The Monty Hall problem

[69dodge]

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.)

Can you say more about this?

If we approach the problem from the point of view of game theory, and suppose that Monty is trying to make you lose, I don't see why you should ever switch.

By never switching, you can guarantee that you'll win at least 1/3 of the time. And by never allowing you to switch except when you've picked the car, Monty can guarantee that you'll win at most 1/3 of the time.

 

[Rolfe]

And you have to assume that

* someone doesn't switch things around behind the doors after you've chosen

Indeed, but that would be outwith any normal game parameters

* that the objects don't quantum tunnel into another universe

Assumption of normal terrestrial physics continuing to apply is not usually required to be stated.

* that gravity continues to work, and the prize does not float away before the door is opened.

Assumption of normal terrestrial physics continuing to apply is not usually required to be stated.

* that if you choose to switch, gunmen won't burst out of a door and shoot you dead

Indeed, but that would be outwith any normal game parameters

* that Monty is not a powerful magician, and won't turn you into a newt if you pick the prize.

Assumption of normal terrestrial physics continuing to apply is not usually required to be stated.

* 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

Indeed, but that would be outwith any normal game parameters

* on to infinity....

I think not.

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:

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

It's pretty silly to object to people pointing out that your way of looking at it involves unwarranted assumptions, by launching into a list of preposterous suggestions including quantum wormholes and magic.

The fact is, if the assumptions are as you would have them, switching doubles your chances of winning. That's easy to show. However, that's only the start of the exercise. These assumptions are not stated in the original formulation of the problem. Other "rules" are perfectly possible, such as one that has Monty picking which door he opens at random, or even one where he only offers the switch if you already picked the car.

A full exploration of the ramifications of the puzzle is only possible if these other, plausible, scenarios are also explored. And if you explore them you find that they lead to different answers, all of which are valid answers to the original problem, which didn't state the exact scenario in operation.

And no amount of bluster about magic or wormholes changes that.

Rolfe.

 

[humber]

It's pretty silly to object to people pointing out that your way of looking at it involves unwarranted assumptions, by launching into a list of preposterous suggestions including quantum wormholes and magic.

The fact is, if the assumptions are as you would have them, switching doubles your chances of winning. That's easy to show. However, that's only the start of the exercise. These assumptions are not stated in the original formulation of the problem. Other "rules" are perfectly possible, such as one that has Monty picking which door he opens at random, or even one where he only offers the switch if you already picked the car.

A full exploration of the ramifications of the puzzle is only possible if these other, plausible, scenarios are also explored. And if you explore them you find that they lead to different answers, all of which are valid answers to the original problem, which didn't state the exact scenario in operation.

And no amount of bluster about magic or wormholes changes that.

Rolfe.

The answer is not disputed. The premise it that unless forced to, the host will not reveal to the contestant where the car is.
Monty Hall is a real person, and host of the game show "Let's Make a Deal"
http://www.letsmakeadeal.com/

 

[Rolfe]

The answer is not disputed. The premise it that unless forced to, the host will not reveal to the contestant where the car is.

At the risk of repeating myself, who says?

Monty Hall is a real person, and host of the game show "Let's Make a Deal"
http://www.letsmakeadeal.com/

For goodness sake, we know that.

The real Monty didn't behave in a consistent manner as far as anyone I've encountered has been able to discern. Not even the premise you stated as if it were fact.

For the purposes of the puzzle, it does no good to say, it was a real game show. You can't insist on following the "Real Monty" rules, because there was no consistent real Monty rule. And indeed, the statement of the puzzle doesn't specify that either. The first time I heard the puzzle, the name Monty Hall wasn't even mentioned.

State the ground rules, and the problem is soluble. Omit them, and the best an answer can do is say, this is so, IF the ground rules are as I assume.

Rolfe.

 

[Marquis de Carabas]

What if the contestant opens door 0.9...?

 

[William Parcher]

In Soviet Russia, door picks you.

 

[69dodge]

What if the contestant opens door 0.9...?

He wins a car with probability 0.333..., and a goat with probability 0.666... .

 

[humber]

At the risk of repeating myself, who says?




For goodness sake, we know that.

The real Monty didn't behave in a consistent manner as far as anyone I've encountered has been able to discern. Not even the premise you stated as if it were fact.

For the purposes of the puzzle, it does no good to say, it was a real game show. You can't insist on following the "Real Monty" rules, because there was no consistent real Monty rule. And indeed, the statement of the puzzle doesn't specify that either. The first time I heard the puzzle, the name Monty Hall wasn't even mentioned.

State the ground rules, and the problem is soluble. Omit them, and the best an answer can do is say, this is so, IF the ground rules are as I assume.

Rolfe.

Of course, one can play with all the variants, but, surely, only those where the host does not reveal the prize are of any interest. Monty or not, the expectation is that the solution will not be so banal as for the host to reveal where the prize is. If that were not relevant, then why bother with the host and so forth?

The "original" problem may not have been framed that way, which is why I said the Monty version is contrived. The intent of Vos Savant's version, is to create an apparently "surprising" result, when in fact it's a simple case of conditional probability. That does not help explain the way "it works" to those who don't have your knowledge of statistics.

 

[hgc]

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.

That's the whole point of the problem -- to always behave a certain way so that the answer can be deduced through the application of logic using the information provided. It's a math problem, not a quiz about real life or behavior.

