The Monty Hall problem

[bruto]

I'm not sure what you mean by "none if he opened the first". In both cases you mention, the probability of winning if you switch is different than it is in the case where Monty opens the same door but we know that he chooses randomly when he has a choice.

If we are guessing that he's opening, not randomly, but the first available door in a set order, then when he opens the first it could be either the only one or one of two. If he skips to the second, then we can surmise that he skipped the first because it has a prize in it.

That's reasonable, but it's not the same as saying that, even if we had such knowledge, it wouldn't make any difference anyway, which is what I thought you meant.

I don't think that's what I meant...I think I meant that if we really know anything, it's better than not knowing anything, but that the puzzle assumes that we cannot know more than is stipulated.....I'm confusing myself by now, so I think it's time to quit and go watch a movie.

 

[marplots]

3. The situation is incredibly contrived. Where else, outside a game show, you have to make a decision based on a partial information, from a knowledgeable source, who is deliberately withholding part of information? It is just not a common occurrence.

This is how God works; why God has to have a personality; and generates the phrase, "He works in mysterious ways."

 

[bruto]

This is how God works; why God has to have a personality; and generates the phrase, "He works in mysterious ways."

And you have to build an altar and waste half the goat.

 

[humber]

I can think of three reasons people have such hard time with Monty Haul problem.

1. Conditional probability is something most people never learn. If you understand conditional probability, it is much easier.

2. A lot of people, including mathematically educated ones, tend to think of "probability of an event" as some objective quantity of said event, whereas in reality it is just a measure of one's ignorance. A tossed coin has "traditionally" 50% of landing on either side. But if you knew precisely the coin's velocity, rotation, aerodynamic properties, air density, and air currents, you would be able to tell "heads" or "tails" with 100% certainty. If you had only some of that information, you could say "80% chance of heads" (for example). As your knowledge of a situation changes, "probability of event" changes. And in Monty Haul you receive additional information when the host opens a door and shows you the goat.

3. The situation is incredibly contrived. Where else, outside a game show, you have to make a decision based on a partial information, from a knowledgeable source, who is deliberately withholding part of information? It is just not a common occurrence.

Exactly. It's all about what you don't know, and what Monty does, and the additional information he provides.
Replace Monty with a randomly selected audience member, and that exposes how contrived the puzzle is.

 

[69dodge]

Are you named after the suspension bridge in England?

 

[BenBurch]

Exactly. It's all about what you don't know, and what Monty does, and the additional information he provides.
Replace Monty with a randomly selected audience member, and that exposes how contrived the puzzle is.

Except the audience member has a chance of accidentally picking the car for the reveal.

 

[epepke]

Oh, this old chestnut again?

The question, as stated to vos Savant, is unanswerable. vos Savant's answer is based on assumptions. I no longer try to explain those assumptions, because people just get all huffy about them. I have a way of winning drinks and dinners from people who are impressed with themselves because they think they are very clever to come up with the "you should always switch" answer. One is not supposed to give away magical secrets here.

 

[69dodge]

Suppose the contestant picks door 1, and then an audience member opens door 2 and reveals a goat.

Let c1 mean "the car is behind door 1", c2 mean "the car is behind door 2", and c3 mean "the car is behind door 3". Let a2 mean "the audience member opens door 2 and reveals whatever's behind it". Let a2g mean "the audience member opens door 2 and reveals a goat". (If the car isn't behind door 2, then a2 and a2g are equivalent. Also, the probability of a2 doesn't depend on where the car is, because the audience member doesn't know where the car is.)

We are interested in the probability that the car is behind door 1, given that the audience member opens door 2 and reveals a goat. That is, we're looking for P(c1 | a2g).

We use Bayes's Theorem.

P(c1 | a2g) = P(c1) P(a2g | c1) / P(a2g)
= P(c1) P(a2 | c1) / P(a2g)
= P(c1) P(a2) / P(a2g)
= P(c1) P(a2) / [P(a2g | c1) P(c1) + P(a2g | c2) P(c2) + P(a2g | c3) P(c3)]
= P(c1) P(a2) / [P(a2) (1/3) + (0) (1/3) + P(a2) (1/3)]
= P(c1) / (2/3)
= (1/3) / (2/3)
= 1/2.

