Monty Hall Problem

[Dr Adequate]

Maybe you can point me to a statement of the problem where he always unveils one goat? I don't think I've ever seen one in the wild.

That just is what the "Monty Hall Problem" is. No account of it says "and then the host chooses whether of not to open a door without a goat behind it", they all say along the lines of:

Finalists in a tv game show are invited up onto the stage, where there are three closed doors. The host explains that behind one of the doors is the star prize - a car. Behind each of the other two doors is just a goat. Obviously the contestant wants to win the car, but does not know which door conceals the car.

The host invites the contestant to choose one of the three doors. Let us suppose that our contestant chooses door number 3. Now, the host does not initially open the door chosen by the contestant. Instead he opens one of the other doors - let us say it is door number 1. The door that the host opens will always reveal a goat. Remember the host knows what is behind every door!

The contestant is now asked if they want to stick with their original choice, or if they want to change their mind...

If it's optional, then we have a completely different problem.

Let's go back to the street hustler version. Let's say it's a given that this is a fair shell game, that the hustler makes his money because he moves the shells so quickly that anyone has only a 1/3 chance of guessing correctly. You put down your money, make your pick, and then the hustler doesn't reveal your cup, but instead turns over an empty cup and offers you the choice to switch. Should you?

You'd be a fool to switch in this case, I hope that's obvious. And this problem is no different in substance from the Monty Hall problem. The only difference is what your assumed motivations for the game host are.

You should switch if he does that every time, street hustler or Monty Hall. If you watched, and he only did it sometimes, then you'd get suspicious.

 

[Number Six]

"since no matter whether you initially picked the correct door, Monty will show you a door with a goat."

If we knew that beforehand then yes, it is correct to switch. But note that MathWorld only told us that Monty would definitely show us a goat in their explanation of the correct solution. That is, they told us that _after_ they told you that it was best to switch.

In order to determine if it is best to switch we need to know at the time we're making the decision whether Monty was going to reveal a goat no matter what door we chose. If we don't know Monty's motiviations at the time we're making our decision then we can't tell what is best _unless_ we make assumptions about Monty's motivations.

 

[gnome]

If we knew that beforehand then yes, it is correct to switch. But note that MathWorld only told us that Monty would definitely show us a goat in their explanation of the correct solution. That is, they told us that _after_ they told you that it was best to switch.

In order to determine if it is best to switch we need to know at the time we're making the decision whether Monty was going to reveal a goat no matter what door we chose. If we don't know Monty's motiviations at the time we're making our decision then we can't tell what is best _unless_ we make assumptions about Monty's motivations.

I have always favored the version where he always reveals a goat because stated that way, it is a simple puzzle with a non-intuitive answer. It illustrates something curious about probability. I don't think it's a stretch to suppose that the author of the puzzle intended it this way.

 

[BillHoyt]

If we knew that beforehand then yes, it is correct to switch. But note that MathWorld only told us that Monty would definitely show us a goat in their explanation of the correct solution. That is, they told us that _after_ they told you that it was best to switch.

In order to determine if it is best to switch we need to know at the time we're making the decision whether Monty was going to reveal a goat no matter what door we chose. If we don't know Monty's motiviations at the time we're making our decision then we can't tell what is best _unless_ we make assumptions about Monty's motivations.

Do this experiment. Read your monitor. Now move in closer and read it again. Now closer. Closer still. Notice how there is a point, past which, closer reading is counterproductive? You're beyond that point.

The problem statement doesn't say "maybe" Monty reveals the goat. It says he reveals the goat. Period.

 

[garys_2k]

I had NO idea this thread would do this! Amazing, it's really amazing.

 

[CurtC]

Now, the host does not initially open the door chosen by the contestant. Instead he opens one of the other doors - let us say it is door number 1. The door that the host opens will always reveal a goat. Remember the host knows what is behind every door!

If it's optional, then we have a completely different problem.

But in the example you yourself use here, it is given that the host always reveals a goat. In that case, the correct answer is "switch." I have no quarrel with that one.

I just don't think it's reasonable to assume, the way the problem is usually stated (see the OP of this thread), that Monty is constrained to offer you the switch always. Maybe that's 'cause I used to watch the show as a kid, and I know how freewheeling and impromptu it was. But all you know from most problem statements is that you've been offered a choice this time, from which it's quite a leap of assumption-making to arrive at the conclusion that the game is always played this way.

