Talk:Monty Hall problem/Archive 2

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I've moved the existing talk page to Talk:Monty Hall problem/Archive1, so the edit history is now with the archive page. I've copied back the most recent thread. Hope this helps, Wile E. Heresiarch 04:40, 10 Aug 2004 (UTC)

Why the probability is 2/3 in fewer words

Recall that the host has opened a door, and we know that door contains a goat. The contestant was either RIGHT or WRONG in the first place. In the former case, we switch to a goat and lose. In the latter case, switching guarantees a victory: we picked Goat 1 originally, and Monty has released Goat 2. So the only probability we need to know is that of being right or wrong in the first place, which are 1/3 and 2/3 respectively. 1/3 chance you will lose if you switch, 2/3 chance you will win.

Just a stupid question. Suppose you have the three doors

Door 1: Selected Door 2: Opened(goat of course) Door 3: Unknown

Since the contestant knows behind 2 doors is a goat, the Goat One from the opened door could be romoved. That leaves a fifty-fifty percent chance that he/she is correct in thier guess, so switching would not increase it by 2/3 because there are only 2 unkown doors left. I've proabably not followed this enough so tell me why I'm wrong soon please.

The point is that opening the door is not done randomly, and leaves the two doors with unequal probabilities. If a random door were opened, the car might sometime be revealed. This is not what happens. Opening the door has no effect on the probability of the initial choice (1/3). Since the probability must sum to 1, and there's only one other door, the probability of that door must be 2/3. Please try the experiment with cards suggested in the article, and elsewhere on this talk page. -- Rick Block (talk) 21:48, July 24, 2005 (UTC)

How does it leave 2 doors with unequal probabilities? Of two doors left, one has to have the goat and one has to have the car, becuase the host always opens up the goat.

Ah, but just because there are two doors, one with a goat and the other with the car doesn't mean there's a 50/50 chance. For example, if you shuffle a deck of cards, give me one, look at the rest and discard 50 of them that aren't the ace of spades there are two cards left. These two cards do not have a 50/50 chance of being the ace of spades (and if you think they do have a 50/50 chance please try this about 10 times and see how it works out). The same effect is at work here. When you choose a door, Monty effectively has the other two. He knows what's behind them (like looking at the cards) and opens one (like discarding a card). Opening a door when you know what's behind it (or discarding a card you can see) doesn't change the initial probabilities. When you first pick, it's 1/3. It's still 1/3 that you picked the car after the door is opened. There's a 100% chance the car is behind one of the (now) two doors, so it has to be 2/3 behind the other one. One way to look at it is if you didn't initially pick the door with the car (2/3 chance), by opening a door Monty is showing you where the car is. This is just like the card experiment, and using 52 cards makes it much more obvious. After dealing one to me, if you discard 50 that aren't the ace of spades and you have the ace (which you will with a probability of 51/52) you've shown me which card it is. -- Rick Block (talk) 01:20, July 25, 2005 (UTC)

I still don't really understand because this is different from a deck of cards. A deck of cards cannot be compared to this because a deck of cards has: a) more choices b) be more then one, say door.

If I put up fifty doors, and I had, lets just say, ten variations, four suites, then there is a much different probabilty then knowing that threre are only two doors, of which on is the goat and one is the car. The problem does not need the third door any more. If, say you had 34 aces of spades, and 17 aces of hearts you still would not get the same equation because it is mroe variables. You know there is a fifty fifty percent chance the door that you first seleceted has the car because the host opens a goat. You either have or don't have the goat, so fifty-fifty chance. Now if you siwtched, it would not imrvoe your chances.

If the host opens the door first, then you pick, there's a 50-50 chance. When you pick first, and then the host opens a door it's a different problem. Do you agree it's the same with only 3 cards (one being the ace of spades)? Deal one to me, you look at the other two and discard one that isn't the ace of spades. I think you're saying it's now 50-50 who has the ace. Right? Why is it any different if you use the entire deck, deal one to me, you keep the rest, and discard 50 that aren't the ace of spades? I think your argument about the doors applies to the cards (at the end there are two left, the ace must be one of them, I either have or don't have the ace, so fifty-fifty chance). The point of making it 52 cards instead of 3 doors is because this is something you can easily try, and the odds are so skewed in this case that trying it, say, 10 times will make it very obvious that it's not 50-50. If you agree it's the same with 3 cards, and you have the patience, you could try this maybe 90 times and keep track of who ends up with the ace. If the probability is 50-50 you'll end up with the ace "close" to 45 times (maybe 40, maybe 50). If the probability is 1/3-2/3 you'll end up with the ace "close" to 60 times (maybe 55, mayber 65). -- Rick Block (talk) 23:33, July 25, 2005 (UTC)

But wait... if you discard a card forever, now there will be only two cards left making it one out of two.

IMHO - Just Mathematics - probability is 1/2

Bayes' Theorem Approach in the main page:

P(C2|03)= P(03|C2)*P(C2) / P(03) (True)

But

P(C1|03)= P(03|C1)*P(C1) / P(03) (True also)

and what do we have: if P(C2|O3)=1*1/3 / P(03) were 2/3, in the same way P(C1|03)= 1*1/3 / P(03) = 2/3. Summ is greater than 1. Impossible!

Let's proove that P(C2|03)= P(C1|O3)= 1/2

P(C2|O3)-P(C1|O3) = (P(03|C2)*P(C2)-P(03|C1)*P(C1)/P(03))=0

because P(O3|C2)=P(O3|C1)=1; P(C2)=P(C1)=1/3

so P(C2|O3)=P(C1|O3)

What's wrong?

P(C2|03)= P(03|C2)*P(C2) / P(03) = 1*1/3 / P(03),

but P(03) is not 1/2 as it says on the main page. Imho it is 2/3.

P(03)=P(E3)*1+P(C3)*0 (~)

where P(E3) - 3rd is empty, if it is, so Monty opens (1) it

if there is a car (P(C3)), Monty doesn't (0) open it.

and P(E3)=2/3, just because we are in the begining of the show and no box is opened yet.

so P(C2|O3)=1/2=P(C1|03)

Am I right or Am I right or Am I right?

The only problem here can be only in (~). Am not really good at mathematicain english to proove now that it's okay. But can

try to do it later.

Best regards, Fade <fade@list.ru>

--213.148.18.178 10:57, 17 Jan 2005 (UTC)

Except do not forget, Monty's choice is constrained, since he MUST open a door revealing a goat, in the above P(O3) = 1, not 1/2. I have deleted the sequel, as it is superfluous given my previous statement. -Dwee I have restored that text, since it's easier to see where the mistakes in a text are when you can actually look at the text. -- Antaeus Feldspar 02:54, 7 Jun 2005 (UTC)

two envelope problem

I was talking to a friend about the Monty Hall problem today and he told me about a similar problem, and I don't think there is a Wikipedia article on it and I'm not sure how to present the solution either, but anyway here is the problem:

You are on a gameshow and the host holds out two envelopes for you to choose from A and B. So you choose an envelope (A) and it's got $2000 in it. The presenter then says that one of the envelopes has twice as much money in it as the other one and offers you the chance to switch. So you think about it this way... "If I switch I will go home with either $4000 or $1000, by not switching I will go home with $2000. There is a 50/50 chance that I will double my money by switching. A normal 50/50 bet results in me either doubling my money or losing it all, whereas here I will only lose half. Therefore this is a better than evens bet so I will make the swap." You are just about to swap envelopes when you think about the problem some more - "Surely this can't be right... ". Mintguy (T) 16:13, 15 Jul 2004 (UTC)

Interesting, but not really similar. In the Monty Hall problem, there's just one possible positive payoff, so the only concern is maximizing your chance of getting it. This problem is more complicated. It is true that if you switch, the expected value of the new envelope is (1000*0.5 + 4000*0.5) = 2,500, so if all you care about is the average amount of money you'll take home, you should switch. However, most people in the real world are risk averse, meaning that they may prefer the sure 2,000. Isomorphic 02:58, 16 Jul 2004 (UTC)

Yeah but you're wrong, you see, there is no advantage to switching. How can there be? . If you didn't know how much money was in the envelope, then you might make the same analysis and switch, but then after switching the same analysis would lead you to switch again.Mintguy (T) 07:41, 16 Jul 2004 (UTC)

After I open the first envelope and choose to switch, but before I open the second, here is my position: The other envelope has $2,000. The one I'm holding has 1,000 or 4,000 with (we are assuming) equal probability. My expectation if I switch back is $500 less than if I hold. I'm holding. Dandrake 18:56, Aug 26, 2004 (UTC)

What do you mean, "there's only one positive payoff"? Isn't the contestant given the goat behind their door?!? :) -- tmegapscm 2005-07-23

[Tired of nesting the blocks deeper and deeper, like bad C code.] Let's pause to list assumptions. My choice of an envelope is not correlated with the loading of the envelopes, so that I'm equally likely to have the good or the bad envelope. The host also is statistically unbiased: his telling me the news is not correeated with my initial choice of good or bad envelope. Oh, and he's telling the truth.

This is not to say that the expectation argument is right. This is the paradoxical part: that the argument has no apparent flaw in itself, but it gives the nonsensical result that one should choose and then change, even though no new information has come along to cause a change. The host's information is new, or seems to be, but how would my course of action be different if I had known it all along? Bottom line: the expectation argument leads to a silly result, but I don't believe that its flaw has been shown. Maybe this paradox deserves its own article. Dandrake 19:35, Aug 26, 2004 (UTC)

I think I'm not understanding this problem correctly. The way I read it, you either pick an envelope with X or 2X dollars in it with equal probability. Given the option to switch, this expands into four cases

	Picked X	Picked 2X
Keep	X	2X
Switch	2X	X

Doesn't this mean that each of these events occurs with equal probablity, and that it doesn't matter? I understand the expectation argument, but I can't reconcile it with this simple grid. Cvaneg 23:13, 26 Aug 2004 (UTC)

This problem now has its own page at Envelope paradox Mintguy (T)

I'm copying this to talk:Envelope paradox. Further discussion of this subject should continue there, but I will also leave the text above so that people can follow on from here. Mintguy (T)

Removed Assumptions

I decided to be a little bold and remove the assumptions section. If you read the problem as stated, then you don't need to make any of the assumptions listed. E.g., it doesn't matter whether or not you assume "Monty always opens a door," or that there is "always a goat behind the door Monty opens," because the problem clearly states that Monty opens a door and reveals a goat. That is all you need to know in order to determine the correct answer. Other points in this section also showed an incorrect understanding of the problem. --Simoes 15:53, 24 Jul 2004 (UTC)

No, you entirely misunderstand. He is known to open the door this time, as you say, but calculating the probability requires knowledge, or assumption, about what happens in general. What, for instance, was the probability of picking an Ace out of a deck of cards, if you know that I took one, and it turned out to be an Ace? Maybe you think it's 1 in 13, but in fact I was using a pinochle deck. The specific probability calculation requires general knowledge of the conditions. For more detail, showing how the assumption is necesary for this problem, read the assumptions section, which I am restoring to its rightful place. However, if it is unclear, let us by all means fix it. Also, you have not addressed others, such as the obvious matter of the value of a goat. Dandrake 22:31, Jul 24, 2004 (UTC)

Monty's knowledge is critical to the problem

It's critically important to the problem that Monty knows what's behind each door. In the actual correct problem situation, there are three scenarios for what's behind each door:

Scenario	Door 1	Door 2	Door 3
Scenario A	Car	Goat	Goat
Scenario B	Goat	Car	Goat
Scenario C	Goat	Goat	Car

Let's presume that the player always first chooses Door 1 (a safe assumption, since we can always just choose how the doors are numbered after the door is chosen.) If we assume that Monty knows what's behind all doors, and opens the door he does because it has a goat behind it, then our three scenarios look like this (with the revealed goat crossed out because the player knows to avoid it):

Scenario	Player's first choice	Door 2	Door 3
Scenario A	Car	Goat	Goat
Scenario B	Goat	Car	Goat
Scenario C	Goat	Goat	Car

If on the other hand we leave the problem too vague, by just saying "Monty opens a door that has a goat behind it", we may create the impression that it is just chance that, this time, the door had a goat behind it -- and this changes the problem completely. Because it simply makes it look as if some of our original scenarios could not have happened:

Scenario	Player's first choice	Door Monty opens	Door 3
Scenario A	Car	Goat	Goat
Scenario B	Goat	Car	Goat
Scenario C	Goat	Goat	Car

The faulty logic thus goes "If Scenario B had been correct, there would have been a car behind the door Monty opened; but there wasn't, so therefore Scenario B is not correct. The only remaining scenarios are A, where the player's first choice contains the car, and C, where the player's first choice contains the goat; therefore the chances are 50/50 whether the player switches or stays." -- Antaeus Feldspar 20:44, 21 Nov 2004 (UTC)

Standard Solution is Illogical

The standard solution seems illogical for 2 reasons. First, probabilities describe chances, and since Monty has removed one of the uncertainty by opening up a door, he changes the conditions which the original set of probabilities described. This is to say, the 2/3 probability should not be transferrable, and the new probability designed to describe the new conditions should state that the car has a 50-50 chance of being behind either of the unopened door. Second, say the standard solution is logical, and say the car is behind door 1, and the player will always choose between door 1 or 2, and Monty will always open up door 3: if the player had originally chosen door 2, then he should change his selection to door 1; if the player had originally chosen door 1, then he should change his choice to door 2: the results of the 1st situation contradicts that of the 2nd, and vice versa; the contradiction shows that the standard solution is wrong, and it doesn't make a difference whether to stick to the original choice or not.

There's no contradiction. Door 1 and Door 2 are just labels, without any further information they are possesed of exactly the same qualities. Whichever door he chooses he should switch. Swap the labels on the doors and the solution is the same. Once more, by far the simplest way to look at the problem is to recognize that if the player randomly chooses a goat, and switches, he will win the car every time, and he has a 2/3 chace of choosing a goat. Now I wish someone could explain the two envelope problem to me. Mintguy (T) 17:22, 25 Aug 2004 (UTC)

I'd love to explain the two envelope problem to you, but the author of that article won't let me, as it seems. :-( --INic 22:37, 14 Jan 2005 (UTC)

Remove "assumptions" and rewording for clarity

I found the section on "Assumptions" to be a little confusing and to obscure the true spirit of the puzzle/problem. I have therefore removed that section and reworded the problem in the introduction to make it clearer that the host knows what is behind the doors and that he always opens a door with a goat. Then the assumptions are redundant and the intentions of the host, benign or otherwise, are not a factor. I think it is obvious that the contestant will want a car, not a goat, and there is no need to state this. I have also reworded the answer to try to make it clearer and more concise.

P S K --213.122.39.201 17:00, 25 Oct 2004 (UTC)

I'm in agreement with the changes you made; the article was wandering into original research on variations of the problem. I think your recent edits have substantially tightened and clarified the article. By the way it would be terrific if you would make a user name and log in. Regards & happy editing, Wile E. Heresiarch 21:40, 25 Oct 2004 (UTC)

Hi P S K, you made the much needed changes to one of our important articles. -- Sundar 04:40, Oct 26, 2004 (UTC)

A clearer explanation?

I have written up what I think is a clearer explanation of the solution, at Monty Hall problem/temp1. I'd like others' feedback, to find out if it's as clear as I think it is and can go in the article. -- Antaeus Feldspar 01:38, 1 Dec 2004 (UTC)

Everyone's making a mountain out of a molehill, or a Lexus out of a goat

The Monty Hall problem is not a problem in people's skill at using probability and mathematics. It is not a problem of gullible people falling prey to tricks. It is not even a paradox. It is a problem in communication and rhetoric.

As commonly stated, the Monty Hall problem is ambiguous as far as how the host chooses the second door.

Whether information about the third door can be inferred by the host's actions, depends on whether the host uses information about what's behind the third door to make his decision.

If the host chooses the second door randomly among the two doors not already selected by the player, and it has a goat behind it, then there is no advantage to the player switching.

If the host chooses the second door because it has a goat behind it, then the probability that the third door has a car behind it jumps to 2/3, and the player should switch.

If someone were to ask me the Monty Hall problem, and it were not clear whether the host used information from behind the third door (or, equivalently, information from behind both the first and second doors) to make his decision, then I would come back and say that the problem was ill-defined. Neither of the two common solutions would be acceptable.

Actually, a third scenario is possible: If the host chooses the second door because the third door contains a goat, then the player should never switch.

The point is that you cannot infer information about the third door unless you can infer information about the host's decision-making process. Unless this is spelled out clearly (which it most often isn't), the problem is ill-defined.

Neither side in this "debate" has done bad mathematics, and mathematicians who righteously scoff at those who express contrary views, calling it bad mathematics, or mathematical illiteracy, add nothing to our understanding.

I'd like a humanities professor's analysis of this problem, because it is not fundamentally mathematical. I'd like to see a detailed analysis of all of the different precise phrasings of this problem, and how they parse against accepted English grammar. What assumptions are socially "valid"?

An analysis of rhetoric is more important than an analysis of probability, in this problem.

(FWIW, I'm a computer scientist who writes mathematical software.)