 

[bruto]

I think some people are putting too much thought and speculation into this. It's a contrived problem, a game. You play a game by its rules, arbitrary or not. There are no what-if's here. What is specified is what the game consists of. As I recall, the problem as stated is simply that Monty, who knows what's behind the doors, opens a door to reveal a goat, then offers you the chance to switch. There is no audience picking, no way to know how Monty decides which door to open if both have goats, etc., and speculating on these things makes no more sense than speculating on how we should bet in a poker game if there were 5 kings in the deck.

 

[Rolfe]

The point epepke is making is that people are assuming rules without them being stated, and these assumptions are not necessarily valid.

If Monty always opens a second door, your chances aren't harmed by switching. If that second door always reveals a goat, your chances are doubled by switching. You can choose to assume either or both if these rules when you solve the puzzle, but neither is in fact stated.

Rolfe.

 

[humber]

Are you named after the suspension bridge in England?

No, I am that bridge. The constant painting...oh, the painting.
No, the name has no particular relevance, 69Dodge,

And....the puzzle is another of Vos Savant's contrived party tricks.

 

[zooterkin]

No, I am that bridge. The constant painting...oh, the painting.

Are you thinking of the Forth bridge?

And....the puzzle is another of Vos Savant's contrived party tricks.

The puzzle was proposed by Steve Selvin, according the the wikipedia page. I don't think anyone would disagree that it's simply a maths puzzle, but you seem to be implying something else in describing it as a 'contrived party trick'. What are you trying to say?

 

[JoelKatz]

It would be unfair if he did open a door for some people and not for others because he would be giving some people odds of 1/3 of choosing a car, and other people greater odds.

Not if he, say, flipped a coin to decide whether to offer you the choice of switching or not. That would still be fair. For the problem to have a definite answer, we must understand the rules Monty follows. (Otherwise, the correct answer is to randomly decide whether to switch or not. I think you switch 2/3 of the time, but I don't recall exactly. If you never switch, you never get the benefit of the additional information. If you always switch, Monty can make it so you always lose when you switch.)

As I've said, though, I can't see any time that the problem as you describe it would ever be presented as a mathematical puzzle. What you seem to be describing is a scenario where someone would set out a version of the problem in the hope that someone, recognising the problem would say the same things that people in this thread have been saying in order for the person who set the problem to be able to leap out from behind a bush and shout "gotcha!" at them.

If a person doesn't understand precisely what it is about the problem that makes "switch and you double your odds" the correct answer, then they haven't grasped the point of the question. That answer is only correct if Monty is required to show you a goat door no matter what door you pick. If you attempt to apply this to a situation where Monty's rules are different, switching can make your odds worse. (For example, suppose Monty only offers you the choice of switching if he knows your first choice was correct.)

If the point is to learn how to use mathematics to make real-world decisions, perhaps the most important thing is understanding in what situations the reasoning applies.

 

[humber]

Are you thinking of the Forth bridge?

Oh, right. The Humber bridge is relatively modern, and not in Scotland.
You have blown my cover.

The puzzle was proposed by Steve Selvin, according the the wikipedia page. I don't think anyone would disagree that it's simply a maths puzzle, but you seem to be implying something else in describing it as a 'contrived party trick'. What are you trying to say?

Well, naughty Marilyn...
Perhaps she ripped it off. This answer may support that.

http://www.parade.com/articles/editions/2006/edition_11-26-2006/Ask_Marilyn

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

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

Her reply is wrong, because it does not matter if the goat is revealed by chance or by intent.

It's contrived because of hidden assumptions that are pandered to, and the limits and conditions of a game show. If Monty were to reveal the car, it would not really serve the game show's purpose.

 

[hgc]

Not if he, say, flipped a coin to decide whether to offer you the choice of switching or not. That would still be fair. For the problem to have a definite answer, we must understand the rules Monty follows. (Otherwise, the correct answer is to randomly decide whether to switch or not. I think you switch 2/3 of the time, but I don't recall exactly. If you never switch, you never get the benefit of the additional information. If you always switch, Monty can make it so you always lose when you switch.)

