Monty Hall Problem

[TeaBag420]

Nevermind, I'll do it....

The Monty Hall Dilemma was discussed in the popular "Ask Marylin" question-and-answer column of the Parade magazine. Details can also be found in the "Power of Logical Thinking" by Marylin vos Savant, St. Martin's Press, 1996.

Marylin received the following question:
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 number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?
Craig. F. Whitaker
Columbia, MD

 

[Robin]

What if I could write it on a secret ballot?

It would save a lot of time and space if everyone would agree to address the problem as stated, and not if it were another totally unrelated problem.

 

[Robin]

Nevermind, I'll do it....

The Monty Hall Dilemma was discussed in the popular "Ask Marylin" question-and-answer column of the Parade magazine. Details can also be found in the "Power of Logical Thinking" by Marylin vos Savant, St. Martin's Press, 1996.

Marylin received the following question:
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 number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?
Craig. F. Whitaker
Columbia, MD

I quoted this a few posts back but it is worth seeing again. The conditions of the problem clearly state that (i) the host opens another door and that (ii) the door he opens has a goat behind it and (iii) that he offers a second choice.

The answer does not depend on the intentions of the host.

Or am I being a 'tard?

 

[insomneac]

Marilyn vos Savant popularized this problem decades ago. One of her books (can't remember its title) contains her solution to the problem and the results of nationwide school trials. In trial after trial, switching resulted in winning the car almost twice as often as staying.

 

[rppa]

Here's an excerpt from that web page:<blockquote>What's being said is that if I pick #1 and monty shows #3, then 2/3 of the time #2 will win. So If I had picked #2 to start with and monty opens #3, then the odds go 2/3 to #1. Why would they change? What if I don't have to tell monty? What if I could write it on a secret ballot?</blockquote>

DaveW hasn't told us his spreadsheet formulas so we don't know where his simulation went wrong. I think I can address this excerpt though.

We're not saying that 2/3 of the time #2 will win. We're saying that 2/3 of hte time Monty's door will win.

If I pick door #1, there are three equally likely possibilities:

1. I've got the winner already. Monty picks #2 or #3 with equal probability.
2. I've got a goat and the car is in #2. Monty opens #3.
3. I've got a goat and the car is in #3. Monty opens #2.

In 2 of these 3 possibilities, the other door is the winner. The odds aren't 2/3 that #2 is the winner, they're 2/3 that the door Monty didn't open is the winner. 1/3 on #2, 1/3 on #3.

If I picked #2 to begin with, there is still a 1/3 chance of the car being in #1, #2, or #3. But in the case that it is in #1 or #3, the winner will be the door left after Monty's move.

If I don't tell Monty what I picked, then his move is no longer dependent on mine, and we're in the 50/50 situation. In fact, he might choose to open the door I already picked, something that can't happen in our version of the game.

 

[69dodge]

Originally posted by Robin
It has nothing to do with what Monty feels like doing as long as he behaves as stated in the problem.

epepke is interpreting the problem statement simply as a description of what Monty does in one particular instance. You are interpreting the problem statement not only as a description of what Monty does, but also as a description of what the contestant, before playing, knew Monty was going to do. You're both right, given your respective interpretations.

 

[Robin]

epepke is interpreting the problem statement simply as a description of what Monty does in one particular instance. You are interpreting the problem statement not only as a description of what Monty does, but also as a description of what the contestant, before playing, knew Monty was going to do. You're both right, given your respective interpretations.

No, it does not matter if the contestant knows what Monty is going to do, or what Monty's intentions are. It makes no difference.

If the conditions as set out in the problem are satisfied then the answer is "Yes - you double your chances of winning by switching".

Somebody please show me a case where Monty can affect the outcome.

 

[insomneac]

Edited because Robin said it better.

 

[CurtC]

I quoted this a few posts back but it is worth seeing again. The conditions of the problem clearly state that (i) the host opens another door and that (ii) the door he opens has a goat behind it and (iii) that he offers a second choice.