Sorry, but I can't force myself to agree wtih your solutution. When the host opens the door he not only skips over the dorr you choose, but the other door. There is a 2/3 chance it was the goat...right? But no remove a goat, just remove the door the host opened. You would not, in your right mind, choose a goat against a car. Now there are two doors left. One is your door with car or goat, and other a door with same. The problem with most of the tables and charts here are that they use goat twice in two rows, making it two chance. Now, goat one and two are interchangable:

Player Chooses Goat One:Remaining Choice: Car

Plyaer Chooses Car: Remaining Choice: Car

You should not add:

Player Chooses Goat 2: Remaining Choice Car

That would not happen. There are three doors, of which one of them is open, which goat DOES NOT MATTER EVERYONE. There will always be two doors left open, not matter if the players chooses goat one or two. There will alwys be a car and a goat remaining. If the player choose goat one and goat 2 is opened, you can't make another choice be player chooses goat two and goat one is opened. With this amount of doors they result in the smae thing unless you did it more often. Try it.

Door One: Door 2: Door 3: Goat Goat Car

Now to help you if you still don't get it: Door One or Door two can be opened and reslult in the smae outcome. Which door they open does not matter. Goat

Ah, thank you very much. Your explanation made this clear to me. -- PenguinPirate 23:45, 24 Jan 2005 (UTC)

Very good explanation. Probability applies to a theoretical problem abstracted away from the Monty Hall Problem itself. But, it would be really interesting to have humanities' point of analysis. -- Sundar 09:35, Jan 25, 2005 (UTC)

Yep, it's impossible to reason about probability after the fact without some prior assumptions about the process. It's like someone who made a bet with a gambler and found that the coin came up 100 heads in a roll and claim that the coin is biased. It's entirely possible that a fair coin produced those outcomes, it's just as likely as any other sequence.

Simple Problem

It has nothing to do with what Monty knows. Like the solution states, there's a 2/3 probability that the car is behind one of the doors you didn't pick. It doesn't matter if the doors are purple, have aliens behind them, are invisible or any of that other nonesense. There's a 2/3 probability that the car is behind one of the doors you didn't originally pick.

Wrong. The fact that Monty knows is vital to the problem.

Let's define whichever door the player picks as door P, whichever door Monty picks as door M, and whichever door is left over as door L. For clarity, let's also distinguish all three values for what's behind the door: C for the car and G1 and G2 for the goats.

Now, just looking at prizes and doors, and not about the constraints of the problem, there are six possibilities for how three prizes can be arranged one-to-one behind three doors:

P	M	U
C	G1	G2
C	G2	G1
G1	C	G2
G1	G2	C
G2	C	G1
G2	G1	C

Now, if we add the constraint that Monty knows what is behind each door, we can see that even if the player chooses wrongly at first and leaves Monty both a door with a car and a door with a goat, Monty can always choose a door with a goat:

P	M	U
C	G1	G2
C	G2	G1
G1	G2	C
G1	G2	C
G2	G1	C
G2	G1	C

But if you establish that Monty does not know what is behind each door, and then say that he picks a door with a goat behind it, there is only one way to interpret that: that there are six possibilities for how the prizes are distributed behind the doors, but we are only considering those four where the condition that "Monty's door has a goat behind it" is true:

P	M	U
C	G1	G2
C	G2	G1
G1	C	G2
G1	G2	C
G2	C	G1
G2	G1	C

Just as in the old Lewis Carroll problem about "a regular coin and a double-headed coin are tossed in a bag; you take out one coin and see a heads; what is the chance that the other side is heads?" if we say that Monty does not know which door contains the car but somehow always avoids it, we can only conclude that we are artificially removing those possible outcomes where he did choose a car. This leaves us with the unsurprising result that if the player receives no information whatsoever about what is behind any of the doors, his chances stay the same no matter which door he picks. It is only because Monty knows what is behind each door, and eliminates a false choice for the player, that switching becomes an option more likely to be rewarded with success. -- Antaeus Feldspar 08:22, 24 Jan 2005 (UTC)

Could somebody answer this question, please?

I realize that I am probably wrong. I mean, if this phenomenon weren't mathematically solid it wouldn't have survived for this long, but something just occurred to me.

How can my picking a certain door affect it's probability?

Consider the following:

It's the exact same set up, with three doors etc. etc. In my mind I pick door 3, but I tell the host I pick door 2 and he believes me. In his mind I've picked door 2. He opens door number 1 and it is a goat.

Which door has the two third probability and which has the one third?

How can the exact same scenario bring up two probabilities based on the door I "picked?" Picking out one door in my mind doesn't mathematically substantiate anything as it has no relevance to anything mathematically.

A good question. The answer is that the significance of "picking" a door and communicating that pick to the host, is that the host will not give you any information about that door. In your example, you say that you pick door 3 in your mind, but tell the host you pick 2, and he opens door 1 to reveal a goat. This could happen, but it could also happen that he opens door 3 to reveal a goat; he has no way of knowing that you "really" picked door 3. But whichever door you tell him you picked, that has the effect that the host will not give you information that affects the probability of what's behind that door. In the example you give, if it is door number 1 the host opens, the door with the 2/3rds probability must be door 3, and the door with the 1/3rd probability is door 2, because the host thought you picked that one. -- Antaeus Feldspar 05:08, 3 Feb 2005 (UTC)

Disputed: It Is 50/50

The problem is not asking us to plot out an entire course of action beforehand. At the point that the decision in question is being made, Monty has eliminated one of the initial possibilities and presented the contestant with an essentially new problem: There is a car behind one of these two doors, pick one. Anything that has gone before is irrelevant. Referring to the decision as "switching" is a red herring.

Please disprove this before reverting the article. -- El Mariachi

It already has been disproved, but you don't seem to be following it. The crux of your misunderstanding seems to be the mistaken notion that "Once Monty has revealed one of the goats, the contestant's initial door choice and the probablilities therein become irrelevant, just as the results of previous coin flips are irrelevant to any future coin flips." The second part is correct: the results of previous coin flips are irrelevant to any future coin flips; the outcome of one random event does not influence the outcome of future random events.

But this has no bearing at all on the problem, because there isn't a second "coin flip": there is exactly one random event in the problem, which is the distribution of car and goats behind doors in the first place. Suppose you were given a set of N doors, and were told that there were either goats behind them all, or cars behind them all. You open one door and find a goat, but you have N-1 doors still left. Which is true: The rest of the doors might have cars behind them, because once you've opened one of the doors, what's behind the rest is "an essentially new problem"? Or that opening one of the doors gives you more knowledge about what's behind the rest, because it isn't? -- Antaeus Feldspar 01:30, 10 Feb 2005 (UTC)

Since the contestant doesn't know what's behind each door, the door choices are completely arbitrary and thus effectively random.

Initially, each door has a 1/3 chance of being the one with the car. As soon as Monty reveals the goat, the chance of the car being behind that door drops to zero. The probability is now distributed evenly between the two remaining doors -- neither door has more of a claim on the orphaned 1/3 than the other. (Your example of all cars vs. all goats is a false analogy.) By your logic, if there had initially been 100 doors and 99 goats, and Monty revealed 98 of the goats after the first pick, the probability that the car is behind the unpicked door would be 99%.

The situation at hand is this simple: A contestant is standing before two doors being asked to pick one. All else is misdirection. -- El Mariachi

OK, try this. Make a 10-by-10 grid. On the first line, put a "C" in the first square and "G" in all the others on that line. On the second line, put the "C" in the second square and "G" in all the rest . On the third line, put the "C" in the third square, and so on for all ten lines.

Now draw a nice thick line between the first column and all the other columns. The first column is what's behind the door the player first picked. As expected, only one time in ten is the car behind that door.

Now we simulate the host's opening of doors. The host does one of two things, depending on whether the player picked the car on their first try:

• If the player picked the car on the first try, the host reveals eight goats, leaving one door which also contains a goat. Cross off eight of the nine goats on the right side of the thick line.
• If the player picked a goat on the first try, the host reveals eight goats, leaving the one door on the right of the first column that contains the car. Cross out the eight goats on the right side of the thick line.

Now look at your ten lines -- the ten different possibilities for how the goats and car were originally distributed. In each of those ten cases, there are two doors left, the one that the player first picked, and one other; however, only in one of those cases does the door the player first picked have the car behind it. To say "there's only two possibilities: car behind the player's door, or goat behind the player's door, and therefore those possibilities are equal in probability" is clearly false when there are clearly nine ways to get "goat behind player's door" and only one to get "car behind player's door". -- Antaeus Feldspar 08:32, 10 Feb 2005 (UTC)

Your grid example, as well as the "empirical" solution, persist in the illusion that the contestant's initial choice has any bearing. Let me introduce a parallel problem to demonstrate why it does not:

The parallel problem is identical to the original except that the initial choice is removed altogether. The contestant stands silently before three doors, one of which conceals a car, the other two goats. Monty Hall opens one of the doors to reveal a goat. The contestant is now asked to pick one of the remaining doors. Would you agree that this contestant has a 50% chance of picking the car?

The parallel contestant's situation at that point is absolutely identical to the original contestant's (excepting psychological nuances.) So the original contestant's chance of picking correctly is also 50%.

(As an aside, one assumption in the stated question of "should you switch?" has not been mentioned yet: that the car is more desirable than the goat. Perhaps the contestant lives on a small island or a remote mountain village inaccessible by road. Maybe the goat is a valuable rare breed such as the Tennessee Fainting Goat and the car is a Yugo or something. But since the chances are 50/50 this really doesn't matter.) -- El Mariachi

I strongly suspect at this point, more than before, that you are trolling, given especially the pointless digression about whether perhaps the goat is the truly desirable prize. But even if you are insincere, we may still get useful means of explaining the counter-intuitive nature of the result -- why the chances of winning by switching in the problem as given are in fact 2/3 and why empirical testing bears this out.

The contestant's initial choice clearly has a bearing; it identifies the one door about which the host will not reveal any information. No matter how many doors there are (call this N), the host will open all but two of them. The host will always leave the door the player picked unopened, for the reason that the player picked it. However, the other door has a N-1/N chance of remaining unopened because it's the door that the car is behind. There is only a 1/N chance that it remains unopened even though it doesn't have the car. -- Antaeus Feldspar 07:11, 11 Feb 2005 (UTC)

(I was sort of kidding about the Tennessee Fainting Goat, but I am quite sincere about the rest.)

The only thing the initial choice even has a possibility of affecting is which goat Monty reveals. But the goats are interchangeable, so that makes no difference. There will always be two doors about which no information is revealed. There will always be a goat behind one and a car behind the other. You are correct that there is a 1/N chance, but N = 2 after all but one of the goat doors has been eliminated. This seems to be where your misunderstanding lies. N is not a constant, N is the number of doors still available to choose from.

You need to show how the choice presented in the original problem differs from that in the parallel I gave previously. -- El Mariachi

Nope, that's already been shown and you refuse to accept it. There is only one random event, and that is the initial distribution of 1 car and N-1 goats behind N doors. There are N possible such distributions; N-1 of them result in a goat behind the player's door and only 1 results in the car behind the player's door. Just because you reduce the number of unopened doors to two does not mean that you have somehow re-randomized the probabilities to one-in-two; the random event has already happened. -- Antaeus Feldspar 01:24, 12 Feb 2005 (UTC)

No, it has not been shown. The player's choices are effectively random events because they are arbitrary. Probability-wise, it makes no difference whether the door is chosen by the player or determined by a die roll. You already admit that the goat revelation changes the remaining probabilities, otherwise you'd have to be arguing that the revealed goat still has a 1/3 chance of actually being a car. I repeat: You need to show how the choice presented in the original problem differs from that in the parallel I gave earlier. Do that and you'll have proved your case. -- El Mariachi

Our case has been proven, all you've asked for us to explain has been explained, and at this point you are trolling. Don't bother. -- Antaeus Feldspar 18:43, 12 Feb 2005 (UTC)

This is the most counter-intuitive solution and hence hard to accept. In order convince ourselves to try to see the merit in the solution see Empirical solution of the Monty Hall problem. -- Sundar 04:55, Feb 10, 2005 (UTC)

One last attempt to explain

Here's one last attempt to explain -- not to trolls, but to anyone who still honestly doesn't see why the probability works out as it does:

Player A and Player B take the 13 diamond cards out of a standard deck of cards. The cards are shuffled, and then Player A receives one card face-down, which he may not look at. Player B gets the other 12 cards, and he may look at them. Both players are trying to wind up with the Ace of Diamonds in their hand.

Question: Player A received one card. Player B received twelve cards. What are the chances that the Ace is currently in Player B's hand? Answer: Twelve out of thirteen.

Player B has twelve cards which he can look at. At least eleven of them are not the Ace. Player B takes out eleven non-Ace cards from his hand and lays them down face-up.

Question: Player B did not discard the Ace. No cards have moved from one hand to the other. Therefore, if the Ace was in Player B's hand at the beginning, it is still there now. What are the chances that the Ace is currently in Player B's hand? Answer: Twelve out of thirteen.

Player A now has an option: he can stay with his current hand, with the one card he was originally dealt, or he can switch his hand with Player B's hand, which was originally dealt twelve of the thirteen cards.

Question: If the Ace is currently in Player B's hand, Player A will win by switching hands. What are the chances that Player A will win by switching hands? Answer: Twelve out of thirteen.

The only differences between this problem and the canonical Monty Hall problem is the scale (13 instead of 3), the objects (doors to be opened instead of cards to be turned over) and the fact that Player A has his initial card/door assigned, rather than selecting it himself. None of these factors makes a difference to the central point, that the chances are better of winning by switching. The only one that comes close is Player A being assigned his initial choice, and since it violates the problem rules to posit that Player A could somehow pick the winning door/card initially at a rate better than chance, it does not make a difference.

Anyone who still believes that once Player B has discarded eleven non-Ace cards from his hand, the chance that he's still holding the Ace drops from 12/13 to 1/2 -- come and see me in person, and we'll take the diamonds out of a deck and play, and every time the Ace is in Player B's hand, you pay me $100, and every time it's in Player A's hand, I'll pay you $200. And I'll be merciful and quit after I've taken $1000 off you. =) -- Antaeus Feldspar 20:45, 13 Feb 2005 (UTC)

What happens if Player A gets none of the cards before 11 non-Aces are discarded, and must then choose between the two remaining? Are his odds in that case 50/50? -- El Mariachi

That would be a completely different problem, since the randomizing event would be happening after the number of cards has been reduced from 13 to 2, instead of before. It has no bearing on the problem at hand. -- Antaeus Feldspar 01:00, 16 Feb 2005 (UTC)

You left out the "yes." But which event are you referring to? In both problems, the cards have been laid out, after which Player A has his pick. The only difference is that in one version, the cards are further away from each other. You're arguing that the act of sliding a card face-down across a table somehow lowers its chances of being an Ace.

When the preconditions of the problem clearly spell out for everyone to see that the only cards which get slid face-down across the table are those which are not Aces, then yes, the chances that the eleven cards which are slid face-down across the table are not Aces is in fact 100%.

I was talking about the one card slid across the table to Player A. As long as cards are face-down, they have a chance of being the Ace. The eleven cards must be revealed before they can be discarded with any certainty that they aren't the Ace. -- El Mariachi

I figured out the missing link in both our arguments, by the way: Both problems contain a random event with a forced result. The chances of flipping three heads in a row is (1/2)*(1/2)*(1/2) = 1/8, right? Now, when you have already landed the first two heads, what is the chance of flipping a head a third time? You would have to answer 1/8 or 7/8 to remain consistent with your answers to the Monty problem and the thirteen diamonds problem, but I think you know it's 1/2. The remaining head side doesn't get preferential treatment simply due to being grouped with the others, just as Player B's card doesn't get any from having been grouped with the discarded 11.

It may help to consider this: All of the problems have suppositions (the forced results) that can be rephrased as "ifs" without changing the correct outcome. Contestant picks a door, then Monty opens one of the remaining doors; if that door reveals a goat... Player B discards 11 cards; if none of those are the Ace... You flip three coins; if the first two come up heads... In each case, the "if" makes it clear that the problem has presupposed a specific outcome to a random event. Please try to be open about this and run through the problems using the "if" phrasing. -- El Mariachi

No. You are trolling. Anyone who was actually trying to understand the problem, instead of just having fun at everyone else's expense, would have read the problem and the various explanations and would already understand that Monty opens the door because he knows it contains a goat; the eleven cards are discarded because they are known to be not the Ace. You are trolling. -- Antaeus Feldspar 23:58, 17 Feb 2005 (UTC)

The statements "Monty reveals a goat," "Player B discards 11 non-Aces," and "two coins have already come up heads" all serve the same purpose. You can't specify any of those events without affecting the resultant probabilities.

Incorrect. "Two coins have already come up heads", as you yourself pointed out, does not affect the outcome of future coin flips.

However, as has already been explained to you, there is no relation between this principle and the problem at hand, since there is only one random event, and that is the one and only distribution of cards between hands/one separation of doors between the set of "door the player chose" and "doors the player didn't choose".

It doesn't matter whether Monty knew there was a goat behind his door; the problem has established it as a fact that there was.

It makes all the difference in the world. Why do you ignore things that have been clearly explained numerous times and then pretend you're innocent of trolling? If the problem was "Monty opens a door at random from the two left, and happens to find a goat", then obviously we are dealing with a subset of all the possible outcomes of the initial distribution; we ignore those outcomes which would have an ignorant Monty opening a door to reveal a car. But the problem specifies that Monty knows where the goats are, and deliberately reveals a goat; the only outcomes of the distribution that we could eliminate are those where Monty has no doors that he can open that have goats behind them. There are no such outcomes, since the set of the doors the player did not initially pick always contains at least one goat.

What is the probability of flipping three heads in a row? 1/8. If the problem specifies that two heads have already come up, it makes no difference if they've come up randomly or through some manner of psychic coin control (the equivalent of knowing what's behind the doors or letting B see all his cards.) The previous events have left the equation. The probability of flipping a third head has become 1/2.

