Monty Hall Problem

[hgc]

First, you get an Australian homo puffy boy.....the sort who is disposed to manipulate large mammals to a state of turgidity.

Homo says what?

The origin of Teabag's name

 

[rppa]

I am prepared to confess I jumped at one conclusion, are the "it depends on the intentions of the host" crowd prepared to admit they jumped at the other?

Since you posted a table proving that it depends on the intentions of the host (giving probability as a function of intentions of the host), I'm confused as to what you could mean by this statement.

What conclusion is being jumped to by saying "the probabilities depend on the intentions of the host"?

 

[garys_2k]

The origin of Teabag's name

Youch! That would force me to get a new ID if it applied.

 

[CurtC]

And I suppose the "420" part refers to this.

 

[rppa]

The recent flap about the "Read Books, Get Brain" ad campaign in New York really made me feel old and out of touch.

http://www.reuters.com/newsArticle.jhtml?type=oddlyEnoughNews&storyID=6731922

 

[RussDill]

Good point, dodge. We can generate random integers between 1 and n, but then each has a probability of 1/n and half are even. But we cannot generate random integers between 1 and infinity, so it's difficult to run the experiment.

Why can we assume the probability is zero for picking real numbers, but not for integers? The set has to be uncountably infinite?

Cabbage, we assumed a probability of zero during a conversation started in one of Interesting Ian's threads. It's this gargantuan monster, I think:

http://www.internationalskeptics.com/forums/showthread.php?s=&threadid=43483&highlight=probability

~~ Paul

Random integer = (coin flip 1) + (coin flip 2) * 2^1 + ...(coint flip n) * 2^(n - 1)


Only the first coin flip has a bearing on whether or not the integer is even or odd, but you can do you infinite coin flips to come up with your number.

 

[69dodge]

Originally posted by RussDill
Random integer = (coin flip 1) + (coin flip 2) * 2^1 + ...(coint flip n) * 2^(n - 1)

Only the first coin flip has a bearing on whether or not the integer is even or odd, but you can do you infinite coin flips to come up with your number.

All integers are finite. But this process will, with probability 1, yield an infinitely large "integer". So it doesn't quite do what you want it to.

 

[RussDill]

All integers are finite. But this process will, with probability 1, yield an infinitely large "integer". So it doesn't quite do what you want it to.

Its no accident that with probability 1 it picks an infinitely large integer. This is true anytime you randomly pick an integer.

 

[Robin]

No, because which case holds depends on what the hosts wants. Are you saying that the idea that the host has free will is an "assumption"?

No I was saying that the idea that the rules allow the host room for strategy is an assumption. We must all assume that the host will follow the rules. If the second choice is part of the game then what the host wants is irrelevant.

It all depends on whether you read the question as a statement of how the game is played or as just one scenario.

 

[Robin]

Since you posted a table proving that it depends on the intentions of the host (giving probability as a function of intentions of the host), I'm confused as to what you could mean by this statement.

What conclusion is being jumped to by saying "the probabilities depend on the intentions of the host"?

The conclusion that the host did not have to offer a choice.

Again I would have to ask people to think of the difference between rules and strategy. The rules delineate what game is being played, strategy delineates how it is played.

Strategy can be altered during the game, rules cannot.

 

[69dodge]

Originally posted by RussDill
Its no accident that with probability 1 it picks an infinitely large integer. This is true anytime you randomly pick an integer.

There's no such thing as an infinitely large integer; that's why I put it in quotes. All integers are finite.

 

[Art Vandelay]

No I was saying that the idea that the rules allow the host room for strategy is an assumption. We must all assume that the host will follow the rules. If the second choice is part of the game then what the host wants is irrelevant.

But the rules depend on what the host wants.

It all depends on whether you read the question as a statement of how the game is played or as just one scenario.

Exactly.

you can do you infinite coin flips to come up with your number.

No, you can't.

Paul

All future statements of the problem should be worded precisely. However, I stick with my claim that that will not help most people see the light, because it is not any possible ambiguity that is the barrier to understanding.

I think that there are wordings of the problem that make it even more difficult to not get the right answer. For instance, suppose it were worded as follows:

"You are shown three doors. Behind one of them is a car, behind the others, goats. You ask the host 'Do you know where the car is?' The host says that he does not. You then ask 'Could you look behind the second and third doors, and then make a statement of the form "If the car is behind one of these doors, it is behind door..." '. The host agrees, and a few seconds later announces 'If the car is behind one of these two doors, it is behind door number two'. Assuming that the host is honest, which door should you pick?"