Cecil Adams put it well in his column on the subject:

Your analysis of the game show question is correct, Bobo, only if we make several assumptions: (1) Monty Hall knows which door conceals the prize; (2) he only opens doors that do NOT conceal the prize; and (3) he always opens a door. Assumptions #1 and #2 are reasonable. #3 is not.

...

Cecil is happy to say he has heard from the originator of the Monty Hall question, Steve Selvin, a UCal-Berkeley prof (cf American Statistician, February 1975). Cecil is happy because he can now track Steve down and have him assassinated, as he richly deserves for all the grief he has caused. Hey, just kidding, doc. But next time you have a brainstorm, do us a favor and keep it to yourself.

 

[Number Six]

If the Monty's decision to reveal a goat to you is unrelated to whether you guessed correctly then you should switch.

That means that if Monty decided to reveal a goat to you before you announced your decision then you should switch.

Or if Monty decided to reveal a goat to you after you announced your decision but the basis on which he made his decision to reveal the goat was independent of whether your choice was right, then you should switch.

But if Monty's decision to reveal a goat to you is related to whether your choice was correct then you have to know _how_ his decision and your choice were related in order to decide if you should switch.

 

[rppa]

The problem statement doesn't say "maybe" Monty reveals the goat. It says he reveals the goat. Period.

You're not the first person to say this gives you useful information about how Monty acts in general. But nobody so far has said WHY this is useful.

Please show me your analysis by which this is adequate information to assess the optimum strategy.

 

[TeaBag420]

I wonder what the original wording was in Ask Marilyn.

As I've mentioned several times, I have posted that text, with attribution to Whitaker/Whittaker/whatever on this thread.

I leave it as an exercise to you to find it. It's here somewhere.

On the other hand, if only there were a network of computers where one could search for such information.

 

[CurtC]

As noted in Marilyn is tricked by a game show host, the question as posed to her did NOT say that the rules to the game are that Monty must always show a goat and switch. According to Wikipedia, the question was:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

By the way, by the time Marilyn published her column with this puzzle, I was already familiar with it - it was a weekly puzzler on Car Talk before that. However, at that time I wasn't yet hip to the subtlety of Monty's motivations.

 

[Yaotl]

If Monty only offered the switch choice when you had originally picked the correct door...

I meant that both situations say that it's best to switch.

 

[Paul C. Anagnostopoulos]

CurtC said:
Maybe you can point me to a statement of the problem where he always unveils one goat? I don't think I've ever seen one in the wild.

Agreed, but since the problem makes no hint at its being conditional, I think it's fair to assume it's not. And, as Dr. A said:

If Monty Hall had made a practice of offering the switch only when the first guess was right, that would have been a fairly poor strategy. And the answer would be "never switch". And no-one would be in the least interested in the problem.

The point is to solve the problem and learn something, right?

garys_2k said:
I had NO idea this thread would do this! Amazing, it's really amazing.

You're too easily amazed. If we simply agreed, what in hell would we talk about? We all agree, so we expand the topic to include something we can argue about. We're skeptics, man!

~~ Paul

 

[Cabbage]

Here's a question for those of you who still claim (incorrectly) that enough information is given in the problem (as stated in the OP or in Marilyn's original column) to conclude that your probability of winning is 2/3 when you switch:

Say you randomly pick a positive integer. What is the probability the number you pick is even?

Now this question seems to be pretty clear in what it's asking. I know that it might not be clear how this relates to the car and goats problem, but if you'll humor me and respond, I promise I'll try to make the connection shortly in the course of the discussion.

 

[Cabbage]

Oh, and I forgot to mention; please justify your answer.

 

[RussDill]

Its a lot easier to think about if you think about picking goats instead of cars. Intuitively, you have a 66% chance of picking a goat. Monty has a 100% chance of picking a goat. You also cannot both pick the same door.

What is the probability that you both selected goats? 66%

 

[epepke]

Here's a question for those of you who still claim (incorrectly) that enough information is given in the problem (as stated in the OP or in Marilyn's original column) to conclude that your probability of winning is 2/3 when you switch.

I admire your tenacity, Cabbage, but seriously, it's a lot easier and more satisfying to win free drinks and meals off of these people. I've found empirically to about a 90% confidence rate that they just don't listen. And so, you can win a free drink or meal nine times out of ten, and who will give you better odds than that?

 

[Paul C. Anagnostopoulos]

Cabbage said:
Here's a question for those of you who still claim (incorrectly) that enough information is given in the problem (as stated in the OP or in Marilyn's original column) to conclude that your probability of winning is 2/3 when you switch.