Irrelevant, as has been explained, since there is only one random event in the problem.

Player B has 12 of the 13 cards, therefore a 12/13 shot at holding the Ace. He discards one card, which happens not to be the Ace, leaving him with 11 of the 12 remaining cards, thus a 11/12 chance that he's still holding the ace. Rinse and repeat ten more times, and he's down to 1/2 cards. Each discard was one of his 12 chances.

As has already been explained, and as you have consistently ignored, B can see his cards, just as Monty knows what's behind the doors. B is not discarding one card which happens not to be the Ace; he is discarding a card that he can bloody well see is not the Ace. Monty is not opening a door that just happens to have a goat behind it; he is opening a door that he knows has a goat behind it.

As for your accusations, nobody trolls for one-word answers.

I will repeat -- anyone who actually wanted to understand the problem would have read the various explanations of the problem which very clearly state repeatedly that Monty knows what is behind each door -- nullifying your sudden statement to the contrary that it is only the result of chance that Monty picks a door with a goat. As has been explained before, this is not the problem we are discussing.

But it's easier than actually addressing the objections, especially when your argument has fallen apart.

Your argument hasn't just fallen apart, it never stood in the first place. It is based entirely upon a stubborn adherence to a false statement of the problem. You should be discussing this at [[Problem that is sort of like the Monty Hall problem in superficial ways but is entirely different because I refuse to read the statement that the person removing non-prize items from the set does in fact know that they are non-prize items and is not just removing items by chance that happen to be non-prize items]].

Obviously it's time to set dispute resolution in motion. Look, I even signed in. Truce, please. -- El Mariachi

If you had any true interest in understanding which answer to the problem was correct and why, you could have and would have taken the most obvious step: actually reading the problem. Just because you don't do so and misstate the problem doesn't mean you can elevate your misunderstandings to a "dispute".

If you want any sort of truce or resolution or anything like that, do one thing first: Acknowledge that in the problem, it is clearly stated that the host knows what is behind each door. That is part of the problem statement; one has to go back to October to find a revision in which it is not stated in the first paragraph of the article. Acknowledge that. If you dance around it, you're virtually admitting to trolling. -- Antaeus Feldspar 03:57, 18 Feb 2005 (UTC)

I acknowledge that the host knows the contents of the door. I assert that it doesn't matter. Whether he reveals the goat through his knowledge or through blind luck, either way, he reveals a goat. Before he does so, that door -- to the contestant -- still has a 1/3 chance of containing the car. Afterwards, it's 0/3. The dispute is over what happens to that orphaned 1/3.

This is incorrect. There is all the difference in the world between the host revealing the goat through his knowledge or through blind luck. Iff the host was opening a door at random with no knowledge of what was behind each door, and the door he opened happened to contain a goat, then the chances for the two remaining doors would be 50/50. Why? Because there are three possible places the car could be, but when the host opens a door to show that it contains a goat, one of them is eliminated. The other two are equally probable but only one of them has the car behind the player's door. This, however, is not the Monty Hall problem.

There are three doors, thus there are three possibilities for which door has the car behind it. However, no matter which of these three possibilities has happened, the host can always satisfy the problem statement: he can always open a door which a) is not the door the player initially picked and b) has a goat behind it. There are still three possibilities for where the car is; just because two of the possibilities look identical to the player (the host opened one door from the set of doors the player didn't pick, and the other one from that set turned out to have the car) doesn't mean that they count together as one possible outcome; if you painted each door a different color, you'd see that there are three possible outcomes (the car is behind the White door that the player picked, and the host picked the Red or Green door at random; the car is behind the Red door and the host had to open the Green; the car is behind the Green door and the host had to open the Red.) Only in one of those three outcomes is the car behind the door the player first picked.

The contestant's initial choice is now obviated, since he is being given a new choice. All the initial choice can affect is which of the goats is revealed by the host. Both remaining doors are still closed. The initial choice is not binding, and has given the contestant no additional information about the still-closed doors. The group at hand, that which inherits the orphaned 1/3, is not "doors the contestant didn't initially choose," it's "doors that are closed."

The only way that you can say that the orphaned 1/3 probability goes to "the set of all closed doors" instead of "the set of doors the contestant didn't initially choose" is if you can show how the door with the car behind it could migrate from "the set of doors the contestant didn't initially choose" to "the set of doors the contestant did initially choose." The car is behind one door, not one-thirds behind each of three doors: the set of all the doors is divided into the "set of the one door the player picked" and the "set of doors the player didn't pick"; the car is in one set or the other, and there is no way it could leave one set and move to the other.

Probability is subjective. When we speak of "probability" we are speaking of a means of quantifying results which are subjectively unknown. You're conflating the probabilities from Monty's POV with those from the contestant's. That's a mistake. The host is not playing against the contestant, the host is an omniscient observer. From a truly omniscient perspective, no event has any "probability" other than 0 or 1, because the results of all events are known with perfect certainty. Of course the car has really already been placed behind exactly one door, but probability addresses the unknown, not just the future. Future events are simply a subset of the set of all unknown event results. Those results which are still unknown to the player are the only ones which concern us here. This is why it doesn't matter if Monty revealed a goat by chance or on purpose, and this is why your set of "doors the player didn't pick initially" is not germane. All the player knows is that there are two doors, and that there's a car behind one of them and a goat behind the other. That is all we need to know to determine the probability from the player's perspective.

Completely false. You are still engaging in the "probability is always equally distributed among the apparent alternatives" fallacy, under which people might reason they have a 50/50 chance of winning the lottery, since either they win or they lose and the probability of those two "unknown" outcomes "must be" 50/50. Similarly, you are arguing that at the time of the player's decision to stay or switch, there are only two doors, and therefore the probability of which door contains the car "must be" 50/50. However, probability doesn't work like that. There are a million different ways to distribute one winning ticket and 999,999 losing tickets and thus the probability is one in a million, not one out of two. Just the same, there are three different ways that the car and the goats could be distributed initially; only one of those three ways places the car behind the player's initial door. No further randomizing events take place, the location of the car cannot be changed during the game, and no information is revealed which makes it impossible for any of those three ways to have happened -- and so the probability of the car being behind the player's door initially can never rise above 1/3. Continuing to argue this idea that because the player only sees two doors at the end, and therefore the probabilities must be equally because probability represents the unknown -- this is just like the would-be lottery winner saying "I don't actually see the 999,999 other people who bought tickets; therefore they are one amorphous lump to me! And the chances of the winning ticket going to me or to that amorphous lump are clearly 50/50; since I artificially class all non-me ticket buyers into one entity, obviously probability does too!" The fallacy is, again, in not realizing that whether you are enumerating them individually or not, that "amorphous lump" is representing 999,999 out of the million ways to distribute the tickets that would leave it in the hands of someone in that group of 999,999. In just the same manner, the player gets one door and Monty gets two; that makes the chances 2 to 1 that the car is in Monty's set. Even if Monty removes a door that doesn't have the goat behind it from his set, the position of the car cannot change and therefore the chances are still 2 to 1 that the car is still in his set of doors. Saying "Well, from the player's perspective, he knows jackshit about what's behind each door, so it must be 50/50" is bunkum. Probability doesn't work like that. -- Antaeus Feldspar 15:23, 1 Mar 2005 (UTC)

I also want to point out that if there is a second contestant who picks a different door than the first, and neither contestant's door winds up being the one opened by Monty, that by your reasoning both players then have better odds by switching to the other player's door. -- El Mariachi

A) this is not the Monty Hall problem, B) it is not even possible to play the game in the way you describe, since one would have to violate either the precondition that Monty's set is initially larger than the set of any other player, or violate the precondition that Monty can always choose a door to open from his set that does not contain the prize. -- Antaeus Feldspar 15:23, 1 Mar 2005 (UTC)

Now it is clear that the initial probability of the car being in the "set of doors the player didn't initially pick" is 2/3, because two of the three doors are in that set. Does that probability change if we remove doors from that set that are by definition not the door with the car behind them? The answer is no. If you suggest that it could be "yes", then I invite you to contemplate an equally ludicrous possibility: In the middle of the game, after the car and the first two goats have already been placed behind the original three doors, Monty calls out the stagehands and the zookeepers, and has them build seven new doors, each of which are left open and each of which has a goat placed behind it. There: If reducing the set of doors-the-player-didn't-pick by one door which does not have the car reduces the chances of that set containing the door with the car from 2/3 to 1/2, then does adding seven new doors that don't have the car behind them raise the chances of the car being in that set from 2/3 to 9/10?

See the preceding response. The relevant set is unknown results, i.e. closed doors. Adding open doors changes nothing. -- El Mariachi

Neither does opening and removing doors that are known not to contain the prize. Wherever the car is placed initially, nothing happens that can move it to a different set, and therefore the probability that it is in a given set cannot change. Saying that the probability is altered by the removal of doors that don't contain the car is ludicrous. -- Antaeus Feldspar 02:22, 2 Mar 2005 (UTC)

The proof of this is that there is no difference in the situation faced by the contestant who had been given the initial choice and that of a contestant who was given no initial choice at all. As I said days ago, if anyone can prove a concrete difference between those two situations, you win. -- El Mariachi

The concrete difference is already there. In the situation you posit, the host has only one set of doors to eliminate a choice from, and that is "the set of all doors". In the actual Monty Hall problem, the host cannot open the door that the player picked initially, even if he knows that there is a goat behind it; he must open a door from the set-of-doors-the-player-did-not-pick. You have been suggesting that the two situations are equivalent, that the doors in the Monty Hall problem are not divisible into the two sets of "the door the player picked initially" and "the doors the player did not pick" but you have offered no good reason why this would be the case. -- Antaeus Feldspar 05:28, 19 Feb 2005 (UTC)

Your attempts at attackign a correct answer with aimless little jabs of "you're just trolling!" are sad. You have failed to make any logical point. You are the troll here. The chance of switchign then picking either car or goat are 50/50 PERIOD. Yes the initial "draw" determines where each object is. Irrelevant. Human action or choice plays no part...whether a random die roll or a person chooses the doors, is also irrelevant. Nothin to do with "uintuitiveness". Just simple, correct logic & math. Anything else is garbage 7 a sidetrack & no one has ever remotely come near confronting the absolute empirical black & white fact that it IS a simple 50/50 choice. Exactly because of the initial dispersion of objects, you choice & if you switch or not make no impact. -- unsigned comment by 71.115.25.22 (talk · contribs)

Simply saying "PERIOD" and "garbage" and "sidetrack" and "absolute empirical black & white fact" will not convince anyone. If you want to convince anyone that you're not a troll, then please respond to which of these statements you think is incorrect:

1. The chance is 1/3 that the player's door contains the car.
2. The chance is 1/3 that the player's door contains goat number one.
3. The chance is 1/3 that the player's door contains goat number two.
4. The chance is thus 1/3 that the player's door contains the car and 2/3 that it doesn't.
5. After Monty has opened a door with a goat, there is one door with a car and one door with a goat remaining.
6. There is no way that Monty opening a door with a goat can change whether the player's door contains the car or not.
7. There is also no way that Monty opening a door that he knows contains a goat eliminates any cases.
8. If the player's door contains the car, the player wins by staying.
9. If the player's door does not contain the car, the player wins by switching.
10. The chance is thus 1/3 that the player wins by staying and 2/3 that the player wins by switching.

If you disagree with any of these statements, simply explain which statement(s) you disagree with and why. That will show good faith, more than simply asserting without backup what is relevant and irrelevant. -- Antaeus Feldspar 22:52, 23 July 2005 (UTC)

Best, most concise explanation yet

Ok, I've been reading the explanations for the probabilities for this Monty Hall Problem for quite some time, and I think I've come up with a much easier way to understand the answer than what I've heard so far. Rather than type out long, drawn-out mathematical equations and using all these other instances, let's look at it logically. The way I understand the problem, the contestant picks a door, the host opens a wrong door, and then the contestant is given a chance to switch. Ok, if the contestant picks the right door, with the car in it, and then switches, the contestant will be lose. There is 100% probability of this. If the contestant picks either wrong door, and then switches, then he/she will win. 100% probability. So let's look at the original odds. The chances of initially picking the right door are 1/3, and since this is the only way to lose if you switch, your chances of losing are indeed 1/3. In the same way, your chances of picking a wrong door initially, and therefore wi nning, are 2/3. How could anyone still not understand this problem?

The original probability is not inherited once the situation has changed. The contestant's chance is originally one of three doors, aka 1/3. One door is eliminated. Now his choice is between one of two doors, aka 1/2. There's no reason for one door's probability to be cemented by a nonbinding choice while the other's is free to double.

"There's no reason for one door's probability to be cemented by a nonbinding choice while the other's is free to double." This is incorrect. The prize winds up either in the set of doors the player picked originally, or in the set of doors the player didn't pick originally. Changes may happen to those sets, like the one stated in the problem of a goat-containing door being opened, but if they cannot change the location of the car, they cannot change the probability that a given set contains the car. -- Antaeus Feldspar 05:55, 19 Feb 2005 (UTC)

In the standard Monty Hall problem, the revelation of the goat does not change the probability that the car is behind the originally selected door. However, there is no principle through which this is true in general. Numerous counterexamples exist. Suppose, for example, that both of the other doors are opened. The probability that the prize is behind the selected door must now be either 0 or 1.

The "long, drawn-out mathematical equations" (I assume this refers to an application of Bayes' Theorem) and lists of events with equal probability are appropriate, valid ways of arriving at the conclusion that the opening of one door does not provide information about whether or not the prize is behind another. As far as I can tell, the other explanations and arguments are only useful as aids to intuition. (And intuition seems to frequently fail people with regard to this problem.) -- Wmarkham 00:15, 9 Jun 2005 (UTC)

Actually, the original probability DOES exist. We're talking about the probability of winning when you do switch vs. the probability of winning when you don't switch. Once separated into these two categories, the ONLY decision that has any bearing is the first, original one with the three doors. Two of those doors will result in a win (by choosing the wrong one and then switching to the right one). This is already determined, and there is no longer any chance involved. In the same way, one of those doors will result in a loss, by choosing the correct door initially and then switching to a wrong one. Again, it is only the original decision that determines the outcome. Therefore, 2/3 of the possibilities will result in a win, and 1/3 will not.

give me a try explaining

You, the candidate, pick a door. Probability dictates that in 1 case out of 3, you picked the car. This case will now be called (C). In 2 cases out of 3, you will have picked a goat. This will now be called (G).

(C): The host will open one of the two remaining doors, revealing a goat. Changing doors will let you loose, as you picked the door with the car already; STAYING with the first choice will always win in case (C).

(G): The host will have no choice which of the remaining two doors two open, as only one contains another goat, which he is bound to show. So in case (G), CHANGING the door will always win, while staying with the goat is a clear loss.

Conclusion: It can be seen easily that in 1 out of 3 cases, staying with your originial choice will be a sure win, while in more probable 2 out of 3 cases changing doors will be a sure win.

790 12:52, 19 Feb 2005 (UTC)

Empirical demonstration

Here, for those of you who don't get the math (or think you do but still think there's a 50/50 chance at the end): Monty Hall applet. Play the game over and over. It keeps track of your wins, and whether you won by switching or staying. Still not convinced? Try the iterative applet. Run it for a few thousand iterations. The statistics are borne out. -- Wapcaplet 00:20, 2 Mar 2005 (UTC)

Paradox

Saying that this is a paradox simply because to some people it is counterintuitive, is like saying "My good friend Jim, I thought you were a natural blond, but now I understand that you have died your hair! I didn't expect that at all; what an amazing paradox." A paradox has a precise mathematical meaning; anything and everything that isn't what you expect isn't a paradox. The monty hall problem is perfectly reasonable and contains no mystery or internal contradictions whatsoever.

A paradox arises when there are two (at least partially) valid and contradictory conclusions. The intuitive answer of "50/50" is demonstrably wrong and completely indefensible. Therefore even if it could have been regarded as a paradox, it is entirely resolved for anyone who takes the time to think about it. To continue to claim that it is paradox, you either have to demonstrate that something is still a paradox even after it has been resolved to the satisfaction of all mathematicians and logicians, or claim that the two answers both have validity.

In casual usage it might be regarded as a paradox. --a

I'm sorry, but if you were to actually check the definition of paradox -- the one that you insisted everyone else must not know if they were describing the Monty Hall problem as one -- you would learn that it is you who are mistaken. "A set of statements which, taken together, are self-contradictory" is a valid definition of paradox, but not the only valid definition. That's all there is to it; insisting that the word should not have two definitions is a fruitless railing against the general nature of language, and insisting that the word does not have any meanings other than the one you are familiar with is a fruitless railing against reality. -- Antaeus Feldspar 00:38, 24 Mar 2005 (UTC)

A Google Scholar search of "Monty Hall" and "Paradox" reveals that it is almost always not referred to as such in scientific papers, since nearly all the results returned talk about Monty Hall and also happen to mention something else, which provides the paradox part of the search. The most common way to refer to it in mathematical papers is "The Monty Hall dilemma." The conspicuous exception is "Increasing working memory demands improves probabilistic choice but not judgment on the Monty Hall" by T Ben-Zeev, J Stibel, M Dennis, S Sloman, the purpose of which is to argue that the chance of winning is actually 1/2. This reinforces my point: It only seems like a paradox to those who do not accept that the answer is unequivocally 2/3 and not 1/2.