So, I say: Monty ALWAYS reveals a goat. If you don't assume that, then you've missed the point entirely of the "Monty Hall problem" and have ventured into a digression.

 

[Rolfe]

You're not reading. That's the simple answer, and may be summarised as "Oh God not again, switch", which was my first post in the thread.

Boring.

Understanding that this is only a subset of the possible answers to this problem is necessary in order to take the debate to the next level.

Rolfe.

 

[Vorpal]

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.

You are quite wrong. If the host is clueless, then the goat tells you nothing except that particular door is wrong. But if the host knows and is constrained by your pick, then his choice carries more information. This is clear from the "pick two but say the third" equivalency if the host knows, which does not work if the host does not.

You can also go through the cases or write a simulation if you wish. You'll see that both staying and switching win half the time that the host didn't accidentally reveal the car. I've done exactly that in a previous thread on this problem.

 

[69dodge]

You are quite wrong.

Yes.

He's also trolling, so I wouldn't continue to argue with him after pointing out once that he's wrong.

 

[JoelKatz]

If we approach the problem from the point of view of game theory, and suppose that Monty is trying to make you lose, I don't see why you should ever switch.

You're correct. My analysis was for a different version of the problem where you can switch whether or not Monty shows you a goat. In that case, if Monty can't show you a goat only when you have a winning door, because if he does, you'll know to switch when he doesn't show you a goat.

 

[JoelKatz]

You are quite wrong. If the host is clueless, then the goat tells you nothing except that particular door is wrong. But if the host knows and is constrained by your pick, then his choice carries more information. This is clear from the "pick two but say the third" equivalency if the host knows, which does not work if the host does not.

You can also go through the cases or write a simulation if you wish. You'll see that both staying and switching win half the time that the host didn't accidentally reveal the car. I've done exactly that in a previous thread on this problem.

Vorpal is 100% correct. If the host is clueless, you will win 1/3 of the time whether you switch or not. It's obvious why you will win 1/3 of the time if you don't switch. Here's what happens if you do switch:

There are three cases:
1) 1/3 of the time, you pick the winning door initially. Whichever door Monty opens, he shows you a goat, since both doors have a goat. You switch to the other goat, you lose.
2) 2/3 of the time, you pick a losing door initially. There are two sub cases:
2A) 1/3 of the time (half the time you pick a losing door), Monty, being clueless, opens the winning door. You are stuck with the losing door (whether you switch are not). You lose.
2B) 1/3 of the time (half the time you pick a losing door), Monty shows you the other losing door. You switch, you win.

So as you can see, you lose in case 1, case 2a, and case 2B. So if Monty is clueless, and you switch every time you can, you win 1/3 of the time.

 

[humber]

You're not reading. That's the simple answer, and may be summarised as "Oh God not again, switch", which was my first post in the thread.

Boring.

Understanding that this is only a subset of the possible answers to this problem is necessary in order to take the debate to the next level.

Rolfe.

Well, that's right. The question is not fully delineated, but that does not necessarily allow all variations to be considered.
There must certain expectations around a game show that are commonly shared, or the puzzle would make no sense.

 

[Ray Brady]

Personally, I'm aghast at the suggestion that winning a goat is the equivalent of losing. Goats are awesome. They've got square pupils--how cool is that?

 

[JoelKatz]

Here's another way to understand Vorpal's point. If Monty doesn't know which door has the prize, he has no additional information to give you.

Say Monty says to you, "Hey, I want to help you. So before you choose, let me tell you what I'm going to do, since I don't know anything you don't know. If you pick the first door, I'll open the second. If you pick the second, I'll open the third. If you pick the third, I'll open the first. Now, pick any door you want."

If you pick the first, Monty will open the second, you will switch and get the third. So picking the first and switching is *precisely* the same as picking the third and not switching. Your odds are 1/3 either way -- you win if and only if the prize is behind door three.

 

[ftl]

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.

At the risk of repeating myself, who says?

That's the definition of "The Monty Hall Problem."

Perhaps the OP did not state all of the details of the problem, assuming that the reader would know them all, but if you look up the problem on Wikipedia, they're there. http://en.wikipedia.org/wiki/Monty_Hall_problem .

How Monty actually behaved in his actual show isn't really relevant at this point - "The Monty Hall Problem" is a well-defined problem with a counterintuitive but well-defined solution.

 

[humber]

Personally, I'm aghast at the suggestion that winning a goat is the equivalent of losing. Goats are awesome. They've got square pupils--how cool is that?

I agree. Who wants the car?
Goats can assuredly "walk" around a Ford Escort at the window level.
They make the very best of sacrificial animals, and we all know how important that is.

 

[Rolfe]

Well, that's right. The question is not fully delineated, but that does not necessarily allow all variations to be considered.
There must certain expectations around a game show that are commonly shared, or the puzzle would make no sense.

I agree. Which is why I was mocking the suggestion that someone might switch the prizes behind the scenes after the final choices had been made, or that a quantum wormhole might open up and suck in the prize, and so on.

However, there are a number of perfectly reasonable variations which might be assumed, and which are not specified in the original formulation of the problem - most notably, when Monty chooses the second door, is he flipping a mental coin, or is he deliberately choosing a goat?

Consideration of these possibilities, remaining within the permutations of what might generally be considered reasonable for a game show, extends the scope of the puzzle and adds to its interest.

Rolfe.