Aaaaarrrggh! All right, I give! Uncle! Uncle! No informal consideration of math problems shall be allowed. All problems will be specified to the degree that they are rendered pedantic and uninteresting.

Say you randomly pick a positive integer. What is the probability the number you pick is even?

This is poorly specified. Is "randomly picking" the same as "picking randomly"? Am I picking from among all positive integers or only a finite subset? Does "even" mean divisible by 2?

~~ Paul

 

[Cabbage]

Aaaaarrrggh! All right, I give! Uncle! Uncle! No informal consideration of math problems shall be allowed. All problems will be specified to the degree that they are rendered pedantic and uninteresting.

But being specific is a fundamental part of mathematics! The specificity doesn't make the problem uninteresting, it makes the problem solvable. Are you claiming that simply adding the phrase "Mony always opens a door and offers a switch" would make an otherwise interesting problem pedantic and uninteresting?

One problem I've always noticed in discussions between people skilled in math and people with little mathematical experience is that people lacking experience are often unable to ask a mathematical question in a meaningful way.

For example, a friend of mine may come up to me and tell me of some astounding coincidence like, "I started talking to this stranger on the bus today, and it turns out he was from the same hometown as my mother and used to date her! What are the chances of that?" In some cases, they may even take such a coincidence as a meaningful providential intervention.

What are the chances of that? Who can say? I don't know. In order to ask a meaningful probability question, a framework must be established.

What span of time are we looking at? I mean, are you interested in the probability of that happening today, or just at some point throughout your entire lifetime? Was the event significant to you because the guy had formerly dated your mother, or would any mutual connection between you and the stranger have been significant? All such details (not limited to the two specifically listed here) must be laid out in order to establish this framework.

On that note, this is exactly the sort of issue I have with the Monty Hall problem in its typical form (i.e., without specifying that Monty always opens a door and gives a choice).

Now honestly, don't get me wrong, I'm not at all bothered by those who choose to interpret it this way Marilyn interpreted it, and claim odds of winning as 2/3 if you switch. What baffles me is why many people don't even realize that they are, in fact, making an assumption in proposing that solution. That assumption, as has been stated many times before, is that Monty always opens a door and offers a switch. This was never stated in the problem. If you want to argue that you are making the assumption in order to make sense of an otherwise unanswerable problem, fine; let's just be clear that you are in fact making that assumption.

This is poorly specified. Is "randomly picking" the same as "picking randomly"? Am I picking from among all positive integers or only a finite subset? Does "even" mean divisible by 2?

I don't see the difference between "randomly picking" and "picking randomly"; as I understand them, they mean the same thing. Could you clarify the distinction as you understand it?

Yes, you are picking from among all (infinitely many) positive integers, and, of course "even" means divisible by 2.

 

[Cabbage]

To be fair, I must admit I'm kind of feeling like a playground bully right now for bringing up that problem regarding picking a positive integer at random. I feel I must admit that, like my earlier chess example, it's not analogous to Monty Hall in a probabilistic sense, but in the sense that both are vague.

If anyone wants to continue to discuss it, I'll certainly go along, but the details running around in that problem run much deeper than the Monty Hall problem. To be fair, I should at least post a link to Kolmogorov's probability axioms first, so that everything is out in the open. These are the standard axioms for a probability function P.

 

[Paul C. Anagnostopoulos]

Cabbage said:
But being specific is a fundamental part of mathematics! The specificity doesn't make the problem uninteresting, it makes the problem solvable. Are you claiming that simply adding the phrase "Mony always opens a door and offers a switch" would make an otherwise interesting problem pedantic and uninteresting?

It's not the problem that's pedantic, it's this discussion. By all means, though, clarify the standard wording of the problem. You're right, it won't hurt. But to claim that the problem as worded was ambiguous and thus could not serve its purpose to surprise people about their intuitions is ... well ... pedantic. In fact, I daresay that such a clarification would do absolutely no good for the very people the problem is meant to educate.

... let's just be clear that you are in fact making that assumption.

Absolutely clear. All I was saying is that seems like precisely the default assumption, given no mention of any conditions whatsoever.

I don't see the difference between "randomly picking" and "picking randomly"; as I understand them, they mean the same thing. Could you clarify the distinction as you understand it?

Just yanking your chain. Actually, though, "randomly picking" has the connotation of picking sometimes and not picking other times, whereas "picking randomly" has the connotation you want. But maybe that's just me.

But continue on. I have no idea why the probability of picking an even integer isn't .5.

~~ Paul