And anyway, I am aware that there is a common definition which means that if something is simply surprising, then it is a paradox (although it should no longer be surprising after the mathematics behind it are demonstrated). What is your problem with the current revision? If you do revert, PLEASE for the sake of accuracy take out that utter crap about it being a paradox by the "mathematical definition," which is precisely what it is NOT a paradox by.

If you are going to cite evidence, use outside material, not wikipedia. If you'll check the discussion page for paradox you will find that there is considerable controversy about many of them (including Monty Hall) and even about the formal definition that is given in the first paragraph (the "apparently" part is bullshit, as several have pointed out). --a

You know, I'm really sorry for you that you don't know what you're talking about, but it doesn't change the fact that you don't know what you're talking about. Whether you like it or not, "paradox" has multiple definitions. In the field of logic and philosophy, the most common usage is "statements which logically combine to create a self-contradictory result". In the field of mathematics, the most common usage is "a situation in which mathematics proves the answer to be at odds with what common intuition would suggest." All your argumentation is circular reasoning that starts at and loops back around to your mistaken notion that only the first of those definitions is a real definition of the word "paradox". You want a reference other than Wikipedia? Fine, try a dictionary. (by the way, I like the fact that you don't regard the Wikipedia article as a valid source but have no problems citing the discussion page for support...) You think you've proved that the Monty Hall problem cannot be regarded as a paradox, because you Googled on "Monty Hall" and "paradox" and didn't get a lot of hits? Hardly exhaustive evidence. Why don't you try Googling for "birthday paradox" and try explaining why about 14,300 hits call it "the birthday paradox" when, according to you, nothing which actually works out in a mathematically verifiable way can be a paradox? Again, it all comes back to your mistaken notion that the word "paradox" applies only to a logical self-contradiction. -- Antaeus Feldspar 06:28, 24 Mar 2005 (UTC)

Oh, now all of a sudden Feldspar wants to allow for broad usage of terms that have multiple meanings? Hmmm... I suppose it's ok then to call John Kerry's 1st 1st "wound" minor, eh? This unsigned comment is by 216.153.214.94, determined by a ArbCom ruling to be a sock puppet of Rex071404.

Feldspar, you apparently do not know the distinction between a Google search and a Google Scholar search. Check out the difference please :) --a

I have moved the italicized text out of MY comment and made it singular as it should be. I don't know who is responsible for these "arbcom rulings" but if you check my IP you will see that it is 24.28.69.216... I have no f'kin clue who that Rex guy is and I resent being misidentified. I didn't make the John kerry comment though and I have no idea what that's all about, so for all I know that guy may have been 216.153.214.94/rex. I've come back and signed all MY comments with --a.

Thank you; that's very much appreciated. It wasn't actually hard to tell who was making which comment -- if you go to the "history" tab you can flip through the history, edit by edit, and see who was responsible for which change. This would have told me which user created which edit, even if Rex's continuing quest to blather about John Kerry's "1st 1st" wound being 'minor' on 100,000 talk pages wasn't a tipoff.  ;)

There's actually an easy way to sign the username or IP address you're logged in as: just type three tildes, ~~~ for your username/IP, or four tildes (the preferred form) for your name and the time of your edit. -- Antaeus Feldspar 16:11, 25 Mar 2005 (UTC)

Hey -- Thanks! I was wondering how that was done 24.28.69.216 17:53, 28 Apr 2005 (UTC)

From The Collaborative International Dictionary of English v.0.48 [gcide]:

paradox \par"a*dox\ [F. paradoxe, L. paradoxum, fr. Gr. para`doxon; para` beside, beyond, contrary to + dokei^n to think, suppose, imagine. See {Para-}, and {Dogma}.]

A tenet or proposition contrary to received opinion; an assertion or sentiment seemingly contradictory, or opposed to common sense; that which in appearance or terms is absurd, but yet may be true in fact. [1913 Webster]

So that's what I found in the dictionary, for what it's worth. -- Wisq 18:41, 2005 Apr 24 (UTC)

Marilyn vos Savant

The issue of the Marilyn vos Savant article was not that she was "wrong" but rather that she was mis-stating the problem. The key word she used was "guess" in regards to making the 2nd selection. By definition, "guessing" requires one to choose without making calculations. When one calculates before "guessing" it's no longer a mere guess, but an educated guess.

The other problem with her article was that the concept of odds has a dual layer in this "Monty Hall" problem which is often overlooked. The odds of finding the car do increase by switching. But the statistical likelyhood that it exists, does not. Additional information can always increase the likelyhood of a successful choice, but it does not affect the underlying statistical facts. If there are 100 piles of horseshit in your backyard and a single can of caviar buried in one of them, the singleness of the can of caviar never changes, only the odds of finding it. Suffice it to say, I have found that most people who argue about Monty Hall problems, overlook this point. Please see this variant [1] of the Monty2.gif from the article page. I suggest that this puzzle would be less puzzling if the location of the car was referred to as "odds of finding card here" rather than "odds of car being here". 216.153.214.94 06:04, 24 Mar 2005 (UTC)

1/2

I agree the probabilities are 1/2 when picking the second door, and it doesn't matter why monty picks his door because if he does reveal the prize then the game ends with you winning anyway.

From the article (emphasis mine):

". . . after Jane has selected a door but before she actually opens it, the host (who knows what is behind each door) opens one of the other doors to show that there is a goat behind it . . ."

Monty opening a door with a goat is part of the premise. That is why a) there is no provision for what happens if Monty opens a car door (which he won't), and b) the odds are 2/3rd.

I had trouble understanding this myself, until I looked it as a simple 'one of three' problem. There are three possible situations for the first choice: You pick the car (Monty picks randomly); you pick goat #1 (Monty picks #2); you pick goat #2 (Monty picks #1). In the first case, if you switch, you lose. In both the other two cases, if you switch, you win. Hence, out of three possible scenarios, switching will win two. That's two thirds, and that's why you should switch. -- Wisq 16:57, 2005 Apr 28 (UTC)

Why are the best explanations at the discussion page and not in the article? ;-) --Abe Lincoln 17:42, 28 Apr 2005 (UTC)

Increasing the number of doors

I added the following statement, but it was removed with the hint, that though it is accurate, it would be "polarized and confusing" (quote Antaeus Feldspar).

---

Often it is suggested to increase the number of doors to help to understand the problem, for instance to 100. This approach is illegal, since when transforming the special case t=3 into the general case t=n, the order to the show host can be interpreted in at least two different ways:

1. Open one of the n-1 remaining doors, but not the door with the prize!
2. Open all but one of the n-1 remaining doors (i.e. n-2 doors), but not the door with the prize!

Only for n=3, both interpretations are exactly the same.

The supporter of this approach refer to the second interpretation. In this case, the chance raises from 1/n to (n-1)/n, so for n=100 from 1% to 99% (in 99 out of 100 cases the candidate does not choose the prize, and in all of these cases a change to the one remaining door will lead to the prize).

The first interpretation is nevertheless just as legal and more than that also more evident than the second one. Even here the chance raises, but only slightly from 1/n to (n-1)/n x 1/(n-2), so for n=100 from 1% to ca. 1.01% (in 99 out of 100 the candidate does not choose the prize, and in 1 out of 98 cases the prize is then behind one of the many remaining doors).

For n=3, both terms - (3-1)/3 x 1/(3-2) and (3-1)/3 - result to 2/3.

So the change is a good choice for both general interpretations, as well for the one special case t=3. But it is illegal to refer to the alleged obviousness of just one general interpretation to illustrate the special case t=3.

---

I'm sorry if it seems polarized, but it refers to a very common approach, that is unfortunately absolutely wrong (please proof if I'm wrong). And unfortunately this approach is used in a very offensive way, like "you are wrong, and you can easily see it if you change the problem a little bit". If you change a problem, then you just CAN'T transfer the solution from the changed problem to the original problem. This is a very common mistake when dealing with mathematical problems. And unfortunately this whole thing IS about accuracy!!! And I am sorry if it is confusing, but mathematics is hard to understand some time. I hope I don't appear arrogant or something, but I think it would be a shame if such a prominent and good article contains a wrong explanation.

Maybe it is confusing because my English is bad. But in my opinion, it is better fpr an encyclopedia to be confusing than to be wrong!!!! So please discuss or improve my text, but PLEASE don't let wikipedia become a propagator of mathematical "superstition" or something!!!! :'-(

--Abe Lincoln 14:49, 29 Apr 2005 (UTC)

I agree completely (as I understand you). It's very simple to understand in a three door situation, once you know what to look for. Maybe we should remove the hundred door thing altogether, since it's not even really part of the three-door original problem. Even the old statement ('it helps to think of 100 doors instead of 3' or something) made absolutely no sense to me... it just needlessly complicates things. -- Wisq 15:20, 2005 Apr 29 (UTC)

Important addendum: Even if there would be only one interpretation, inferring from one special case to another special case is not a valid mathematical method for proofing a solution. It is only legal to infer from a general case to a special case. Though the general case has the "change is good"-solution for n>3 for both interpretations (see above), it is not legal to illustrate the special case n=3 with the more obvious special case n=100 with just one random interpretation. So this approach seems wrong in at least two directions:

1. It does not generalize the problem, but just switches to another special case, which is not valid (sorry, but this problem DOES deal with accuracy).
2. The chosen special case suggests that there is only one class of problems to which this problem belongs to. Indeed it belongs to at least two classes of problems, of which one is far less obvious than the other and even less obvisious than the special case n=3.

--Abe Lincoln 15:26, 29 Apr 2005 (UTC)

Abe, sorry for reverting, rather than doing a more thorough edit as I would have done if I wasn't heading out the door to catch a train in just a few minutes. Learning that English is not your first language (which I don't think I would have suspected, because your English is very good) helps to explain things. Some of the things you wrote might be read as a lot stronger than you meant them (the use of the word "illegal" in particular.)

I've done informal research among friends and relatives (many of them now run away swearing or start whimpering if they hear me mention goats and doors) and I've had a great deal of success with the following illustration:

You and Monty take all the hearts out of a normal deck of cards, shuffle them, and then deal one card to you and the other 12 to Monty. Monty now looks at his hand and discards eleven cards that are not the Ace. Which gives you the better chance of getting the Ace: sticking with your own hand, or switching hands with Monty?

Most people 'got it' at that point. I asked them what was the difference between the goats-and-car-and-doors version and the hearts version that helped them see the principle. I had been expecting that most of them would say it was the use of cards instead of doors, since it's easier to perceive that the cards fall into 'hands'. To my surprise, however, most people said that the increase from 3 to 13 really helped it "click in".

I think some of the issues Abe is addressing could be handled by suggesting that the classic problem can be thought of as the n=3 case of a general problem where the player always gets just 1 door, and Monty gets the other n-1 doors, of which he opens n-2 that do not have the prize behind them. This eliminates the possibility that people will assume that Monty opens only one door no matter how many doors there are in total. -- Antaeus Feldspar 00:01, 30 Apr 2005 (UTC)

Ambiguity of problem statement not addressed

The following paper is one of the better explanations of this problem that I could find online: http://econwpa.wustl.edu/eps/exp/papers/9906/9906001.pdf I believe that it explains much more clearly why 1/2 can be a reasonable answer, as compared to this current article. -- Wmarkham

It's only a reasonable answer to an insufficiently specific presentation of the problem. -- Antaeus Feldspar 00:57, 3 May 2005 (UTC)

I agree. However, as presented in the current version of the article, that interpretation is entirely valid. In other words, the current presentation of the problem is insufficiently specific. Furthermore, the entire reason that this puzzle is a "paradox" is due to the confusion over what question is being asked, so it seems to me that this article ought to explain it clearly. Currently, I don't think that it does that. -- Wmarkham 23:47, 3 May 2005 (UTC)

Even with my recent modifications, particularly the Problem summary section? I hoped to reduce the whole issue to a very concise set of axioms to eliminate confusion, particularly things I see on this talk page, like 'what if Monty chooses the car', etc. -- Wisq 01:37, 2005 May 4 (UTC)

I think so. Even with the statement "Monty always picks a goat", it is possible that Monty sometimes picks a goat that is behind the door that the contestant has initially picked. -- Wmarkham 02:38, 4 May 2005 (UTC)

After a while, I was able to see how one could, possibly, come to an interpretation of the problem under which the probabilities would be other than 1/3. It would be necessary for the interpreter to believe that a) Monty knows what is behind each door, since that is stated outright in the problem, but that b) Monty chooses randomly from those doors that have goats behind them. This works out to six possible outcomes: three sets of doors-with-goats, two choices for each set. If we then read the part of the problem statement that specifies that Monty chooses a door the player didn't pick as "eliminate all those outcomes where the did", we reduce those six outcomes to four. In those four outcomes, two of them have the car behind the player's door.

To me, this is a quite tortured interpretation. It is technically possible, but not, I believe, an interpretation that anyone would read naturally. Nevertheless, I have revised the problem statement to eliminate this interpretation. -- Antaeus Feldspar 23:53, 4 May 2005 (UTC)

I took a stab at rephrasing the question. I think it is unambiguous, and worded in a more natural manner. If not, feel free to revert. -- Wmarkham 02:19, 5 May 2005 (UTC)

Rephrasing of problem and solution

I've done my best to incorporate some common confusions on this page, including one alternate (1/2) variant (and why it does not apply in this case), which I think also helps to express why the initial choice is so critical -- 2/3rds of the time, it eliminates one potential goat, forcing Monty to reveal the car.

I also added the 'problem summary' section, for lack of a better title -- I was hoping for something more like 'problem givens' or 'given statements' or 'known facts' or whatnot, but none really seemed to sound right. I added it because these are not assumptions, but rather, integral parts of the main problem... things a lot of people seem to miss, in whole or in part, when reading through and forming their conclusions.

I'm hoping all this will make the matter clearer. It's an interesting thought experiment, and I'm guessing a lot of people are so sure it's 1/2 until they see the right explanation for them, something clicks, and they realise why it's 2/3. For me, it was the scenario-based one. YMMV. -- Wisq 03:37, 2005 May 3 (UTC)

---

I think the wording is really essential. It must be distinguished between two different problems:

A. Monty does have to open one of the two other remaining doors, but one that must conceal a goat.

B. Monty opens any of the two other remaining doors, and incidentally a goat is appearing behind the door.

In case A it's better for Jane to switch for the described reasons. In case B it doesn't matter whether you switch, but the car is shown by Monty himself with the probabilty of 1/3. If Monty incidentally does not pick the car, the other both doors still have the same chance 1/3.

To make this a lesson in probability or rather in understandig a problem of probability, it must be worded in a way that the reader might think it is problem B when it indeed is problem A. So it must not be identified too obvious as either of the both.

The current wording is

...the host (who knows what is behind each door) opens another door, to show that there is a goat behind it.

Does anyone have other suggestions? --Abe Lincoln 13:04, 5 May 2005 (UTC)

Some phrasings of the question (see the one in the Parade column) also allow:

C. Monty does not have to open any of the two other remaining doors, and makes the decision of which door to open, if any, based on knowledge both of the player's chosen door as well as the prizes behind the doors.

In those cases, I often find this to be a natural interpretation, given my basic knowledge of how game shows work. In this case, the solution depends on the host's motivations, which have not been stated. Once these have been laid out, the solution is usually arrived at through game theory.

When ambiguously phrased in such a way, the problem can give a good demonstration of why one needs to have a repeatable process in order to assign probabilities. See also Dandrake's comments under #Removed Assumptions, above. Bertrand's paradox (probability) is also relevent. Of course, an ambiguously phrased question is a terrible problem to ask on, say, a college test. However, it might be a suitable topic for an encyclopedia article....

If this article is expected to include a description of that ambiguous question, then the depth of analysis given to one particular interpretation of it might be inappropriate. Currently, the article reads like "here's a problem. here's the solution. here's an ambiguous statement of that problem. here are more explanations of the solution". In my opinion, it is rather the ambiguity itself that is interesting about this problem, so I would prefer to see "here's an ambiguous problem. here's one way of making that problem concrete. here's the solution according to that interpretation. here's another way of making the problem concrete. here's the solution according to that interpretation." Keep in mind that the article is currently listed under Paradoxes.

Of course, that style of organization assumes that there is nothing "special" about the particular interpretation that is currently described. (Which is often called the "standard" Monty Hall problem.) For me, or any one person, to unilaterally make the decision that this article needs to have that organization would certainly be an imposition of a particular POV, of course. Unfortunately, though, I think the current article falls slightly on the other side providing a neutral POV.

From what I've read, I understand that Selvin's original statement of the problem left some ambiguity, and he followed his article up with an explanation of what assumptions he had intended. I'll see if I can copies of the article and followup, to support this. -- Wmarkham 16:08, 5 May 2005 (UTC)

I have realized that there may be a simple way to state the problem:

• The contestent initially selects a door, A.

• The host opens a door, B, to reveal a goat. B!=A.

• The host solicits a final door to open from the contestant.

• Whatever the process by which this game show operates, it must produce these events in every execution.

So, the first four describe just the events of a particular run of the game, but the last one entails the key (sometimes unstated) assumptions needed for the "standard" problem.

-- Wmarkham 00:27, 9 Jun 2005 (UTC)

Selvin's statement of the problem

In "Letters to the Editor", Selvin provides a statement of the problem that describes a dialogue between "Monte Hall" (sic) and a contestant. The basic events are as follows:

The host explains that the keys to a new car are contained within one of three boxes, labelled A, B, and C. The other two are empty. The host explains that if the contestant chooses the box with the keys, the contestant will win the car. The host solicits a selection from the contestant, who selects box B. The host hands box B to the contestant. The host offers the contestant $100 for the box. The contestant declines. The host offers the contestant $200 for the box. The contestant declines. (The audience agrees!) The host provides the analysis that the probability that box B contains the keys is 1/3. The host offers $500 for the box. Once again, the contestant declines.

