You should know there must be a goat even before Monty shows you a goat."

Without wasting any time, Richard Buxton then chimed in with a comment decrying any claims that C.Bond was his creation:

712 Richard Buxton
"Well – this gives a lie to Mr Bond being the alter-ego of Buxton or Saltmarsh…

Mr Bond states…"

Now, it looks fishy enough that Buxton was so quick (eager) to point out that “C. Bond” could not be him by inserting his comment #712. But immediately after this comment, Buxton is equally quick (impatient) to clarify his comment at #711 under “C.Bond” and does so so hastily that he does not notice that his user name is still set as “Richard Buxton” (his hand then caught in the proverbial cookie jar lol #punny)!

At comment #713 as “Richard Buxton”, Buxton-Bond-(and, let’s face it)-Saltmarsh-Jorgensen corrects C.Bond’s comment at #711 as if it were his own when Buxton writes:

713 Richard Buxton
"Re 711 – correcting 707

even when you know 1 of the 2/3 chances is a goat.

SHOULD READ [capitalisation added]

even when you know 1 of the 1/3 chances is a goat"

How could “Buxton” be so presumptuous as to imply that he knows precisely what “C.Bond” really meant unless he were a mind reader (0% probability) or knew implicitly what C.Bond meant, ie because Richard Buxton and C.Bond are one and the same (highly likely).

Naturally C.Bond expressed neither affront nor gratitude for Buxton’s otherwise presumptuous meddling in his post because C.Bond and Buxton are one and the same and C.Bond could not cause his own good self to be grateful or affronted. Naturally “C.Bond” did not want to draw attention to Buxton’s slip up by acknowledging the correction - as would be expected if someone corrected someone else’s post as if it were there own (which of course it was), as an erratum, using words “xyz SHOULD READ” as opposed to taking the opportunity to point out the errors in counterpoint.

Add to that C.Bond is supposed to be in BC Canada 8 hours behind Buxton in Reading UK and yet their posts are often proximate to one another.

So there you have it folks; not only is Buxton capable of spurious arguments, the only way he can hope for his musings to be buttressed in any way is by clumsily invoking alter egos to lend their support. But he has woven a tangled web that has ensnared him one too many times.

He cannot conceal either his strange sense of logic or his pretentious writing style merely by changing his user name and it all falls apart when he forgets to reset the user name to match the alter ego he is using, as clearly happened in comment 713, above.

812

@Jonathan, I’m sure someone, somewhere, appreciates your close reading of the comments and sluething out people who post under multiple names. I don’t see how that matters, though.

I entered my real name and website (typicalprogrammer.com). I’m not hiding behind a first name only, or a character’s name from a Philip Dick novel.

Whatever Richard Buxton wrote, under however many names, has nothing to do with me. I’m not surprised that in the space of over 800 comments several people have made similar arguments. Nor am I surprised that writing styles are similar among people inclined to read and comment on threads like this. But, again, I don’t see how that matters unless you think ad hominem attacks reinforce your own arguments.

813

That may be a way of interpreting it, however…

If you had 1 million doors and thus nearly impossible to get it right on the first guess, why is it then true that you cannot actually make a meaningful switch until the bulk of the doors have been opened?

You already know there are 999,999 goats so that can’t be the reason. He has to eliminate 999,998 doors so that you actually have a door to switch to. Otherwise you would be clueless as to a valid swap door, and if permitted you’d be switching to all of those doors to find the car.

Its easy to visualize this but I’m having trouble explaining it in terms that reach the reader, possibly due to the way we analyze.

Rather, he shows the location of the maximum amount of goats that he can while leaving one door. If your first choice is a goat (most likely), then the car is in the remaining door which means his hand was forced to open all the other doors because otherwise he would reveal that car. Thus the last door is the car.

This problem scales with the amount of variables.

The greater the number of doors/variables, the more benefit you receive from switching. But you can’t switch to one million doors so he has to open every door but 1…The door with the car.

Run this problem with 100,000,000 doors …visualize and then tell me that you gain nothing from him opening all of the doors except for the car. Hes sorting them for you so you don’t have to guess which one of those other 999,999 doors the car is behind. Hes showing you which one to switch to. The switch is inevitable at this many doors…but you need a valid option. He can see behind, you can’t…He gives you a valid swap option. It seems different with more doors but its not…Its only more apparent.

814

Erratum to #802:

712 C. Bond
“Acck!! Edit to comment 707, 2nd last paragraph.

Should read:

“711 C. Bond”

ie, “C.Bond’s” post was #711, not #712.

I guess this is a case in point on how people only correct typos in their own posts, not other people’s.

815

I’ve always found the MHP interesting because of the psychological tricks it plays, and how it takes advantage of common misconceptions about probability and logic. I came across this posting yesterday after reading some other posts by Kalid, and chose to offer my thoughts on how the MHP problem is a logic problem, and the theatrics with goats is just entertainment.

Every time someone writes that the probabilities change they contribute to the confusion. The contestant has 1/3 chance of winning after making their first choice. They still have 1/3 chance of winning after the reveal if they choose to stick with their first choice. It follows that they have a 2/3 chance of winning if they switch. Kalid explained this in the article. The goat reveal’s only purpose is to give the contestant the impression that the game is now a coin toss, and play on the tendency of most people to stick with their initial hunch: once they’ve seen a goat they think they can be more confident in the first choice because one goat has been eliminated. In fact that was always going to happen, it doesn’t change the odds that they chose correctly in the first place.

I see other people have made the same argument, and I’ve read it on other articles on the Monty Hall problem. What I didn’t expect was to make some logical and friendly observations here, only to find that there’s a goat behind every door.

