Monty Hall Problem

[Robin]

We should briefly address the "I use this to win drinks off people so it must be true" test of an argument.

There are no details about how this happens but presumably someone in a bar agrees to gamble for a drink without first asking the rules of the game.

This appears not so much a lack of skepticism as a lack of sobriety, a condition not unknown in bars.

In fact it is not difficult - "I bought the last round, don't you remember?" also works, as does exclaiming "Hey that's mine", at which someone replies "What's yours?" and everybody says "A schooner of 'New' thanks" (or whatever their favourite tipple is).

You probably know the general rule about gambling but maybe someone can tell me who said it:

"A man will approach you with a pack of cards and bet you $20 that the Jack of Hearts will jump out and squirt whipped cream in your ear. Don't take the bet, or you will end up $20 poorer with an earful of whipped cream".

(edit)Hey I broke the 10 - do I get a cookie or something?(/edit)

 

[Art Vandelay]

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

Well, I said the rules "depend" on the host, which just means that he has some say. And how is that a big assumption? Is the idea that Letterman has some control over the content of his monologue a big assumption? Heck, there's nothing in the problem that actually says there are rules. As far as the problem says, there are no rules other than the whim of the host.

hgc

I thought the idea that Monty is a robot (single scenario defines the rules) was a reasonable assumption.

I really don't see how that's a reasonable assumption. If it is, then two description of the same exact scenario would result in completely different answers.

1. "Three doors, yadda yadda. The host opens one of the doors you didn't pick and reveals a goat. Should you switch?"

2. "Yadda, yadda. The host opens door number two, and reveals a goat. Should you switch?"

3. "The host opens one the doors that you didn't pick and shows you what's behind it. How should you react?"

These three descriptions are of the exact same events. A person actually experiencing them would not be able to tell which was the "right" description; only a person who isn't actually faced with the choice, and is merely responding to a hypothetical, would see a difference. What sense does it make for a person actually on the show to not know what to do, but someone reading a second-hand account to know?

Here's another puzzle:
"There are one hundred doors, 60 cars and 40 goats. You pick door number twelve. The host opens all the doors except number twelve and one other one, revealing 39 goats. The host then allows you another choice, this time between the two remaining doors. Does it matter which one you pick?

I don't think this is just a nitpick. A lot of people make decisions based on what they see as being the situation, rather than thinking about whether their viewpoint is skewed.

but I still think that people who didn't get it as worded need a good slice of Occam.

Didn't Occam counsel against making assumptions, no matter how "obvious" they are?

69dodge

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.

Interesting point. But I don't see how this is any more valid than in the original case. Is that what you meant by "I had already planned to make the point I'll make"?

I'll have think about this a bit more. On one hand, no flaw in your reasoning jumps out at me. On the other hand, "Ask this question of the host, then switch" is clearly a better strategy than simply picking and sticking.

 

[Robin]

Art Vandelay
Heck, there's nothing in the problem that actually says there are rules. As far as the problem says, there are no rules other than the whim of the host.

Well you said it not me. If you think that it is a reasonable interpretation of the problem that the contestant is on a game show with no rules whatsoever except the whim of the host.

I am sure that any activity without rules qualifies as a game but as of now I officially lose interest in this silly debate.

 

[CurtC]

Well you said it not me. If you think that it is a reasonable interpretation of the problem that the contestant is on a game show with no rules whatsoever except the whim of the host.

Did you ever see the show Let's Make a Deal? That's exactly what it was. The whim of the host, a very colorful character, Monty Hall.

 

[hgc]

...

(edit)Hey I broke the 10 - do I get a cookie or something?(/edit)

Behind one of these 3 doors is a cookie...

 

[hgc]

...

hgc

I really don't see how that's a reasonable assumption. If it is, then two description of the same exact scenario would result in completely different answers.

1. "Three doors, yadda yadda. The host opens one of the doors you didn't pick and reveals a goat. Should you switch?"

2. "Yadda, yadda. The host opens door number two, and reveals a goat. Should you switch?"

3. "The host opens one the doors that you didn't pick and shows you what's behind it. How should you react?"

These three descriptions are of the exact same events. A person actually experiencing them would not be able to tell which was the "right" description; only a person who isn't actually faced with the choice, and is merely responding to a hypothetical, would see a difference. What sense does it make for a person actually on the show to not know what to do, but someone reading a second-hand account to know?

Gadzooks! I just fell out of my chair!