The host says "I'll do you a favor and open one of the remaining boxes". The host opens box A, revealing it to be empty. The host provides an analysis that the probability that box B contains the keys is 1/2. The host offers the contestant $1000 for the box. The contestant counters this offer with the offer, "I'll trade you my box B for the box C on the table".

Selvin asserts that "The contestant knows what he is doing!"

-- Wmarkham 18:06, 5 May 2005 (UTC)

Of course, it depends on whether offering increasing sums of money and opening another box are normal parts of the program. If they're not, the behaviour above sounds like desperation to maintain the show's bottom line. ;) -- Wisq 21:45, 2005 May 5 (UTC)

Proposal

I think that some of the recent edits have been heading in an unfortunate direction, making the initial problem statement so detailed in order to prevent misunderstandings that it's becoming hard to find the essential meat of the problem in the problem statement.

I'd like to propose instead that we start with a very simple statement of the problem situation, and the fact that the probability of winning by switching is 2/3, then follow up with something like:

"Sometimes ambiguous phrasing in the problem statement causes the problem to be misunderstood. If the teller does not specify that Monty knows what is behind each door, the listener may believe that Monty picks the door he opens at random, and that those cases where he accidentally picks the car are being discounted. Similarly, the listener may take "Monty opens one of the other doors" to mean that Monty picks randomly between the two doors that contain goats, but that those cases where he picks the same door that the player picked are being discounted. Under such readings, the probability may be 1/2 instead of 2/3, but these are misreadings of the problem.

"More frequently, people understand the problem statement but mistakenly think that the probability must be 1/2 because Jane is choosing one of the two doors that are left at the end."

-- Antaeus Feldspar 23:48, 5 May 2005 (UTC)

I think the additional sentence about the restrictions on the host (must open a door, must make the offer) is necessary. It wouldn't hurt to add a section about common misinterpretations due to imprecise problem statements, but (IMO) the initial problem statement here should be precise and unambiguous. There's currently a redundancy (the statement that the host knows what is behind each door appears twice in the first paragraph), but other than that I think this is one of the best wikipedia articles I've run into. I see this article has been previously nominated for peer review - has it been nominated for WP:FAC yet? I think it should be. -- Rick Block 01:20, 6 May 2005 (UTC)

I agree it should be unambiguous, and my edits have been to that end. As believers and understanders of the 2/3rds answer and why it is true, and speaking as a recent introductee to the problem, I think we can lose track of the true paradoxical nature of the problem, which is that it triggers a natural rejection in the mind -- "No, of course not. The final decision is between two doors, so why should switching matter?"

I think however detailed the problem statement is, it will still retain its 'fun' nature of being initially confusing and ultimately obvious at the same time. The only thing that remains, then, is to make sure that the reader cannot (completely justifiably) refute our solution based on their assumptions about unstated non-rules, or accept our solution but feel 'cheated' by not being given all the rules up front. (Or am I taking this all too seriously?) -- Wisq 03:19, 2005 May 6 (UTC)

Nydick/Nudick

Section "The problem" says "Nydick." This author is listed in "References" as "Nudick, Robert L." Which is correct? Graue 01:47, 25 May 2005 (UTC)

Nydick. He's collaborated with Liberatore on other publications. There's a reference to the paper from here. -- Rick Block 03:07, 25 May 2005 (UTC)

nominate for featured article?

Anyone think this article is NOT ready for WP:FAC nomination? I haven't worked on it much, but I'd be willing to nominate it if no one else is interested. -- Rick Block (talk) 19:04, Jun 12, 2005 (UTC)

Flash of insight

Here's how I finally understood it in my soul.

There is a 1/3 chance that the contestant's original door has the car, because he chooses randomly. There is a 0 chance that the door Monty opens has the car, because it always has a goat. The probabilities sum to 1. Therefore, the probability that the car is in the remaining door is 2/3, so switching is good. People think the probabilities are equal because they miss the fact that Monty deliberately, not randomly, avoids choosing the door with the car. By not picking a door, he shows it to be more likely to contain the car. Superm401 | Talk July 1, 2005 06:59 (UTC)

You are WRONG(or at least I think). Admitedlly, in the start there is a 1/3 percent chance. But then Monty skips the door you chose, along with the other one. He selects one, so he also skipped over the door you picked. Now, the goat has been revealed (or the first one) so it is fifty fifty, switching doesn't matter.

Proposed to replace "effect of opening a door" and "combining doors"

I believe the central points of these two sections might be replaced by the following explanation:

If the player got the prize just for correctly guessing which set of doors — his own set of one, or the host's set of two — contained the car, then a math-savvy player would always choose the host's set, since the chance of the car being in the player's one-door set is only 1/3.

If the restriction was then added that the player must also guess which door in a set holds the car, then there would no longer be any clear strategy: selecting a single door from the host's set would first risk the 1/3 chance that the car was in the player's set after all, and then risk the 1/2 chance that the car was in the host's set but behind the other door. The total chance of guessing right both times would be only 1/3 -- the same chance as for the single door of the player's set.

When the host opens (and removes from his set) a door that contains a goat, however, it eliminates the need to guess between doors in a set -- and eliminates the 1/2 chance of guessing wrong that went with it. The player once again has a clear winning strategy that pays off 2/3rds of the time, just as in the simplified game where he only needed to guess the correct set: now that selecting a single door is the same as selecting its entire set, that is essentially the game now being played.

I think maybe this captures the essential points of those sections? -- Antaeus Feldspar 3 July 2005 23:00 (UTC)

I think this line of thinking is now covered using somewhat different words in all three of the diagram, combining doors and opposing player sections. Reading these sections do you think more is needed? I think I'd prefer to keep the combining doors section sharply focused on that alternative. -- Rick Block (talk) July 4, 2005 00:18 (UTC)

Yes, I do -- I'm trying to address a specific intuitive block that people have. They don't see how one door could be any more probable than another, if the host hasn't given away any information about what's behind any of the doors. They might get it if they look at it from the other perspective, that the host has removed a need for information. -- Antaeus Feldspar 4 July 2005 21:41 (UTC)

Religious War

This argument comes down to two mutually exclusive views:

1) The initial conditions of the game, (three doors, two goats, one car) have everything to do with the final choice.

2) The initial conditions of the game, (three doors, two goats, one car) have nothing to do with the final choice.

You can argue the niceties of the statement of the problem until the goats die from the heat produced by the studio lights, but proponents of either side will never convince the other. This is known as a religious war.

Never-the-less, I will add my two cents.

I am in the camp number 2. The statement of the problem, no matter how carefully crafted, is a classic example of a RED HERRING. Proponents of solution number one are falling prey to the "too much information" trick. Most of the explanations above are superfluous, as they just don't apply to the issue. The only purpose of the three doors stage of the game is to provide drama and indirection.

The final choice the contestant has to make is between two doors not three. Pure and simple, two doors, 50% chance either way, end of story. What has gone on to get to that point is IRRELEVANT. Monty may have just as well arranged to have one of the goat doors open when the curtain is pulled aside to reveal the doors. It doesn't matter if you had a million doors, and opend 999,998 - at the end you still have 2 doors, one with a car and one with a goat.

There is indeed one randomizing event, and the 1/3 versus 2/3 probability applies correctly to the initial red herring part of the game, however, THAT IS NOT THE GAME. As soon as Monty opens a door with a goat and offers the contestant a new choice, a choice between two doors, it is a new set of conditions. The 1/3 chance of the opened door does not 'magically' get transferred to the unpicked door, nor does it get 'magically' spread across the two remaining doors. The entire original situation is made null, and a new set of conditions apply.

Computer simulations designed to show the results of solution number one will always support solution number one. I can just as easily devise a computer simulation that will support solution number two (and I have done so - big deal). Both simulations, by their very nature, are biased to the solution they are meant to support. -- unsigned comment by 144.140.62.104 (talk · contribs)

I agree because the intial conditions do not matter at all. After the host opens the goat, you know behind the door you choose could be one of two things, a goat or a car, and same for door 2. You know know your door, for sure is either the right or the wrong one, compared to what you knew at the start. The problem INVOLVES changing your conditions. To make it clearer, say behind a peice of paper, you had three sets of numbers, of which tow are the same. The contestant must guess the one set, not the two other identical sets, Now, the contestant in this game knows the sets of numbers, so he has a one in three getting the correct set of numbers. But know suppose one of the set of wrong numbers was revealed. It would leave the player with one right answer and one wrong answer. It is like someone asking you the distance from here to "a" and from "m" to "n" you not given any of the vasriables. You may as well take a random guess. It is not like the host skipped over yours, even if you choose the goat the host would simply open the other. Get it?

Well, the only way you can program a computer simulation that will support "solution" number two is to program in something that is not the Monty Hall problem. Your belief that it is a "new set of conditions" after Monty has opened a door from his set of two that he knows to contain a goat is incorrect; if the car was behind the player's door all along, it's still there when Monty opens the goat-door, and if the car was not behind the player's door, nothing Monty can do will put it there.

Now, if you were to disregard the condition that Monty knows what's behind each door, then you could make an argument that the chances have become fifty-fifty, because stating "Monty opens a door to reveal a goat" is equivalent to stating that we are discarding all the combinations of "door-the-car-is-behind" and "door-that-Monty-opens" where they are the same door, and once those combinations are discarded, the car is behind the player's door in half of those that are left. However, that is not the Monty Hall problem.

You ridicule the idea that the 1/3 chance of the opened door gets 'magically' transferred to the unpicked door, but you're proposing that the chance gets 'magically' split between all the doors still unopened, which does not seem in any way less ludicrous. In truth, because the door Monty opens is selected because it doesn't have the car behind it. The '1/3 chance' does not get transferred because it never existed; what existed was the 2/3 chance that the car would be behind one of Monty's two doors. The mere fact that it looked to us, before we knew which door Monty would open, like each of his two doors had a 1/3 chance of holding the car, does not mean that all remaining doors now have a 1/6 chance transferred to them when Monty shows us that this door has no chance whatever. -- Antaeus Feldspar 4 July 2005 01:05 (UTC)

If you could share your computer simulation, I suspect we could point out where it violates the stated assumptions. Consider a card game where you are dealt one card and I'm dealt the remaining 51. I get to look at my hand and must turn 50 cards face up. Is it a 50-50 chance which of us has the ace of spades? If you think the answer is yes, I strongly suggest you not spend any time in Las Vegas. In fact, if you think the answer is yes, I'll gladly play this game (let's say at least ten times) with you for real money giving you 2-1 odds. -- Rick Block (talk) July 4, 2005 01:25 (UTC)

Sorry, but you still miss the fundmental, unchanging, unarguable point. The contestant is finally and uncontrovertably given a choice between two doors. Not three. Two. I'll repeat that once again. Two Doors. There are only two doors. There is only one goat (unless the car is a Pontiac GTO ;-) ). The contestant has to make a choice between two doors. Do you get it yet? Three doors is a red herring. There are only two doors in the game.

I don't have to supply you with my computer simulation to demonstrate the point, you would point out where my simulation is biased to my solution by makeing the decision that only the last choice is relevant, while I would point out where your simulation is biased to your solution because it makes the decision that the first choice is relevant.

Consider the computer game MineSweeper (available on Windows and every Linux distribution I have seen. Fundmentally you have to 'step' on a square which may or may not contain a 'mine'. If you miss the mine, a number is exposed which tells you how many mines are in the adjacent 8 squares (or 5 for an edge, or 3 for a corner). As you progress through the game, if you step on a square that does not have any mines adjacent to it, the game will automatically clear the surrounding squares until the exposed space is bounded by squares that do have adjacent mines.

Suppose at the beginning of the game, we choose a non-edge square, there is no mine, and the number '1' is exposed. Clearly, each of the adjacent 8 squares has a 1/8 chance of hiding the mine. If we randomly choose one of the 8 and it is clear, each of the remaining 7 squares has a 1/7 chance of hiding the mine. So far I think this description is matching your arguments above, because we are eliminating the clear squares not the computer or Monty Hall.

Now suppose that as we progress the computer clears some blank space that it knows does not contain the mine. We may find situations where there is a number '1' surrounded by 7 cleared spaces and one hidden space. The chance that the remaining one space contains the mine is 100% by my rekoning and I doubt that you would argue otherwise. Now we may also find situations where the computer has exposed a number '1' with 6 cleared spaces and two uncleared spaces. Disallowing overriding information from other squares on the board (in a real game you have other clues to decide which square to pick most of the time), you have two squares to choose from. I argue that the odds are 50/50 on each one.

I know this does not describe the process of pre-picking one of the squares, so there is no room here for consideration of the alternate solution of 1/8 versus 7/8 on the two squares. But that is my point: the pre-picking of the door is irrelevant to the final choice the contestant has to make, and the MineSweeper example exactly describes the conditions and choice the contestant has to finally make. The contestant does NOT have to pick between 3 doors. End of story. (Sorry I haven't figured out how to put in a proper signature yet - Keith Latham)

I'll just add one more half cent then I'm through. In a previous edit, Antaeus said "what Monty knows is crucial to the problem". Well of course he does have to know where the goats are to safely remove one of the losing doors from the game. But what is more crucial is what the contestant knows. And what the contestant knows is that there are two doors, one of which hides a car. Removal of one losing door gives zero information about which of the remaining doors hides the car. --Klatham 4 July 2005 11:13 (UTC)

Minesweeper is far more of a red herring than anything you're complaining about; you'd have to show us screenshots to demonstrate exactly the principle you're trying to show and even then you're talking about a completely different game whose strategies are hard to relate to what is really a very simple problem. If you have any intention to approach this honestly (i.e., rather than as a religious war where you fight fiercely for what you have put blind faith in) please follow this demonstration:

  P  1 2
 +-++-+-+
P!C!|G|G|
 +-++-+-+
1!G!|C|G|
 +-++-+-+
2!G!|G|C|
 +-++-+-+

That grid represents the three possible arrangements of goats. Along the top are the labels for the doors: Player, Monty's 1st door, Monty's 2nd door. (To avoid pointless quibbling, we'll just pretend the doors are on a lazy Susan so that whichever door the player chooses, Monty's 1st door is automatically the one to the right of that, and Monty's 2nd is the one to the right of that.) The three rows then represent the three possibilities for where the car is, relative to the designations affixed by the player's choice.

  P  1 2
 +-++-+-+
P!C!| |G|
 +-++-+-+
1!G!|C| |
 +-++-+-+
2!G!| |C|
 +-++-+-+

Now we've just fulfilled the condition that Monty opens one of his two doors that he knows to contain a goat; the empty space represents that door being eliminated as a choice. Each of the three rows still represents one possible way that the car and goats could have been distributed in the only random event of the problem. In two of those three, the goat is behind one of Monty's doors. That means that, two out of three times, selecting whichever door Monty still has left is the winning move.