The answer does not depend on the intentions of the host.

Or am I being a 'tard?

You're not being a 'tard, but you're wrong nonetheless. The answer depends greatly on the intentions of the host, because in the original problem statement (as given in the OP) all you know is that the host switched this time, not whether he had to.

My favorite illustration of this is one I came up with myself. Imagine that you're walking down the street and come upon a street hustler offering a game of find-the-ball-under-a-cup. You plop down your $20, and the hustler mixes the three cups around very quickly, so you get lost. You make a guess at random, and the hustler then shows you one of the empty cups, and asks if you'd like to switch your guess to the other one. Would you switch?

You'd be a fool if you did, because the street hustler wouldn't be offering you the choice at all had you guessed wrong initially. If you switch, you have a 100% chance of losing.

The only difference between this problem statement and the OP is who the person hosting is, and what his intentions can be assumed to be.

 

[69dodge]

Originally posted by Robin
No, it does not matter if the contestant knows what Monty is going to do, or what Monty's intentions are. It makes no difference.

Suppose you know, before starting your game, that Monty intends to allow you to switch only if you initially pick the car. If you pick a goat, he'll just give you the goat.

Now, you start playing the game. You pick a door. Monty opens a different door, revealing a goat, and asks if you'd like to switch to the third door. Would you switch?

Of course you wouldn't. The fact that he offered to let you switch tells you that you picked the car.

Monty's actions are exactly as described in the opening post of this thread. What you know about his intentions makes all the difference.

 

[insomneac]

Suppose you know, before starting your game, that Monty intends to allow you to switch only if you initially pick the car. If you pick a goat, he'll just give you the goat.

Now, you start playing the game. You pick a door. Monty opens a different door, revealing a goat, and asks if you'd like to switch to the third door. Would you switch?

Of course you wouldn't. The fact that he offered to let you switch tells you that you picked the car.

Monty's actions are exactly as described in the opening post of this thread. What you know about his intentions makes all the difference.

But you don't know Monty's intentions at all. Therefore you don't take them into consideration.

 

[Robin]

Suppose you know, before starting your game, that Monty intends to allow you to switch only if you initially pick the car. If you pick a goat, he'll just give you the goat.

Now, you start playing the game. You pick a door. Monty opens a different door, revealing a goat, and asks if you'd like to switch to the third door. Would you switch?

Of course you wouldn't. The fact that he offered to let you switch tells you that you picked the car.

Monty's actions are exactly as described in the opening post of this thread. What you know about his intentions makes all the difference.

OK, from the last two posts (yours and CurtC) I see where you are coming from. Let me sleep on it.

 

[TeaBag420]

What Monty thinks, wants, or is required to do is IRRELEVANT.

What matters is what he does.

It's so tempting to call folks retarts, but I struggled with this problem for about an hour before someone beat the correct understanding into my head. So my heart goes out to the retarts.

 

[Cabbage]

I quoted this a few posts back but it is worth seeing again. The conditions of the problem clearly state that (i) the host opens another door and that (ii) the door he opens has a goat behind it and (iii) that he offers a second choice.

The answer does not depend on the intentions of the host.

Actually, the intentions of the host do matter. Here are a couple of examples:

Suppose sometimes the host opens a door after you make your selection, and sometimes he doesn't. Now, when it's your turn to play, he opens a door to reveal a goat. Should you switch? In this example, I don't think it's so clear what the probabilities of switching vs. not switching are. It depends on the motivations of the host--Maybe he only opens a door and offers the switch when you've already picked the correct door. Maybe he just likes to be tricky like that.

On the other hand, say the host always opens a door other than yours, but suppose the host doesn't know where the car is himself. Say you pick door 1, and the host says, "Just to make it interesting, let's see what's behind one of the doors you didn't pick...Door 3. If there's a car behind that door, you lose; if it's a goat, I'll give you a chance to switch!"

He opens the door, revealing a goat. Should you switch?