You give 3 descriptions of scenarios that could conceivably the be same events, but only 1 of them comports the description I gave in the initial puzzle. So what's the point of saying that the other 2 could be the same event? I didn't say the host opens door #2. I did say that the host opens one of the doors you didn't pick. The difference is everything. I left unspecified those details which could change case to case, so that my description would fit with all the possible paths the game might follow, such as what door you picked first and what door Monty reveals. I specified the details that are consistent for every possible path -- Monty reveals a goat. I tried to just precise enough so that a puzzler would infer from the specifics and lack thereof what the rules of the game are -- without being too pedantic in trying to cut off every avenue of expansion beyond the point of solving a probability puzzle.

Here's another puzzle:
"There are one hundred doors, 60 cars and 40 goats. You pick door number twelve. The host opens all the doors except number twelve and one other one, revealing 39 goats. The host then allows you another choice, this time between the two remaining doors. Does it matter which one you pick?

That's too hard. What's the answer?

...
Didn't Occam counsel against making assumptions, no matter how "obvious" they are?

Not exactly. Occams suggests not to manufacture any more entities than are necessary to hypothesize and explanation. Your manufactured scenarios, 2 and 3, are not required.

 

[Art Vandelay]

You give 3 descriptions of scenarios that could conceivably the be same events, but only 1 of them comports [with] the description I gave in the initial puzzle.

What do you mean? All of them comport with your description.

I didn't say the host opens door #2.

You didn't say he doesn't.

That's too hard. What's the answer?

According to your logic, it doesn't make a difference.

Not exactly. Occams suggests not to manufacture any more entities than are necessary to hypothesize and explanation. Your manufactured scenarios, 2 and 3, are not required.

But their negations are not required either. You are hypothesizing that Monty makes a habit out of doing this stuff. I say that it is possible to exclaim the events without reference to such an assumption, so Occam favors my point of view.

 

[hgc]

With another Monty Hall thread already going strong, I first thought it best to leave this one safely buried. But then the masochist in me couldn't let this go unanswered.

What do you mean? All of them comport with your description.

No, no, no, no, no. Convinced? OK, I'll go into details:

Your #2 - "Yadda, yadda. The host opens door number two, and reveals a goat. Should you switch?" I don't say that the host opens Door 2. I say the host opens one of the other doors. The difference is everything. I was not specific about the door because that is a variable from my scenario to other possible scenarios. I didn't ask "should you switch?" I ask what your chances are of getting the car if you should switch. Just so there's no distraction about whether the car is what you want, or the goat that lays the golden egg or whatever.

Your #3 - "The host opens one the doors that you didn't pick and shows you what's behind it. How should you react?" I don't leave out that he reveals the goat; I specify that he reveals the goat. I don't ask generally how you should react. I ask what your chances are of getting the car if you switch your choice.

You didn't say he doesn't [open door #2].

I explained that already. I didn't say it because it's not part of the problem. The problem doesn't depend on which door number is which. It works the same way for doors 1, 2, 3 or doors a, b, c or doors green, blue, yellow; in any order in any scenario.

According to your logic, it doesn't make a difference.

Sorry, but I didn't attempt to solve your puzzle. I hope it wasn't crucial to making your point.

But their negations are not required either. You are hypothesizing that Monty makes a habit out of doing this stuff. I say that it is possible to exclaim the events without reference to such an assumption, so Occam favors my point of view.

No, only required by pedants. All others needn't grow a garden of weedy suppositions and red herrings.

 

[gnome]

Did you ever see the show Let's Make a Deal? That's exactly what it was. The whim of the host, a very colorful character, Monty Hall.

The question is nearly pointless as a probability puzzle if Monty can do whatever he wants. That makes it entirely reliant on his intentions and unsolvable.

So is it unreasonable to assume that in repetitions, the scenario will be the same? If you make any other assumption than that, there's no satisfactory answer at all.

 

[Mason]

One more take on the matter...

Monty presents 3 doors; behind one is a car, and behind the other two are goats. You pick a door. But before Monty opens that door, he reveals behind another door that there is a goat. He then gives you the opportunity to switch your choice. What is the probability that you will get the car by switching, or by staying?

Perhaps I'm mistaken, but the problem is pretty obviously a question of probability, not a question of psychology. It is a word problem to help illustrate a very basic level of probability, not an exercise in psychology.

Again, it is a math problem in the math forum to show basic math functions.

It is not a philosophy problem in the philosophy forum to show elements of psychology.