Now if you think something different happens that makes the chances one in two, not two out of three, show us. As Rick says, show us the program code for your simulation, or show us a grid showing how the three possible distributions of the car and goats become only two distributions. That is, if you're not just trolling. -- Antaeus Feldspar 4 July 2005 15:22 (UTC)

Or, try a simple experiment. Take 3 cards from a deck including the ace of spades. Shuffle them and deal one to me. Look at the other two and turn over any one of them as long as it's not the ace of spades. I think this is the same problem and you're saying the chances are 50-50 that my card and your remaining card are the ace of spades. If you agree, how about if you shuffle the entire deck, give me one card and you keep the rest. Looking at them, discard any 50 of them so long as you don't discard the ace of spades. There are now two cards left (mine, and yours). Is it still 50-50? If you think the answer is yes, please actually do this 10 or twenty times and let me know how it works out. Thanks. -- Rick Block (talk) July 4, 2005 16:05 (UTC)

Rick, this experiment is a good demonstration -- I've noticed that more people seem to "get it" if the problem is expressed as "which hand is more likely to contain the good card, even after a bad card is discarded?" rather than "which door is more likely to contain the car after a door that holds a goat has been opened?" Think we can work this into the article? It's an empirical proof that anyone with a few playing cards laying around can prove to themselves. -- Antaeus Feldspar 4 July 2005 21:50 (UTC)

I'm very interested in the code for this computer simulation that provides a 50/50 result. Could you please display it here, or provide a link to it? I think it would be demonstrative in showing the differences in rules between the problem you are imagining we are talking about, and the problem we are actually talking about. In my experience, there are three kinds of people who disagree with the Monty Hall problem's solution... those who don't yet understand it (but will, usually with increasing n=3 to n=arbitrarily large number), those who misunderstand the problem and think we're talking about another problem entirely (who are usually convinced when the rules are unambiguously stated to their satisfaction), and those disagreeing for the sake of disagreeing (who can never be convinced). I suspect you lie under catagory two, and determining the differences between your problem and the problem we're talking about could be key. Fieari July 5, 2005 19:09 (UTC)

Maybe he's using an extremely poor random number generator? I wrote a Visual Basic program several years ago to simulate the problem and got 57% (neither 50 nor 66!) with the default random function :-) - Fredrik | talk 21:59, 9 July 2005 (UTC)

Yeah, that sounds like either a too-small sample size, or a psuedorandom number generator with some serious flaws. Fieari 03:43, July 11, 2005 (UTC)

I think our anon's issue is a bit less involved than that; since he believes "religious[ly]" that the player faces a new 50/50 chance when the number of doors has been reduced to two, his "simulation" code probably just selects the door after one door has been opened. -- Antaeus Feldspar 00:46, 12 July 2005 (UTC)

Sorry if I'm rehashing here, but I just stumbled onto this. How about this: You walk onto the stage with one door already open, showing a goat, and two closed doors. Are you saying that you have a 2/3 chance of choosing the car? I mean, technically, you only have 1/3 chance, but I don't know anyone who would pick the open door. So you have two doors to choose from. One out of two. Isn't that 1/2 chance? In fact, take away the open door completely, since you're obviously not going to choose it. Two doors. You choose one. How is that not 1/2? --Kbdank71 18:39, 15 July 2005 (UTC)

Or, to make it easier, you are playing the game as noted in the article (three doors, you pick one, Monty opens one to reveal a goat), and I walk onto the stage with one door already open. It's stated in the "solution" secion of the article that in my situation it would in fact be 1/2. But because you chose one door already, it's 2/3. Now look at the situation. We are both standing in front of three doors, one open showing a goat. Take a picture of both situations and show it to someone, they wouldn't be able to tell who picked and who didn't. I challenge anyone here to explain how my 1/2 chance is the same as your 2/3 chance. --Kbdank71 18:59, 15 July 2005 (UTC)

And the more I think about it, the more I believe the anon is correct. Three doors is a red herring. You do not have three choices. You have two: Either you choose a goat or you choose a car. If you choose a goat, Monty shows you a goat, and you have the choice between your goat or the car. 1/2. If you choose the car, Monty shows you a goat, and you have the choice between your car or a goat. 1/2. Add as many goats as you want, it doesn't matter, because Monty is going to remove all of the choices except for one goat and one car, leaving you with 1/2. --Kbdank71 19:17, 15 July 2005 (UTC)

Eh, never mind, I think I figured it out. You have a 2/3 chance of picking a goat in the beginning. And if you switch your pick, you win a car. Which is still 2/3. Unreal. --Kbdank71 19:51, 15 July 2005 (UTC)

Yes, exactly. You've hit the key to the solution: there may be only two possibilities for where the car can be, once we're at the stage when one door has been opened, but there are three possibilities for how we got from three doors down to two, and only one of those three possibilities has the car behind the player's door. -- Antaeus Feldspar 01:27, 16 July 2005 (UTC)

And, if you walk onto the stage during the game, after I've picked a door and the host has opened a door, the two remaining doors aren't 50/50 for you either. Yes, there are two doors. But the one I initially picked is 1/3 (for both of us) and the other one is 2/3 (for both of us). -- Rick Block (talk) 03:02, July 16, 2005 (UTC)

As a layman who just read all this (and it's fascinating), I would like to state that Anteus' grids are extremely informative, since I now understand what he's been trying to convey for months in statistical terms. However, it seems to me that the Monty Hall problem is paradoxical in that while it is true that switching increases the statistical chance of winning from 1/3 to 2/3, ruling out a door gives the illusion that there is a 50% chance that the car is behind either door.

Let me see if this makes sense:

Anteus' grid clearly demonstrates that the smartest decision that a player can make is to switch from the door he's chosen, since in 2 out of 3 cases, he'll win by doing so. However, at that point 1 door has a goat and 1 door has a car, thus giving the illusion that the choice is 50/50, which it really isn't.

Does any of that make sense? (JRVJ - I also don't know how to sign my name yet).

Simulation

I disagree with including simulation results in the article. I don't know where people are going with this. The simulation results don't increase the value of the article since the solution is already nicely explained with probabilities. One could argue that these results are against Wikipedia's no original research policy. 64.229.142.80 02:10, 23 July 2005 (UTC)

I can sortof see removing the results as original research, but mentioning simulation should be kept in, since it does empirically verify the presence in a mostly irrefutable way. On the other hand, given that we're mentioning the simulation for this purpose, a sample result seems reasonable... it's not original research, it's quoting what anyone can (and many have) verify for themselves. Fieari 02:17, July 23, 2005 (UTC)

People stop listening to anyone but themselves quite often with this problem. The other proofs can be argued for days, but this can't. The results themselves may be original research, but the method of verification itself has been around for years. — 131.230.109.211 03:05, 23 July 2005 (UTC)

Well, I think the addition of the "simulate it yourself by playing" -- ahem -- "three-card Monty" section really removes the need for including our own simulation results. People who aren't getting why switching is a 2/3 strategy are probably not even going to trust that our computer code isn't jiggered, let alone trust us when we say "these were the results of our simulation, and they prove that we're right! =D" But I have to think most people, if they sit down and actually start dealing out the cards, realize "wait a minute, any time that the ace of spades isn't the one card the player gets, they win by switching..." and suddenly they grok it. -- Antaeus Feldspar 04:24, 23 July 2005 (UTC)

KEEP! I didn't believe this article until I saw the computer program results of 2000 to 1000 or whatever it is. Computer program results are easier to understand then math results.

The Chester Theorem

The probablility of the goat sitting still is quite small. Therefore, the contestant will have the following possibilities after selecting a door:

-Monty picks a door where there was a goat

-Monty picks a door where there are now two goats mating

-Monty picks a door with a goat that is sleeping, going potty, or is too indifferent to move

As such, the door the contestant picked and the other unopened door will have one of the following:

-A messy stage where a goat once was

-Two goats mating

-A stall with a goat that is sleeping, going potty, or is too indifferent to move

-A car that has the seats chewed, the bodywork dented, and the paint scratched by (a) rampant goat(s)

Therefore, the contestant's best choice would be to Not give up the prize they won earlier in the show and stay out of the Final Prize Round of Let's Make a Deal.

-Pf

Rephrasing The Given Solution to show 50/50

The answer to the problem is no; the chance of winning the car is the same when the player switches to another door.

There are four possible scenarios, all with equal probability (1/4):

• The player picks goat number one. The game host picks goat number two. Switching will win the car.
• The player picks goat number two. The game host picks goat number one. Switching will win the car.
• The player picks the car. The game host picks goat number one. Switching will lose.
• The player picks the car. The game host picks goat number two. Switching will lose.

In the first two scenarios, the player wins by switching. The last two scenarios the player wins by staying. Since two out of four scenarios win by switching, the odds of winning by switching are 1/2.

Why?

If we make a distinction between goat one and two when the user picks a goat, we need to do the same when the user picks a car. Why would we do otherwise? They are both possibilities aren't they? User:Oyejorge (07:49, July 23, 2005)

I took the liberty of signing your message for you. In the future, you can do this yourself by putting four tildas ( ~~~~ ) after your message, which will both sign and date your message automatically. In responce though, the last two events you have listed there are actually, mathematically, the same event. There's no real difference. To rephrase one of the "Aids to Understanding" placed on the main page, consider the following slightly modified situation...

There are three cards. One queen, two Jacks. They are shuffled and placed face down on the table. One player picks one of these cards at random, the other person picks the remaining two cards. Who has the better chance at having the Queen?

Now add one more element...

There are three cards. One queen, two Jacks. They are shuffled and placed face down on the table. One player picks one of these cards at random, the other person picks the remaining two cards. The player with two cards then discards a single Jack (which he is garaunteed to have at least one of). Now who has the better chance at having the Queen?

I'd like you to first answer the two questions I've posted above, and then if you still doubt the validity of the solution, tell me how the second problem differs from the Monty Hall problem. Fieari 08:08, July 23, 2005 (UTC) I just considered something more... if the problem was reworded to constrain Monty to picking ONLY Goat #1 if the car is picked, it wouldn't change the problem at all. In the card version, we'd have a Queen, the Jack of Hearts, and the Jack of Spades. Player two, when picking up the remaining two cards, MUST discard the Jack of Hearts if he has it, and the Jack of Spades otherwise. No randomly picking which one to discard. I propose that wording it this way doesn't change the problem at all, and neatly eliminates your last option of the four you listed. Fieari 08:21, July 23, 2005 (UTC)

Remember that the two last ways have just half the possibility as the two first. And are mathematically seen the same event as one and two. But one could say that the possibilities for the events are 1: 1/3, 2: 1/3, 3: 1/6,4 1/6.Gillis 08:19, 23 July 2005 (UTC)

To illustrate Gillis's good point, here's a bullet-chart of the probabilities:

• Chance that the car is behind one of the three doors: 1/1
  • Chance that the player picks goat number one: 1/3
  • Chance that the player picks goat number two: 1/3
  • Chance that the player picks the car: 1/3
    • Chance that the player picks the car and the host picks goat number one: 1/6 (1/2 of 1/3)
    • Chance that the player picks the car and the host picks goat number two: 1/6 (1/2 of 1/3)

The last two items are actually sub-probabilities of the player picking the car, so they share equally of that probability. -- Antaeus Feldspar 21:50, 23 July 2005 (UTC)

Ok, I've been humbled. I don't think I should do this sort of thing late at night. Here's a diagram I used to understand the problem... It's the same info as what's in the article, just presented a little differently, maybe it'll help other neighsayers. Thanks for the clues Fieari and Gillis,

First Choice	Probability	Monty Hall Takes Away	Switching Wins?
Goat 1	1/3	Goat 2	Yes 100% of the time
Goat 2	1/3	Goat 1	Yes 100% of the time
Car	1/3	Goat 1 or Goat 2	Yes 0% of the time

Oyejorge 17:44, 23 July 2005 (UTC)

I fixed the last cell in the table, which used to read "No 0%." Also correct would be "No 100%," but that's not as straightforward. Old Nick 19:22, 25 July 2005 (UTC)

Theorem history

I remember once reading that this riddle/theorem whatever is much older then ever mentioned in the article. i understand it was atelast from the 19:th century.

I can't verify this with google though. Does anyone have any better knowledge of the issue or better googling-luck? Gillis 08:13, July 23, 2005 (UTC)

I've never even heard of it existing previously than specified... no google-luck here. By the way, I took the liberty to sign your message for you. In the future, you can (and should) sign your messages yourself by placing four tildas ( ~~~~ ) after your message. Fieari 08:16, July 23, 2005 (UTC)

Actually, it seems i found it with another attempt at google, i added in the new information, here is the source http://barryispuzzled.com/zmonty.htm. SSomeone could do some verifying... and ok i'll try to remember always signing my posts Gillis 09:47, 23 July 2005 (UTC)

Game Theory?

I would argue it's a problem in probability. Rich Farmbrough 12:44, 23 July 2005 (UTC)

I have switch the statement back because it is certainly a problem of probability. Apparently MathWorld categorizes the problem under game theory, but in my opinion the connection is tenuous. In the standard statement of the problem there are no "conflicting interests" because the host is not an active player in the sense that he cannot make choices that affect the outcome. Because the problem clearly could generalize into a game theoretical topic, I would have no objections to a Wiki categorization as such. In the text, however, this would require some explanation, so in my opinion it isn't appropriate for the first line. Certainly it does not supersede probability. Davilla 16:21, 23 July 2005 (UTC)

Monty Hall Problem in Politics / Elections

Can showing you that one of the three candidates in a tight race is a loser make you change or question your vote?

I happen to believe that people love to feel included by voting for the popular candidate.

Let's say people were given a choice to vote for three candidates A B C. After the election, you find out that one of the candidates was a loser and the two candidates were so close that a special runoff election was required. Would you be more likely to be with the majority by switching your vote on the runoff?

Consider this election ..

Dino Rossi (R) and Christine O. Gregoire (D) in 2004 ran one of the tightest governor races in history. Christine Gregoire won by a mere 129 votes.

The Republicans wanted a runoff election to solve the inaccuracies.

Do party affiliations play in this problem?

There are quite a number of things to say about a situation like this, as per the mathematical treatment of social choice functions. Your slant is new to me, and interesting if not a bit offensive. I don't think it's necessarily wrong to be in the minority, although probability-wise you may have a point. I'm not much to speculate about the psychological aspects. Davilla 16:36, 23 July 2005 (UTC)

Removed "probability laws" section

This poorly-titled section formerly said this:

When you pick a door (A), the probability of having chosen a door with a car is 1/3 since there is only 1 car and 3 doors. When the host eliminates a door (B), the probability of the car being in A remains constant, but now there are only two possible doors to pick from: A and C. Since one door's probability is 1/3, the other one is 1-1/3 = 2/3. Therefore, if the car is in C, it means it's not in A, and so the probability of the car being in C is the probability of the car being in A negated.

This is circular logic. In support of the idea that door A ends up with a 1/3 probability of having the car, it makes the statement that "the probability of the car being in A remains constant" without any support. One may reasonably ask, why doesn't the probability of the car being in C remain constant instead? This explanation provides no answer. (It also ends with a spectacularly muddled sentence that adds nothing but confusion.) I have remove the section. --Doradus 16:29, July 23, 2005 (UTC)

The statment is, of course, correct, but I see your problem with it, especially the unsupported claim. Rather than deleting it, maybe it could be fixed without making the explanation too verbose. Davilla 16:43, 23 July 2005 (UTC)

1/2 or 2/3 depending...

Probability is 2/3 if all decisions are made before any doors are opened. Probability is 1/2 if a fresh decision independent of history is made after Monty reveals a goat.

This misunderstanding is why the problem seems counterintuitive to people.

So should you switch if Monty gives you the choice AFTER he reveals a goat? Doesn't matter. Should your predetermined strategy for playing these games be to always switch? Absolutely.

Are you sure you understand the problem? You don't have to decide to switch before the choice is offered. A person who switches, even if they don't have a strategy or understand the mathematics, has a 2/3 chance of winning.

Statements like the above exemplify that problem is counter-intuitive. Of course, I don't of know anyone who completely understood the problem the first time they heard it. Davilla 16:48, 23 July 2005 (UTC)

Yes, it is quite clear to me, and you have apparently met the first such person. You are wrong if you say that 2/3 chance is independent of knowledge prior to Monty revealing a goat. It comes down to history. Without the history of Monty revealing a goat, it is merely a decision between two doors, exactly one of which has the prize. That gives you a 50:50 chance, and that is why people intuitively think there should be a 50:50 chance. It is no different than any other 50:50 scenario. That is to say, if in your life's travels you keep coming across a pair of mysterious doors with one prize, it makes no difference which one you choose and how often you change your mind. However, if the game involves the entire history starting with 3 mysterious doors, then consistently switching will increase your chances to 2/3. It is all about the given history. Indeed, if you are given additional history that the prize is behind your first chosen door, then switching will reduce your chances to zero.

The problem is that people don't seem to understand probability is context-dependent--it varies with what information you put into it. Increased knowledge increases certainty. Of course this isn't explicity stated, because why would the entire problem even be presented if it is to be ignored? It is understood that it is to be incorporated into probability calculations. However, folks who don't understand the dependence of history see the Monty problem as equivalent to the 2-door problem. They seem to think that 2/3 is some sort of physical feature of the door, rather than a measure of information and uncertainty.

So I say again, if you make a fresh decision (one ignorant of the past), your chance is 1/2, whether or not you switch. That is why knowledge is a good thing. You can often use it to increase your chances in life.

Multi step problem

This whole thing seems to hing around independent steps.

50/50 result. Some condition is set up; in this case chose 1 of 3; chance of selecting the correct outcome is exactly one third. BUT the next INDEPENDANT step is offered after the removal of one of the incorrect outcomes. What is left ? 2 options; of which one and only one is correct.Proberbility of picking the correct outcome is 1 out of 2 or 1/2.

Please demonstrate very clearly and precisely how any event prior to being offered the 2nd step choice can possibly influance the outcome of that choice.

Or to restate that - you are left with a choice of one out of two.

Somebody mentioned tossing coins; it doesn't matter how many sides previous coins had; or how many coins were tossed prior; the next toss of a double sided coin has what probability of producing a head... yes; exactly 1/2 as PRIOR conditions have zero effect. I presume at this stage no one would bet a million pounds/dollars/euros that following 1 million heads a tail would occur ? (and if you believe it MUST be a tail then you need to re-vise your basic proberbility theory

Please cut all the waffle and explain how any preceeding action affects the 1 out of 2 selection left at the end !

I'm stuck behind a dynamic isp so can only sign Pete-dtm

You've got it exactly backwards: claiming that the chance is 1/2 requires that you explain how the preceding actions do not affect the outcome. See the section below with a million doors: are you saying that after picking a door (with just a 1/1000000 chance of being right) and Monty closing 999998 wrong doors, the chance that your door is still right at the end is suddenly 1/2?! No. It started off as 1/1000000, and it still is. Since we know either this door or the other is right, there's a chance of 999999/1000000 that it's the other. And you'd like to claim tossing a coin would be the same? That's true only if the candidate is stupid and deliberately chooses at random. But we're assuming the candidate knows what Monty's doing. In that case the candidate had better use his brain, not a coin, and pick the other door.

What you're saying is that this is exactly the same as being dragged from your bed and being asked to choose between two doors, with no other information at all, except that one of them is definitely right and the other definitely isn't. That's not what's happening here. Statistically we only know that the chance of one of these doors being right is 1. You're making the mistake of assuming that means that it must be evenly split between the two doors. It's not. Tossing coins has nothing to do with this, as each coin toss, as you say, is independent of the other. But Monty closing all but one door and giving you a very good chance of getting the right one by switching is as dependent as it gets. 82.92.119.11 19:31, 23 July 2005 (UTC)

Well put. hydnjo talk 20:39, 23 July 2005 (UTC)

Which would win?

Jeesh, I can't make up my mind as to which is more interesting (revealing), the article or this talk page! hydnjo talk 18:26, 23 July 2005 (UTC)

Blow up the problem

The corrolation between gravity and time becomes evident at a black hole.