816

@Nemo, The contestant is not given the choice to switch to ANY of the other doors, whether there are 2 or 999,999. The contestant is given the choice to switch to ALL of the other doors. It’s a given that at least one (or at least 999,998 if you like) of those doors hides a goat.

After the contestant has made their initial choice the odds that they chose the winning door do not change. It’s always a 1 in N chance. Whether the host shows you some goats or not doesn’t change the odds you chose correctly at first. In the end you are offered the choice of keeping your 1/N choice or trading for the N-1/N choices, and you know that unless you already picked the winning door all but one of the remaining doors hides a goat and one hides the car.

817

First, a close examination of every post I have made would yield no results of insults to others. Yet it seems ok to personally attack me for explaining how I visualize the problem.

Second, how an analytical mind would miss the connection between my pseudonym which involves “The Variable Man” and a problem involving “variable change”…im not sure, nor the reason for attacking it except to try to reveal one’s identity and put them down to inflate themselves?

Lastly, if all is merely a theatrical spectacle, then I ask you to make the switch independent of him opening any doors. You cannot, because there would still be 2 doors with rotating variables. You need one of those doors to become a constant with a fixed identity. Until then, they are both in a quantum state of being car and goat. The objects in play are not in question…we know the objects, however what we don’t know is which door of the remaining 2.

“My first choice is probably wrong…id like to switch!”

“Great, which of the doors would you like to switch to?”

“Well, Monty there’s still another goat I could end up switching to…”

“Oh, don’t worry see look the goat is behind this door…Why don’t you use the last door as your swap door?”

Thanks Monty, I was about to pick that door as a swap choice before you opened it…good save…"

Now obviously he opens the door first and then asks you but that’s irrelevant since your choice is already flawed on arrival, requiring a switch.

I really don’t understand why its hard to miss that he actually provides a service by giving you a door to swap to. I think its because everyone is stuck on order of events rather than the actual dilemma of bad odds…

818

Jorgensen, your reasoning is remarkably like Buxton-Bond’s.

In the game as posed by Ms vos Savant, the contestant is given the option of sticking with their initial pick or switching to the other closed door.

Only you, Buxton-Bond and Saltmarsh (and he may be the same person) maintain the invitation is to switch to both doors.

For one, the invitation is only extended after the goat is revealed (after which only one other door remains an option) and is expressed in the problem as an invitation to switch from one door to the other.

If, like Buxton-Bond-Saltmarsh, you want to imagine the invitation is to switch from one door to two doors that is a matter for you. However, the invitation to switch is no more an invitation to switch to two doors than the invitation to pick one door in the first instance is an invitation to pick 2 doors rather than one.

I quoted how Ms Vos Savant postulated the Monty Hall problem, above, and restate it below so you might read it carefully and see there is no invitation other than to switch to one other door, not both doors:

"Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you,

“Do you want to pick door No. 2?” Is it to your advantage to switch your choice?"

Source: http://en.wikipedia.org/wiki/Monty_Hall_problem

See that? The host said: “Do you want to switch to DOOR NUMBER 2?”

The Host DID NOT say “Do you want to switch to doors 2 AND 3?”.

The effect might be the same, but that is only because the sum of the probability of door 2 concealing the car is the same as the sum of the probabilities of door 2 and 3. That is because adding door 3 adds nothing because the probability of 3 concealing the car is 0.

If the host were actually offering what Jorgensen and the discredited Buxton-Bond are saying, there would be no paradox or dilemma because everyone would quickly realise two doors are better than one.

It is precisely because the offer is made to choose between ONE door and ONE other door that the misunderstanding, that gave rise to the deliciously counter-intuitive problem and this formerly somewhat more enlightening thread, arose.

End of story, one would hope, unless perhaps if Jorgensen is a face-saving incarnation of Buxton-Bond-Saltmarsh.

As for ad hominem arguments or vitriol, anyone will struggle to find any offensive comments by me despite the abject frustration induced by the likes of Buxton-Bond-Saltmarsh and, more recently, Jorgensen, and which fly in the face of the facts and reason, no offence intended.

Now you don’t need to put that in your compiler, “Greg”, to see if it’s bug-free.

819

So far, judging by replies…no one even understands what I am saying.

If its theatrical, then tell Monty not to bother opening a door but rather just have him point to one of the remaining doors at random and switch to that one. See if you still get the car…

This has schroedingers cat written all over it…

The non-picked doors both have an equal probability of being a goat and a car since that’s what remains. How can you guarantee a car without confirming which of the non-picked doors has the second goat? (Assuming yours indeed has the first)

This is a paradox…he has to reveal something or you’d still be staring at 3 blank doors with no new information than you had when arriving on the scene.

Im a little frustrated at the fact that no one realizes there are 2 other doors you could switch to UNLESS he stops you/shows you why you shouldn’t switch to a certain one.

The switch is already a given just from the odds, so why do you have to wait for him to do anything unless it actually benefits you…Why not just switch immediately on arrival and win a car without him showing you anything whatsoever?

820

@Greg

“Every time someone writes that the probabilities change they contribute to the confusion”. I fail to see why, since the (subjective) probabilities do change. The door that Monty opens now has a 0/3 chance of hiding the goat, whereas before it had a 1/3 chance; the door that Monty didn’t open now has a 2/3 chance of hiding the car whereas before it had a 1/3 chance. How you can dispute that simple FACT is beyond me.
What contributes to the confusion is changing the problem by saying the offer is to exchange 1 door for 2 doors.