 

[Robin]

Art Vandelay
But the rules depend on what the host wants

Now there is a big assumption!

 

[Robin]

The likelihood that Monty will offer you the switch isn't relevant. What is relevant is whether his offer is independent of whether your initial choice was correct.

If his offer is independent of whether your initial choice was correct then you should switch.

If his offer is not independent of whether your initial choice was correct then you can't determine whether you should switch unless you know _in what way_ his offer depends on your initial choice.

Maybe he wants to make you lose, in which case he only offers the switch if your initial choice is correct. Maybe he wants you to win, in which case he only offers the switch if your initial choice is incorrect. Maybe he has decided beforehand to offer you the switch with probably X if your initial choice is correct and with probability Y if your choice is incorrect. (Incidentally, the first two sentences of this paragraph are special cases of that). If you don't know what that probability is then you can't say whether it's best to switch.

But all of this assumes first that Monty is not obligated to make the offer under the rules of the game.

I think that we all understand the conditions that would hold if the host is free to offer or not offer the switch. I have already put a table of a number of the conditions that might hold.

But what some people fail to undertand is that all of that is irrelevant if the host is obligated to offer the switch. So this matter must be settled first before any host intentions become relevant.
(Edited to remove silly rude comment)

 

[Art Vandelay]

Now there is a big assumption!

The idea that the host is not effectively a robot is a big assumption?

 

[Robin]

The idea that the host is not effectively a robot is a big assumption?

Well first off the idea that the host makes the rules is a big assumption.

I think we established earlier that the host cannot cheat. In as much as something is part of the rules then the host is effectively a robot.

If you are saying that the host has the free will to ignore the rules of the game or to arbitrarily vary them then you are adding yet another layer of ambiguity.

 

[hgc]

The idea that the host is not effectively a robot is a big assumption?

I thought the idea that Monty is a robot (single scenario defines the rules) was a reasonable assumption. But too many people were able to manufacture scenarios of host behavior. I am willing to reword the puzzle to remove ambiguity about Monty's intentions (which don't exist), but I still think that people who didn't get it as worded need a good slice of Occam.

 

[Robin]

I thought the idea that Monty is a robot (single scenario defines the rules) was a reasonable assumption. But too many people were able to manufacture scenarios of host behavior. I am willing to reword the puzzle to remove ambiguity about Monty's intentions (which don't exist), but I still think that people who didn't get it as worded need a good slice of Occam.

This is how I understood your original question and you did provide a clarification so I think that inasmuch as we are debating the question put by you, rather than some other version, then the answer is clear.

 

[Robin]

Incidentally for those who are interested in my schoolyard puzzle earlier:

"There is no 'f' in weigh"

And there is no effin' way we will all agree on this one.

 

[69dodge]

Originally posted by Art Vandelay
I think that there are wordings of the problem that make it even more difficult to not get the right answer.

More difficult, yes. But still not impossible. Watch me.

(Actually, I had already planned to make the point I'll make, even before I read your post. But then when I read it, it seemed like the perfect hook.)

For instance, suppose it were worded as follows:

"You are shown three doors. Behind one of them is a car, behind the others, goats. You ask the host 'Do you know where the car is?' The host says that he does not. You then ask 'Could you look behind the second and third doors, and then make a statement of the form "If the car is behind one of these doors, it is behind door..." '. The host agrees, and a few seconds later announces 'If the car is behind one of these two doors, it is behind door number two'. Assuming that the host is honest, which door should you pick?"

Door 3 definitely has a goat, so it's right out. I'd pick door 2, because it certainly is no worse than door 1 and it might be better.

However, we still do not have enough information to assign a definite probability of 2/3 to door 2. The probability that it hides the car might be anywhere from 1/2 to 1. Here's why. If the host sees the car, he has to tell you where it is. But if he sees two goats, he has a choice about what to say. Perhaps he decided, before looking, that if he sees two goats he'll say, "door 2." Then the probability is 1/2. Or, perhaps he decided that if he sees two goats, he'll say, "door 3." Then the probability is 1. If he chooses one or the other of the two statements at random when he has a choice, the probability that door 2 hides the car depends on the probability that he'll choose one statement over the other. Only if he chooses uniformly between them (i.e., the two are equally likely) will the probability be 2/3 that door 2 hides the car.