• What Monty is really doing

Guaranteeing that the correct door is one of the two left.

• Bigger

• One million doors, pick one.
• Monty opens 999998 and leaves one closed.
• Monty guarantees the correct door is either the original you picked or the one he left closed.
• Do you switch?

Contributed by User:Tailpig

• And now what are the odds of winning by the switcher? hydnjo talk 18:58, 23 July 2005 (UTC)

• I don't know, pretty good. Basically, the odds that he got the correct door on his first guess were pretty small. I'm trying to write an explanation without resorting to fractions.--Tailpig 19:28, 23 July 2005 (UTC)

• My question was rhetorical and intended for the non-believers out there. Your expansion of the problem should help them to see the error of their perception (or maybe not). Anyway the odds in favor of the switcher are obviously about 999999:1 but I'm sure that there are some that will still argue for 50%. It turns out that the odds in favor of the switcher are exactly the same as the odds against his making the winning choice at the outset. BTW, your example is an excellent expansion of the problem and should help someone to understand the underlying principle; exaggeration can sometimes be illuminating. Perhaps it should go into the main article. hydnjo talk 19:45, 23 July 2005 (UTC)

This is the clearest way to understand it. Good job! Is it in the main article? --pippo2001 21:46, 23 July 2005 (UTC)

Yes it is right here using 100 doors (not 1000000). I missed that part. Thanks. hydnjo talk 22:08, 23 July 2005 (UTC)

4 star article

This was a great featured article. Congrats to those who contributed.

LegCircus 22:34, July 23, 2005 (UTC)

Adding doors

One point that's not clear: if you have a larger number of doors (say four for simplicity), is the new situation one where Monty Hall opens one door or two doors? Peter Grey 22:46, 23 July 2005 (UTC)

He opens all but one as was the case in the original presentation. After all, if he only opened one of many, then what is it that you would switch to? One of the remaining unopened doors or all of them as a group. The original proposition says that you are left with a choice to switch or not from your original door to the one remaining door. hydnjo talk 23:03, 23 July 2005 (UTC)

The new situation is the one where Monty opens n-2 doors, where n is the total number of doors. When Monty opens his doors, it reduces the number of doors down to 2, which is what produces the illusion: people think that the probability of the car must correspond to the current number of doors, and thus be 1/2, instead of corresponding to the number of ways of getting to that 2-door situation, which is what makes it 1/n. (Contributed by User:Antaeus Feldspar)

• Just hold fast until there is only one alternative door presented to you. You see, the "know-it-all" host doesn't just reveal any door, he always reveals a losing door (otherwise what's the point). Choose the final one he presents and you will be rewarded. Trust me on this. hydnjo talk 03:08, 24 July 2005 (UTC)

Well, even if he opened only one door you would still be better off if you switched, even if only slightly so. It's really no longer a Monty Hall problem then and it certainly doesn't illuminate the problem because the odds that you win get slimmer instead of larger with more doors. However, if there were, say, 10 doors, and Monty would eliminate just one, switching would give you a chance of (9/8)/10 or 1.125/10, while staying would still be 1/10. Junes 09:29, 24 July 2005 (UTC)

Presentation

I've presented the solution as a way to "trick the host". I think beating the house appeals to most people in a way that dry mathematical explanation doesn't. — 131.230.133.185 23:47, 23 July 2005 (UTC)

Your changes are a little too drastic and at odds with our style guidelines to be accepted without consensus first, particularly considering how this is currently a featured article that came to its current state through a lot of collaborative editing. Make your case for such a change first. Postdlf 23:58, 23 July 2005 (UTC)

One of the guidelines on Wikipedia is to be bold, not to wait for everybody and their brother to agree. That policy does not suddenly change because this page is featured; after all, articles are featured to draw people to improve them (see the congratulating comment above) and I was not vandalizing. My anonymity should not count against me as I did nothing wrong, yet that is your primary argument against me in the edit history of the article. I challenge you to show me the guidelines I am violating. I can guarantee that whatever was replaced violates the letter and spirit of far more.

This page has way too many explanations. Wikipedia is not, as far as I know, a repository for every proof ever thought of for every math puzzle covered by Wikipedia. The reason it is long is because it does not adequately convince people, so more and more proofs have been added in hopes of doing so. This has not been successful. Focussing more on convincing wording would be a good idea.

— 131.230.133.185 00:08, 24 July 2005 (UTC)

Monty.gif

I find this diagram at best confusing. Or should I say the diagram looks fine, the numbers at the bottom (which add to more than 1) are confusing. Rich Farmbrough

Fundamental flaw

The current article (which is incorrect, by the way) states the following:

There are three possible scenarios, each with equal probability (1/3):

• The player picks goat number 1. The game host picks the other goat. Switching will win the car.
• The player picks goat number 2. The game host picks the other goat. Switching will win the car.
• The player picks the car. The game host picks either of the two goats. Switching will lose.

In the first two scenarios, the player wins by switching. The third scenario is the only one where the player wins by staying. Since two out of three scenarios win by switching, the odds of winning by switching are 2/3.

This is fundamentally wrong because it leaves out a scenario. It can be fixed in one of two ways:

Fix #1: There are two possible scenarios, both with equal probability (1/2):

• The player picks a goat. The game host picks the other goat. Switching will win the car.
• The player picks the car. The game host picks either of the two goats. Switching will lose.

In the first scenario, the player wins by switching; in the second, the player loses by switching.

Fix #2: There are four possible scenarios, each with equal probability (1/4):

• The player picks goat number 1. The game host picks the other goat. Switching will win the car.
• The player picks goat number 2. The game host picks the other goat. Switching will win the car.
• The player picks the car. The game host picks goat number 1. Switching will lose.
• The player picks the car. The game host picks goat number 2. Switching will lose.

In the first two scenarios, the player wins by switching; in the last two, the player loses by switching.

Once the host opens a door and reveals a goat, the old problem is gone. The new problem is there are two doors: one has a goat and one has a car. What are your odds in picking one? (Picking means changing doors, because the odds of winning stay at 1/3 for the first-chosen door.) The answer is one-half. — Fingers-of-Pyrex 00:01, July 24, 2005 (UTC)

You listed the four possible outcomes but their odds of happening are not all equal to (1/4)

• The player picks goat number 1. The game host picks the other goat. Switching will win the car. (1/3)
• The player picks goat number 2. The game host picks the other goat. Switching will win the car. (1/3)
• The player picks the car. The game host picks goat number 1. Switching will lose. (1/6)
• The player picks the car. The game host picks goat number 2. Switching will lose. (1/6)

--Tailpig 00:37, 24 July 2005 (UTC)

In your "Fix #1", you assert that the two possible scenarios have "equal probability". This is not true. There are two goats and only one car. The player therefore has a 2/3 chance of picking a goat for his first choice and only a 1/3 chance of picking the car.

In your "Fix #2", you assert that all four possible scenarios have "equal probability", which is also untrue. Scenarios 3 and 4 can only happen when the player picks the car, which there is only a 1/3 probability of. Assuming Monty picks randomly between the two goats, the latter two scenarios have a probability of 1/6 each.

Merely enumerating possibilities does not give them equal probability. If it did, I could claim that it matters whether the goat behind the door Monty opens is sleeping or awake, and then claim that there are six equiprobable scenarios: the player picks goat 1, the player picks goat 2, Monty picks goat 1 and it's awake, Monty picks goat 1 and it's sleeping, etc... or I could include speculation on whether the goat is dead and now I have eight possibilities! But the truth is that it doesn't matter whether the goat is awake, sleeping or dead, and it doesn't even matter which goat it is that Monty shows. -- Antaeus Feldspar 00:41, 24 July 2005 (UTC)

After one goat is revealed, though, there is only one goat and one car. Forget about everything else. One goat—one car. 50%. One out of two. 1/2. — Fingers-of-Pyrex 01:04, July 24, 2005 (UTC)

If I ever run for public office I will seek you out, Fingers-of-Pyrex, to be my campaign manager. hydnjo talk 01:19, 24 July 2005 (UTC)

Yes, there is one goat and one car, and that is what produces the illusion that the chances are 50/50 -- because people are used to situations where there are n possibilities and they are all equally probable, giving them the (false) intuitive belief that any time there are n possible outcomes, each of them occurs with 1/n probability. But people continue to insist that seeing n possible choices means 1/n probability even when that's not the n that figured into the probability -- that because there are two doors now, the probabilities must be 1/2, even though there were three doors at the time of the only randomizing event in the whole scenario!

Look, I'll make it really simple; here is the syllogism, and you tell me where you can possibly dispute this:

• Whether the car is behind the player's first-chosen door determines for the rest of the problem whether staying or switching is the correct strategy. If the player picks the car on his first try, he cannot win by switching, and if he picks a goat on his first try, he can only win by switching.
• The player's chance of picking the car on his first try, out of a set of three doors, is 1/3.
• Therefore, the chance that the player wins by switching is 2/3.

If you read that carefully and honestly still argue that the chances must be 1/2 because we're seeing two doors, and the fact that the only randomizing event in the problem involved three doors and not two is not relevant, there's only one step to take -- I'm coming to your house to get rich playing poker with you. =D -- Antaeus Feldspar 02:59, 24 July 2005 (UTC)

May I suggest you try the experiment with cards as suggested in the article? First, do you agree the card experiment is equivalent (3 cards including the ace of spades, shuffle, give me one, you keep two, now you discard either one as long as it's not the ace)? If so, is it still 1/2 if we do this with the whole deck - you give me one, keep the rest, discard 50 so long as none of the ones you discard is the ace of spades? If you really think this, then can you please try it? With 3 the difference between 1/2 and 2/3 may not show up until 20 or 30 trials. With the entire deck I'm relatively certain the difference will be apparent nearly immediately. The reason the result is not 50-50 is because you HAVE TO discard n-2 of the cards (just as Monty HAS TO open one of the doors) and you're not discarding at random (just as Monty is not opening a random door). In both cases, the result is that the remaining cards (remaining doors) do not have equal probability. Opening a non-random door does not change the probabilty of the original choice. -- Rick Block (talk) 01:34, July 24, 2005 (UTC)

Straight Dope article link added to main page

Kinda looks like this talk page. I guess there's no hope, they're everywhere. I'm going to rv the link as it doesn't add anything new to the article except controversy. hydnjo talk 00:50, 24 July 2005 (UTC)

Dispute tag

Methinks all should read Wikipedia:Resolving disputes. We may need to change the article's focus to the Monty Hall problem and then have two different sections for explaining the answer—one wrong and one right, of course :). In the meantime, we should apply the {{dispute}} tag to the article to point readers to this talk page. — Fingers-of-Pyrex 01:21, July 24, 2005 (UTC)

• Methinks not. A featured article has already been through peer review. Where were you then? hydnjo talk 01:31, 24 July 2005 (UTC)

Not fair. This way we can argue that featured articles are perfect and nobody who comes afterwards has any business changing them. I agree that {{disputed}} would be way overblown for this, but I'd rather not see the "it went through peer review so be quiet" argument anymore. 82.92.119.11 09:56, 24 July 2005 (UTC)

"It went through peer review so you can't change anything" is, you are quite correct, unfair. "It went through peer review so you can't change the central premise of the entire page" is, I would say, a very different story. -- Antaeus Feldspar 15:31, 24 July 2005 (UTC)

Of course. But "where were you then" is unnecessarily hostile, and unfair. I'm sure not everybody is thoroughly familiar with peer review. And even if they are, looking up the discussion to see if your concern was addressed isn't exactly easy either. I found it here. Not exactly suggesting that the matter has been thoroughly addressed, is it?

Enough, this is not productive. I'm not challenging the article, after all. I'm sure we all agree that claiming that the article is disputed is spurious; disagreeing with its current form is not the same as challenging its validity. I hope we can also agree that taking the "how dare you even suggest something like that" response isn't productive either. 82.92.119.11 15:53, 24 July 2005 (UTC)

Featured Article protection

• How do we protect or lock a featured article page (at least for a while). This seems a fundamental right for an article that has been through peer review and has gone on to be a featured article. That the article would be troll bait is obvious. So, a reasonable amount of protection during the surrounding time frame would seem prudent. I'm just guessing here but the trolls shouldn't be allowed an advantage just because we have chosen an excellent article to feature. hydnjo talk 02:36, 24 July 2005 (UTC)

The relevant policy is Wikipedia:Protection policy and, I think, none of the stated reasons apply. In particular, it says (regarding high profile pages, like featured ones) it is best not to protect pages in this case. The place to request a page be protected is Wikipedia:Requests_for_page_protection, but given the stated policy I doubt a request in this case will be honored. (sigh) -- Rick Block (talk) 02:57, July 24, 2005 (UTC)

No original reasearch

One of Wikipedia’s policies is No original research. So, please do not try to prove to me what the answer is with your own calculations. Show me your source.

The following web sites claim that the answer is 2/3 probability if you switch:

• http://www.straightdope.com/classics/a3_189.html, assertion by Cecil Adams
• http://econwpa.wustl.edu/eps/exp/papers/9906/9906001.html, assertion by professors Peter R. Mueser and Donald Granberg of the University of Missouri–Columbia
• http://www.cut-the-knot.org/hall.shtml assertion by Alexander Bogomolny
• http://www.letsmakeadeal.com/problem.htm, assertion by Monty Hall – Yes, the game show host after whom this paradox is named

For those that assert that the probability is 1/2, please site your source. -- JamesTeterenko 03:01, 24 July 2005 (UTC)

Re: Monty Hall problem

Hi. It doesn't matter if there are one trillion doors, cards, or whatever. Let me pick one. You show me all the other goats until there are two doors left. I'll have one-trillionth chance of winning the car if I stick with my first choice. I'll have 50% chance of winning if I switch. — Fingers-of-Pyrex 02:53, July 24, 2005 (UTC)

• Just hold fast until there is only one alternative door presented to you. You see, the "know-it-all" host doesn't just reveal any door, he always reveals a losing door (otherwise what's the point). Choose the final one he presents and you will be rewarded. Trust me on this. hydnjo talk 04:14, 24 July 2005 (UTC)

Your analysis would be right, Fingers, if Monty was opening doors without knowing that they were losing doors. If that was the case, then we could have a trillion doors, but we'd only be dealing with a tiny, tiny subset of possible cases, those where the car is located behind one of the last two doors to be opened. Within that subset, the chances of it being behind either one of the doors is, indeed, 50/50.

But that's not the Monty Hall problem. if we were playing the Monty Hall problem with a trillion doors, there would be a trillion cases, one for each of the different doors the player could pick, and in each and every one of those cases, the game would be reduced to two doors, one of which would have the car. But only in one of those trillion cases would the car be behind the player's door. -- Antaeus Feldspar 03:40, 24 July 2005 (UTC)

OK, I made a mistake here. I concede your point if the host reveals 50 cards. Now, answer me this. I shuffle, and deal one to you. I turn over one card that isn't the ace of spades. What should you do? Switch, or not? What are your odds of having the ace of spades if you stand pat? I say they're 1/52. What are your odds of picking the ace of spades if you switch? I say they're 1/51. What do you think about this scenario? — Fingers-of-Pyrex 04:02, July 24, 2005 (UTC)

Look, Monty doesn't just turn over any card, he turns over a losing card. That is the essence of the situation. Three doors or three million doors, he just never reveals a winning card or door. So you see, his hand is always better than your's, so switch to it if given the chance, please. If you were my kid I'd whomp you upside the head with a frying pan for being so stupid as to risk our inheritance. hydnjo talk 04:31, 24 July 2005 (UTC)

OK. If you as Monty are playing by the rules and you are eliminating one losing card from the 51 in your hand because it's a losing card, then my chances are better if I pick a new card from out of your hand at random. The only way I win if I stand pat is on the 1/52 chance that I picked the Ace of Spades on the very first try. To win by switching to a new card picked from your hand after you discarded one, first the Ace of Spades would have to be in your hand (a 51/52 chance) and I'd have to pick it out of the remaining cards (which, since you just discarded a losing card, brings that chance up from 1/51 to 1/50. Multiply 51/52 by 1/50 and you get 51/2600, which is not great odds, but is better than 50/2600 (1/52, the chance of winning by standing pat with the first card you picked, which can never go upwards during the game.) The more cards you discard because they are losing cards, the better my chances get of picking the Ace of Spades from your hand if it's there to be picked. Once you've discarded all but one card from your hand, the chance that I'll win by picking that card from your hand is 51/52. -- Antaeus Feldspar 07:00, 24 July 2005 (UTC)

Listen Finger, the only way to lose by switching is to get the correct door on the first try. This is not a philosophical question. There is no two points of view. Just accept that you don't understand.Tailpig 05:16, 24 July 2005 (UTC)

Think through the problem backwards

Part of what messes up intuition is how to make use of the partial information you have when deciding whether or not to switch. Here's another way to look at it that might help (this is sort of a rewording of "Best, most concise explanation yet") :

Assume the contestant always switches, i.e. ignore the second decision. The contestant's first choice is either a) car (probability 1/3) or b) goat (probability 2/3). In (a), contestant loses, in (b) contestant wins. Thus 1/3 probability of losing, 2/3 probability of winning.

Now assume the contestant always stays. The contestant's first choice is either a) car (probability 1/3) or b) goat (probability 2/3). In (a), contestant wins, in (b) contestant loses. Thus 1/3 probability of winning, 2/3 probability of losing. Peter Grey 04:17, 24 July 2005 (UTC)

The Differing Viewpoints

As far as I can see, the different in the two viewpoint of 2/3 and 1/2 comes from the interpretation of the problem.