If a person doesn't understand precisely what it is about the problem that makes "switch and you double your odds" the correct answer, then they haven't grasped the point of the question. That answer is only correct if Monty is required to show you a goat door no matter what door you pick. If you attempt to apply this to a situation where Monty's rules are different, switching can make your odds worse. (For example, suppose Monty only offers you the choice of switching if he knows your first choice was correct.)

If the point is to learn how to use mathematics to make real-world decisions, perhaps the most important thing is understanding in what situations the reasoning applies.

Thank you. So much of the discussion about the Monty Hall problem is due to misunderstanding the question, either because it was posed ambiguously or people are not paying attention to the rules.

Monty ALWAYS reveals a goat, and then asks if you want to switch. It never goes any other way.

 

[Rolfe]

Monty ALWAYS reveals a goat, and then asks if you want to switch. It never goes any other way.

Who says?

This statement does not describe his behaviour in real life, and there is no such stipulation in the problem as posed in the OP.

Rolfe.

 

[humber]

Who says?

This statement does not describe his behaviour in real life, and there is no such stipulation in the problem as posed in the OP.

Rolfe.

It's not stated, but it is expected that a game show host won't show the car to the contestant.

ETA: That is always the case, so if the contestant's choice is made after the reveal of the donkey, as posed by the OP, then the chance is 50/50.

 

[AvalonXQ]

Oh, right. The Humber bridge is relatively modern, and not in Scotland.
You have blown my cover.



Well, naughty Marilyn...
Perhaps she ripped it off. This answer may support that.

http://www.parade.com/articles/editions/2006/edition_11-26-2006/Ask_Marilyn

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

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

Her reply is wrong, because it does not matter if the goat is revealed by chance or by intent.

Actually, her reply is correct. In this scenario, there are two equally likely possibilities, and one possibility of zero likelihood...
A (probability 1/2): The car is behind door 1. You chose door 1, and the host randomly opened door 3 to reveal a goat.
B (probability 1/2): The car is behind door 2. You chose door 1, and the host randomly opened door 3 to reveal a goat.
C (probability 0): The car is behind door 3. You chose door 1, and the host randomly opened door 3 to reveal a goat.
Because the known facts give equal probability to door 1 or door 2 holding the car, staying or switching doesn't make any difference.

 

[humber]

Actually, her reply is correct. In this scenario, there are two equally likely possibilities, and one possibility of zero likelihood...
A (probability 1/2): The car is behind door 1. You chose door 1, and the host randomly opened door 3 to reveal a goat.
B (probability 1/2): The car is behind door 2. You chose door 1, and the host randomly opened door 3 to reveal a goat.
C (probability 0): The car is behind door 3. You chose door 1, and the host randomly opened door 3 to reveal a goat.
Because the known facts give equal probability to door 1 or door 2 holding the car, staying or switching doesn't make any difference.

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

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

It makes no difference at all if a goat is revealed if he knows or not.
"Much to his relief" suggests why that is.

 

[roger]

The point epepke is making is that people are assuming rules without them being stated, and these assumptions are not necessarily valid.

If Monty always opens a second door, your chances aren't harmed by switching. If that second door always reveals a goat, your chances are doubled by switching. You can choose to assume either or both if these rules when you solve the puzzle, but neither is in fact stated.

Rolfe.

And you have to assume that

* someone doesn't switch things around behind the doors after you've chosen
* that the objects don't quantum tunnel into another universe
* that gravity continues to work, and the prize does not float away before the door is opened.
* that if you choose to switch, gunmen won't burst out of a door and shoot you dead
* that Monty is not a powerful magician, and won't turn you into a newt if you pick the prize.
* that we are speaking English, not English Prime, where the sentences sound like English but you are really agreeing to have your head chopped off
* on to infinity....


What are we arguing about? It's obvious that you have to make assumptions in any puzzle. We all understand that if the assumptions change, the math changes as well. We all understand that there is ambiguity in the way it was worded in the vos Savant column. We also understand that if if it was worded in a fully rigorous, mathematical sense no one but a few math geeks would get any traction on understanding the question. For example, wikipedia offers:

Suppose you're on a game show and you're given the choice of three doors [and will win what is behind the chosen door]. Behind one door is a car; behind the others, goats [unwanted booby prizes]. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one [uniformly] at random. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?

which I suggest is full of assumptions and ambiguity, such as my bulleted list above.