Now's it's 50/50 whether you switch or not. To calculate it:

First, what were the chances of a goat being revealed in the first place? The host's door is picked at random, so there's a 2/3 chance the goat is revealed. (Note that if the host always shows a goat door, this probability would be 100%. This note to be continued to contrast the two problems).

Now, what is the probability that both 1. A goat was revealed, and 2. You picked the right door?

The probability that picked right is 1/3. Given that has happened, the probability a goat is revealed is 100%. So the probability of them both happening is 1/3. (Note, if a goat is always revealed, the chance of these two events happening is still 1/3).

Now, given that a goat was revealed (2/3 chance), what is the probabilty that both a goat was revealed and you picked the right door? (1/3) divided by the 2/3 were normalizing on, (i.e., given that the goat was revealed). So the probability you picked correctly is 1/2--It doesn't matter if you switch or not.

(Note that when the goat is always revealed, probability becomes (1/3) / 1 = 1/3 chance of picking correctly initially (so 2/3 chance of winning if you switch, which answers the question as it's generally intended to be stated--that you know Monty always reveals a goat, (intentionally, of course)).

 

[Robin]

By Cabbage
...but suppose the host doesn't know where the car is himself?

Unfortunately I was looking at another version of the problem, as asked of Marilyn Vos Savant in 1990. The version stipulated that he does. As I said before, the problem is in differing understandings of the question.

 

[Cabbage]

Unfortunately I was looking at another version of the problem, as asked of Marilyn Vos Savant in 1990. The version stipulated that he does. As I said before, the problem is in differing understandings of the question.

OK, I wasn't sure about that.

On the other hand, it still makes a difference in that version of the problem if the host sometimes opens a door, and sometimes doesn't.

 

[insomneac]

Suppose sometimes the host opens a door after you make your selection, and sometimes he doesn't. Now, when it's your turn to play, he opens a door to reveal a goat. Should you switch? In this example, I don't think it's so clear what the probabilities of switching vs. not switching are. It depends on the motivations of the host--Maybe he only opens a door and offers the switch when you've already picked the correct door. Maybe he just likes to be tricky like that.

There's that same mistake. The stated problem doesn't say squat about Monty's intentions or his knowledge about what lies behind each door. So stop making unwarranted assumptions when solving the problem.

 

[Robin]

Cabbage
On the other hand, say the host always opens a door other than yours, but suppose the host doesn't know where the car is himself. Say you pick door 1, and the host says, "Just to make it interesting, let's see what's behind one of the doors you didn't pick...Door 3. If there's a car behind that door, you lose; if it's a goat, I'll give you a chance to switch!"

He opens the door, revealing a goat. Should you switch?

Now's it's 50/50 whether you switch or not. To calculate it:

First, what were the chances of a goat being revealed in the first place? The host's door is picked at random, so there's a 2/3 chance the goat is revealed. (Note that if the host always shows a goat door, this probability would be 100%. This note to be continued to contrast the two problems).

Now, what is the probability that both 1. A goat was revealed, and 2. You picked the right door?

But in this case the odds are still 0.667 if you switch, even if the host didn't know where the car is. The fact that you now see a goat means that if you originally guessed wrong then the remaining door contains the car. And the odds of you getting it wrong are 0.667

 

[Cabbage]

So stop making unwarranted assumptions when solving the problem.

Actually, you are the one making assumptions about the problem--you're assuming he is operating in a "fair" manner, and that he always reveals a goat.

I'm saying I don't know--maybe he reveals a goat sometimes, maybe sometimes he doesn't. Maybe he's operating in a deceptive manner--Always revealing a goat and offering the switch when I'm initially right, and not revealing a goat when I'm wrong.

 

[Robin]

Cabbage,

However you (and others) are right about the matter of whether the host sometimes offers you a second chance and sometimes doesn't. Here I was just assuming given that it was a game show that the problem was stating the rules rather than his actions on one occasion.

In this case it is a problem of imprecision of definition.