• If the question is stated as "What is the probability of winning?" then the probability is "1/2".
• However, the question is stated as "Does switching improve the player's chance of winning the car?" then the answer is "yes" becuase not switching is 1/3 and switching is 2/3.

See the "Full Mathematical Solution" below. aCute 05:22, 24 July 2005 (UTC)

In short, the probability of winning really is 2/3 if you use the right strategy. (It's 1/3 if you stick by your choice, the wrong strategy unless you really like goats.) It's 1/2 if you flip a coin to decide whether to switch. Davilla 15:59, 24 July 2005 (UTC)

Full Mathematical Solution

• Let X1 be the initial choice of the player.
• Let M be the what Monty eliminate.
• Let X2 be the final choice of the player.
• Let C, G1, G2 be the car, goat_1 and goat_2.

P(X1)	P(M l X1)	P(X2 l X1, M)	.	P(X1, M, X2)
P(X1=C) = 1/3	P(M=G2 l X1=C) = 1/2	P(X2=C l X1=C, M=G2) = 1/2	.	P(X1=C, M=G2, X2=C) = 1/12
		P(X2=G1 l X1=C, M=G2) = 1/2	.	P(X1=C, M=G2, X2=G1) = 1/12
	P(M=G1 l X1=C) = 1/2	P(X2=C l X1=C, M=G1) = 1/2	.	P(X1=C, M=G1, X2=C) = 1/12
		P(X2=G2 l X1=C, M=G1) = 1/2	.	P(X1=C, M=G1, X2=G2) = 1/12
P(X1=G1) = 1/3	P(M=G2 l X1=G1) = 1	P(X2=C l X1=G1, M=G2) = 1/2	.	P(X1=G1, M=G2, X2=C) = 1/6
		P(X2=G1 l X1=G1, M=G2) = 1/2	.	P(X1=G1, M=G2, X2=G1) = 1/6
P(X1=G2) = 1/3	P(M=G1 l X1=G2) = 1	P(X2=C l X1=G2, M=G1) = 1/2	.	P(X1=G2, M=G1, X2=C) = 1/6
		P(X2=G2 l X1=G2, M=G1) = 1/2	.	P(X1=G2, M=G1, X2=G2) = 1/6

The probability of winning (not Monty Hall problem) is:

• P(X2=C)

= P(X1=C, M=G2, X2=C) + P(X1=C, M=G1, X2=C) + P(X1=G1, M=G2, X2=C) + P(X1=G2, M=G1, X2=C)

= 1/12 + 1/12 + 1/6 + 1/6 = 1/2

The probability of winning without switching is:

• P(X2=C l X2=X1)

= P(X2=C, X2=X1) / P(X2=X1)

= (P(X1=C, M=G2, X2=C) + P(X1=C, M=G1, X2=C)) / (P(X1=C, M=G2, X2=C) + P(X1=C, M=G1, X2=C) + P(X1=G1, M=G2, X2=G1) + P(X1=G2, M=G1, X2=G2))

= (1/12 + 1/12) / (1/12 + 1/12 + 1/6 + 1/6) = 1/3

The probability of winning with switching is:

• P(X2=C l X2<>X1)

= P(X2=C, X2<>X1) / P(X2<>X1)

= (P(X1=G1, M=G2, X2=C) + P(X1=G2, M=G1, X2=C)) / (P(X1=C, M=G2, X2=G1) + P(X1=C, M=G1, X2=G2) + P(X1=G1, M=G2, X2=C) + P(X1=G2, M=G1, X2=C))

= (1/6 + 1/6) / (1/12 + 1/12 + 1/6 + 1/6) = 2/3

The answer to the Monty Hall problem:

• Since probability of winning with switching is greater than probability of winning without switching, the player's strategy should be to switch. In other words, the player should be free to make any initial choice (X1), but must make sure that the final choice (X2) is different in order to maximize the probability of winning.

aCute 05:22, 24 July 2005 (UTC)

What is and what is NOT the Monty Hall problem

Note: It is my observation to lead me to conclude the following:

• It is often the case that some people argue about the final event (X2) to the exclusion of other factors, which is P(X2=C l X1=*, M=*) = 1/2, and although it is correct, but sadly it is NOT the Monty Hall problem.
• It is also often the case that some people argue about the winning-probability of player in the entire game, which is P(X2=C) = 1/2, and although it is correct, but sadly it is also NOT the Monty Hall problem.
• The Monty Hall problem is to compare the strategy of switching, which is P(X2=C l X1<>X2) = 2/3, versus the strategy of not switching, which is P(X2=C l X1=X2) = 1/3, and to conclude switching is better than not switching.

The Monty Hall problem is NOT to determine the winning-probability of the last event or the entire game, but rather to determine which of the two strategies will maximize winning-probability. The problem is like comparing the subset cases in the manner similar to the Simpsons paradox. This is also like trying to find a winning strategy in Blackjack by counting cards already played previously as additional information (sort of like Monty Hall revealing a goat) in order to adjust subsequent choices to gain an advantage over the house. aCute 06:49, 24 July 2005 (UTC)

Diagram Problem

In "the formal probability diagram" within the project page of the Monty Hall problem, the leaves of the diagram tree has the values "1/3 1/3 1/3 1/3 1/6 1/6 1/6 1/6" which cannot possibly exist since the sum of these values canNOT exceed 1.

The letter "Y" should be "1/2"

The letter "N" should be "1/2"

The final line should be "1/6 1/6 1/6 1/6 1/12 1/12 1/12 1/12"

Please someone change them ASAP. aCute 05:22, 24 July 2005 (UTC)

• You would be right, except for one important fact. We are dealing with two possible situations. The first is where the contestant switches amd if you sum up the the "Y" probabilites, these come to exactly 1 (1/3 + 1/3 + 1/6 + 1/6 = 1). The second situation is where the contestant doesn't switch amd if you sum up the the "N" probabilites, these come to exactly 1 (1/3 + 1/3 + 1/6 + 1/6 = 1). There aren't probabilites attached to the Y or N, because this isn't random. We are asking GIVEN the contestant has(n't) switched, what is the chance he gets the car. (By the way, I made that diagram). I could make the point that I have a degree in maths, and thus know what I am talking about, but some might see that as pretenious. Most problems in probability are exteremly counter-intuitive, and this is no exception. Tompw 10:45, 24 July 2005 (UTC)

Yes I have one of those too, and I found the diagram confusing (as noted above). Perhaps the colour of the YES and associates probabilites could be diffeent from the NO and assoc, probs. Perhaps a caption woul clairfy things. Rich Farmbrough 18:25, 24 July 2005 (UTC)

Okay, now I see how the diagram is constructed. Previously, I failed to see that BOTH situations are placed onto the SAME tree. aCute 19:41, 24 July 2005 (UTC)

Never Argue

"Never argue with someone who knows that he's right" hydnjo talk 05:27, 24 July 2005 (UTC)

Reductio ad absurdum

When this doesn't work well, it's time to go on to something more productive. Buh bye now. hydnjo talk 05:45, 24 July 2005 (UTC)

One More Time

Ok, so stupid me, but I have to do this. Lets take this down to such a basic level that even I understand.

• Three doors and I pick one of them.
• Monty, by the rules, gets the other two doors.

• Forget this step as Monty will always reveal a goat door whether he has one or two goats thus this step makes no difference. In "magic" performance this would be called misdirection as it has nothing to do with the outcome but it will distract you. The hidden rule is that Monty will never open a car door (this is what throws some folks off the trail) - Monty doesn't make a random selection; he knows what is behind every door after all so why bother with this step at all.

• So now, would you trade your one door selection for Monty's two door leftovers or not (remembering that he hasn't even gone to the trouble of opening a (for sure) goat door).

• n.b.: My youngish grandchildren all elected to switch.

Respectfully, hydnjo talk 21:18, 24 July 2005 (UTC)

Look at it this way

The game is fixed from the moment that you pick one door and Monty gets two doors. All else is just fluff and misdirection to make the game last longer. You hold a 1/3 chance of winning and (because he has two doors) Monty has a 2/3 chance of winning. Swap with him at your earliest opportunity. Please. hydnjo talk 02:10, 25 July 2005 (UTC)

• Best explanation yet! Peter Grey 20:04, 25 July 2005 (UTC)

I don't understand the controversy

You get one of three possibilities and Monty gets two of the three possibilities. The fact that you chose first changes nothing. If given the opportunity to swap with Monty well, just jump at it. He did, after all, try to screw you at the very outset. And, if Monty tries to confuse you with more doors, well then hold fast. When he offers a swap (which he will) just take it. Don't forget, you got to pick one door and Monty gets all the others. Lucky lucky him he would like you to believe! hydnjo talk 02:48, 25 July 2005 (UTC)

Analysis of the errors in the intuitive answer

(as opposed to the correctness of the mathematical answer)

It is not enough to describe why the mathematically derived solution is correct. To resolve the paradox to the satisfaction of all, one must also describe why the intuitive solution is wrong. I think this requires three steps.

1) an understanding that the "intuitive solution" is just a dismissive term for a first analysis that turned out to have a flaw

2) a description of the logical steps that were employed in coming to the intuitive solution

3) an analysis of that logic, to find its flaw(s)

(1) I not argue the point, but rather just hope that people agree with it.

(2) I think that the intuitive solution takes the following steps:

• transform the game into a simpler yet completely isomorphic game
• calculate (or intuit) the probabilities for that game
• extrapolate the answer back to the original Monty Hall game

Here is the game that I believe is used intuitively (I’ll call it the Silly game):

• There are three doors, A, B, and C.
• Door A is open and has a goat behind it.
• Doors B and C are closed, and there is a goat behind one and a car behind the other.
• The contestant just guessed that door B has the car behind it, and is now being given a chance to change his mind.

Should he change it or not?

Clearly the answer to this is that the chance is 50% either way, and it does not matter whether the contestant changes his mind.

Now compare the Silly game to an original Monty Hall game at a point half way through, in which the contestant has guessed door B and Monty has opened door A.

• There are three doors, A, B, and C
• Door A is open and has a goat behind it.
• Doors B and C are closed and there is a goat behind one and a car behind the other.
• The contestant just guessed that door B has the car behind it, and is now being given a chance to change his mind

It seems that the Silly game is exactly the same as the original Monty Hall game at this point. To a viewer who has just ‘tuned in’ and does not know what has previously happened, the games look identical. Therefore, logic would dictate that the answer to the original question is that it doesn't matter whether the contestant changes their mind; the probability is 50% either way.

(3) It turns out, of course because the two games are not completely isomorphic. There is a crucial piece of information missing from the Silly game that is needed to make it isomorphic with the Monty Hall game, as follows: Monty (who knows where the car is)

• was required to open a door to reveal a goat,
• was given an opportunity to open door C,
• chose not to open door C.

Monty has thus said something concrete about door C (“I didn’t choose it, perhaps because I couldn’t choose it”), but nothing about door B. This is the source of the asymmetry between doors B and C, and the reason that door C is more likely to not have a goat behind it. Happyharris 20:57, 25 July 2005 (UTC)

Perhaps, what we need in the main project page is an explanation to "why the intuitive answer is NOT the Monty Hall problem". Something like: Extrapolating the probabilities of an isomorphic game back to the Monty Hall game is the cause of much of the controversy of the game, since answering the probabilities of the game with randomized strategy (1/2) is NOT answering the probabilities of each strategy of switching (2/3) and not switching (1/3). aCute 08:50, 27 July 2005 (UTC)

Breaking it down into steps

For the people who still find it hard to see why the probability is not 50/50 when there are two doors left, I hope the following illustration may help. I'm going to show that the Monty Hall problem is a specific case of a more general game; I'll call all the games that differ in their parameters "Hall games" for ease of use.

The basic idea behind a Hall game is this: We start with one large set of secret-hiding items -- they can be doors to be opened, or cards to be turned over, it doesn't actually matter. What matters is that there are n cards, but only one of them is the Prize card. The rest are Null cards.

Step One: The cards are divided into two hands. Each hand must have at least one card. For simplicity's sake, we call the number of cards in the first and second hands h1 and h2.

Step Two: Someone who can see which cards are Nulls can discard some number of Nulls from one hand, the other, or both. We call the number of cards remaining in each hand after the discarding of Nulls r1 and r2 (like h1 and h2, they must be at least one.)

Step Three: The player makes a guess at which of the two hands contains the Prize.

Step Four: The player makes a guess at which card out of the hand he selected is the Prize.

It's clear that to win the game, the player has to make correct guesses in both Step Three and Step Four. What are the chances of picking the correct hand? The first hand is correct h1/n of the time; the second hand is correct h2/n of the time. If we want, we could enumerate the cases: if h1=2 and h2=5, then the Prize could be the first card of the first hand, the second card of the first hand, the first card of the second hand ... et cetera, et cetera.

Now this is the part that many people find counter-intuitive. If we enumerate the cases, and then we reduce Nulls to meet any legal value of r1 and r2, we find that in no case can the removal of Nulls switch the Prize from one hand to the other. This means that the chance of the card being in the first hand or the second always stays at h1/n and h2/n -- even if the sizes of the hands do not stay at h1 and h2. If it's dealt to that hand, it stays in that hand; therefore the chance of it being in a particular hand is always equal to the chance that it was dealt to that hand.

What about the removal of Nulls? Does it affect anything after all? Yes, it does -- it affects the player's chances in Step Four. The chance that the Prize is in a particular hand is dependent upon h1 and h2 -- how many cards each hand started with. But the chance of finding the Prize in a hand (assuming it's the right hand) is based on how many cards that hand contains after Nulls have been removed -- since there's only one Prize, the chance of finding it in a hand of r1 cards is 1/r1.

Now, what are the chances of making both guesses correctly? If no Nulls get removed from either hand, then the chances of picking the Prize are either h1/n x 1/h1 (since r1 is equal to h1 when no Nulls have been removed) or by similar logic h2/n x 1/h2, which also multiplies out to 1/n. If, however, one of the hands -- say, hand 1 -- has been reduced down to one card (r1 = 1), then the chances of finding the Prize in that hand is h1/n x 1/1 -- if you've correctly guessed that the Prize is in that hand, you have a 100% chance of finding it in that hand when it's the only card in the hand.

With this being the general structure of the Hall game, we can see that the Monty Hall problem is really just the case where n=3, h1=1 and h2=2, r1=1 and r2=1. The chance that the Prize is in the player's hand is h1/n -- 1/3. The chance that it's in Monty's hand is 2/3. Before one Null is removed from Monty's hand, the chance of finding the Prize in his hand is 2/3 x 1/2 -- i.e., 1/3. But when the Null is removed, the chance is now 2/3 x 1/1 -- i.e., 2/3! -- Antaeus Feldspar 03:44, 26 July 2005 (UTC)

three prisoners problem

kudos to those contributors to this article. i have been thinking about this for days (despite the fact that i "got it" ... after about 20 minutes or so). i find the "two sets" explanation the most clear, though i'm sure some people will love the bayes' theorem explanation. i also found the talk page very entertaining. some contributors are clearly manifesting what kahneman and tversky (writers on cognitive biases - the former won a nobel prize) refer to as "belief perseverance." geez, sit down with a friend and three cards for 10 minutes and you'd realize you're wrong. anyway... i thought you might find this page interesting:

http://econwpa.wustl.edu:8089/eps/exp/papers/9906/9906001.html

besides elaborating on many of the key assumptions (often unstated) underlying the MHP, the paper presents an interesting (and supposedly earlier) problem of the same form, the "three prisoners problem." here's the upshot...

There are three prisoners, you and Prisoners A and B. Two of you are to be executed, while one will be pardoned. The prison warden knows who will be executed and who will be pardoned (like monty must know where the goats and car are). According to policy, the warden is NOT allowed to tell any prisoner if he/she is to be pardoned. You point out to the warden that if he tells you if A or B will be executed, he will not be violating any rule. The warden says C is to be executed. What is the chance that you will be pardoned?

the answer presented, like that to the the MHP, is that your chances of being pardoned remain at 1/3, while the the chances of B being pardoned, GIVEN WHAT THE WARDEN HAS TOLD YOU, is 2/3.

I found looking at this problem reinforced how *$&%^@# hard it is to get one's head around the MHP, because despite having read about the latter for days, i still had to think about the three prisoners problem for several minutes to get my head around IT, despite knowing that it was essentially the same as the MHP. K-razy.

someone mentioned the idea of blocking edits to featured articles for some period of time. while apparently there is a policy against such things, i agree with the suggestion, at least where articles that are sources of dispute. the featuring of an article doubtless attracts many potential editors, some of who may simply not know enough about the subject to edit appropriately. for example, the first time i read the monty hall page (when it was featured), someone had altered the solution section to express the misguided (given certain assumptions, which too often are unstated) 50/50 approach. needless to say, i was confused by the fact that all other sections of the article contradicted the solution.

good job. Contributed by 24.89.202.141.

The Mueser and Granberg paper is already one of the references, and the three prisoner's problem is the Gardner problem referred to in both the "The problem" section and the "Origins" section (although Grardner's version is not described). I suspect the point of allowing featured articles to be edited is to reinforce the notion that wikipedia is really, no kidding, editable by anyone. I find this to be a very principled, and quite admirable, stance even though it does require some effort (and I suspect protecting the main page was only done with the greatest reluctance). I cannot take credit for much more than nominating the article as a WP:FAC, but thanks for the compliment. -- Rick Block (talk) 01:01, July 27, 2005 (UTC)