For example, "If a train leaves from New York at 12:00 traveling at 100 miles per hour..." Is a common word problem that we all see in early math text books. The problem is designed to illustrate certain aspects of math. We can sit around all day and ask "What if the conductor only says he's going 100 miles per hour, but in fact speeds up once he gets out of the city limits? The problem can't be solved because we don't know what the conductor will do once we lose sight of him!"

Hogwash.

It's a simple problem, anyone who is making more of it is just being obtuse for the sole purpose of being obtuse.

 

[BillHoyt]

I think hgc nailed it with "grow a garden of weedy suppositions and red herrings."

 

[billydkid]

I know this is old news

It took me a little while to grasp this - what is sometimes referred to as The Monty Hall Problem. It has been around for a long time in various forms. It is one thing to know it, but it is something else to actually grasp it intuitively. The most obvious illustration is using an example with a million doors. You are essentially choosing between one door and all of the rest. When looked at that way it is very obvious. Still, virtually nobody I have described the problem to gets it. They can't let go of the 50/50 idea because there are two doors at the end. I have never been great at brain teasers, but it bothers me that it took some effort to wrap my mind around this. I'm more of a spacial kind of guy.

 

[pgwenthold]

It took me a little while to grasp this - what is sometimes referred to as The Monty Hall Problem. It has been around for a long time in various forms. It is one thing to know it, but it is something else to actually grasp it intuitively. The most obvious illustration is using an example with a million doors. You are essentially choosing between one door and all of the rest. When looked at that way it is very obvious. Still, virtually nobody I have described the problem to gets it. They can't let go of the 50/50 idea because there are two doors at the end. I have never been great at brain teasers, but it bothers me that it took some effort to wrap my mind around this. I'm more of a spacial kind of guy.

There are some who can't grasp the math, but the most vocal (and legitimate) objections come from those who claim there are certain assumptions that are made in solving the problem that are not justified, such that there is no correct answer.

If the problem is set up properly, generally to reflect the spirit in which it is intended, then sure, your answer works fine. However, the devil is in the details, and they technically matter.

 

[Mark6]

It took me a little while to grasp this - what is sometimes referred to as The Monty Hall Problem. It has been around for a long time in various forms. It is one thing to know it, but it is something else to actually grasp it intuitively. The most obvious illustration is using an example with a million doors. You are essentially choosing between one door and all of the rest. When looked at that way it is very obvious. Still, virtually nobody I have described the problem to gets it. They can't let go of the 50/50 idea because there are two doors at the end. I have never been great at brain teasers, but it bothers me that it took some effort to wrap my mind around this. I'm more of a spacial kind of guy.

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 with "heads" or "tails" 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.

 

[IMST]

Here's another puzzle:
"There are one hundred doors, 60 cars and 40 goats. You pick door number twelve. The host opens all the doors except number twelve and one other one, revealing 39 goats. The host then allows you another choice, this time between the two remaining doors. Does it matter which one you pick?

With this puzzle you've been shown 59 cars and 39 goats, right?

If I'm reading that right, you had a 60% chance of guessing right to begin with, so you retain that percentage by staying. Switching would give you a 40% chance of getting the car since all probabilities have been collapsed to those two doors.

Of course, this assumes the host deliberately chose to leave one car and one goat uncovered.

 

[Skeptic Guy]

The Marquis de Carabas answer: "With that many goats, everyone is a winner."

 

[gnome]

Not this thread again! NO!!!! WHY, GOD, NO!!!!

We must encase ourselves in an underground bunker and defend it with our best weapons!

I'M RUNNING THIS MONKEY SHOW NOW, FRANKENSTEIN!!

 

[This is The End]

I'm pretty sure I arrived at a decent explanation of both sets of odds in this related topic:

http://www.internationalskeptics.com/forums/showthread.php?t=108001

http://www.internationalskeptics.com/forums/showthread.php?postid=3507440#post3507440

 

[lionking]

I didn't get it until I read the three card analogy in the wikipedia entry on the problem. I can see why so many say 50/50.

 

[billydkid]

Not this thread again! NO!!!! WHY, GOD, NO!!!!

http://www.applefritter.com/images/zombie_1-9316_640x480.jpg

We must encase ourselves in an underground bunker and defend it with our best weapons!

http://www.skepticalcommunity.com/phpbb2/uploads/monkeyshow.jpg

I'M RUNNING THIS MONKEY SHOW NOW, FRANKENSTEIN!!

Come on, it's not the end of the world. You know, I could have started a NEW thread. Count your blessings. On the other hand, SOMETHING compelled you to open it.