Monty Hall Problem... For Newbies

[hgc]

You forgot to mention that if there are two goat doors for Monty to choose from, he chooses randomly between them.

Yeah, about that -- I don't know what difference it makes to the answer to the problem. If Monte is not choosing randomly, does the contestant pick up on some pattern that makes the non-random choice between goat doors distinguishable from the scenario where Monte has a forced choice?

Also, what's wrong with discussing other variations of the problem? Regarding the original problem, we've already finished the discussion: there's no one left posting in the thread who is unclear about the odds of that scenario.

But some people found some other interesting things to talk about. What's wrong with that?

I found the two coin flip problem interesting, for instance.

Nothing wrong with that. It's all the pages devoted to disabusing misunderstandings about the original problem.

 

[jt512]

New rule: In some future year, when someone starts another new thread on this topic, please phrase the problem thusly:

A contestant on The Price is Right is told by Monte Hall that behind Doors 1, 2 and 3 are two goats and one car. THIS IS TRUE. IT IS ALWAYS TWO GOATS AND ONE CAR. The contestant, trying to win the car, guesses a door and tells Monte. Before revealing what is behind the door selected, Monte opens one of the other doors, to reveal a goat, and gives the contestant the choice to switch. MONTE ALWAYS REVEALS THE GOAT. HE KNOWS WHAT'S BEHIND EACH DOOR, SINCE IF HE DIDN'T HE WOULDN'T KNOW WHICH DOOR TO REVEAL. THERE IS NEVER A VARIATION. HE ALWAYS REVEALS A GOAT. Should the contestant stay or switch?

That should clear things up. Or not.

I'm totally confused. I thought the show was Let's Make A Deal. And who is this "Monte" character?

 

[ynot]

It can because they may be essentially mistaking one game for the other when computing the odds.

Really? - Okay

 

[ynot]

You forgot to mention that if there are two goat doors for Monty to choose from, he chooses randomly between them.

Also, what's wrong with discussing other variations of the problem? Regarding the original problem, we've already finished the discussion: there's no one left posting in the thread who is unclear about the odds of that scenario.

But some people found some other interesting things to talk about. What's wrong with that?

I found the two coin flip problem interesting, for instance.

Yep - Ignore me

 

[ynot]

I'm totally confused. I thought the show was Let's Make A Deal. And who is this "Monte" character?

I thought it was called - "Monty with an obsessive compulsive disorder that causes him to have a preference of showing a goat behind a particular door" game.

 

[ynot]

You only need to add the bolded below to the OP . . .

You're on a game-show trying to win a new car. The car is behind one of three doors. Behind the other two doors are goats.

You pick one of the three doors at random and the presenter (who knows which door the car is behind) opens a door that you didn't pick, revealing a goat. (This is standard procedure for the show.)

If two goat doors are available for the presenter to open then the one he opens is selected at random.

The presenter then offers you a chance to change your mind and pick the remaining door instead.

 

[hgc]

I'm totally confused. I thought the show was Let's Make A Deal. And who is this "Monte" character?

Oh Jesus. You're not the only one. Let's be charitable and assume that I once knew these things.

 

[Prometheus]

Yeah, about that -- I don't know what difference it makes to the answer to the problem. If Monte is not choosing randomly, does the contestant pick up on some pattern that makes the non-random choice between goat doors distinguishable from the scenario where Monte has a forced choice?

Yes.

Suppose that, given a choice, Monty always opens Door 2. To see how that matters, you have to look at when he actually has that choice available to him:

1) Contestant chooses Door 1, the car is behind Door 1; Monty will open Door 2. If the contestant sticks he wins, if he switches he'll lose.
2) Contestant chooses Door 1, the car is behind Door 2; Monty will open Door 3. If the contestant sticks he loses, if he switches he'll win.
3) Contestant chooses Door 1, the car is behind Door 3; Monty will open Door 2. If the contestant sticks he loses, if he switches he'll win.
4) Contestant chooses Door 2, the car is behind Door 1; Monty will open Door 3. If the contestant sticks he loses, if he switches he'll win.
5) Contestant chooses Door 2, the car is behind Door 2; Monty can open either Door 1 or Door 3. If the contestant sticks he wins, if he switches he'll lose.
6) Contestant chooses Door 2, the car is behind Door 3; Monty will open Door 1. If the contestant sticks he loses, if he switches he'll win.
7) Contestant chooses Door 3, the car is behind Door 1; Monty will open Door 2. If the contestant sticks he loses, if he switches he'll win.
8) Contestant chooses Door 3, the car is behind Door 2; Monty will open Door 1. If the contestant sticks he loses, if he switches he'll win.
9) Contestant chooses Door 3, the car is behind Door 3; Monty will open Door 2. If the contestant sticks he wins, if he switches he'll lose.

So, if Monty always opens door 2 when he has a choice, and the contestant knows that, then he should pick either Door 1 or Door 3 first. If Monty then does NOT open door 2, then switching brings about an automatic win. I am assuming that some similar conditional rule of thumb can be worked out for any other non-random method Monty might use to choose which door to open.

Nothing wrong with that. It's all the pages devoted to disabusing misunderstandings about the original problem.

For me the funny thing about the purist attitude towards the "original problem" is that, as I pointed out upthread, Monty never played according to the "original problem" on the original game show.

 

[Brian-M]

ETA: Permitted variations -- curtains instead of doors or mules instead of goats, but not both

Plus the host can be any member of Monty Python instead of Monty Hall... but only if they do the Ministry of Silly Walks routine on stage.

 

[ynot]

For me the funny thing about the purist attitude towards the "original problem" is that, as I pointed out upthread, Monty never played according to the "original problem" on the original game show.

The “Monty Hall Problem” was never an actual game show that was hosted and played by anyone. Monty’s name was merely associated with it because of his work on “Let’s Make A Deal".

 

[jt512]

The “Monty Hall Problem” was never an actual game show that was hosted and played by anyone. Monty’s name was merely associated with it because of his work on “Let’s Make A Deal".

Random trivia question: Without looking it up, who was the model (first and last name) on Let's Make a Deal?

 

[Prometheus]

The “Monty Hall Problem” was never an actual game show that was hosted and played by anyone. Monty’s name was merely associated with it because of his work on “Let’s Make A Deal".

Yes, that's my point. There was never a real thing, so it's funny that some people are so loyal to one contrived game over another.

 

[bruto]

Yes, that's my point. There was never a real thing, so it's funny that some people are so loyal to one contrived game over another.

All games are contrived.

 

[ynot]

Yes, that's my point. There was never a real thing, so it's funny that some people are so loyal to one contrived game over another.

For me it’s about keeping the thread on topic with the OP and poll which are about the standard version. Also it’s a matter of simplicity not loyalty or purity. I think introducing multiple scenarios is more likely to complicate rather than clarify. Perhaps I’m wrong.

 

[zooterkin]

And who is this "Monte" character?

Three card door Monte?

 

[Prometheus]

All games are contrived.

...and?

For me it’s about keeping the thread on topic with the OP and poll which are about the standard version. Also it’s a matter of simplicity not loyalty or purity. I think introducing multiple scenarios is more likely to complicate rather than clarify. Perhaps I’m wrong.

If that were a vital concern, the thread shouldn't be more than half a page long. To me, many of the alternate scenarios help to amplify subtle distinctions between the ways different people think about the problem. They're all on topic.

 

[Stamuel]

Consider this: why in the world would a variant where there are curtains rather than doors be on topic? That's not the original problem! The obvious answer is that doors vs. curtains does not affect the probabilities or the underlying logic. Note that this is a claim about the original game: doors/curtains don't matter (as long as they hide the prizes). It is on topic to say that some parts of the original game do not affect the answer and are just there to fill out the story.



As for the other variants: in this very thread, on multiple occasions, people have made a very similar claim about the original game: that it does not matter what Monty knows or which rules Monty follows in opening a door, as long as a goat is revealed. This is an on topic claim about the original game. It is certainly on topic to point out that it is a false claim. Thus, it is on topic to show why it is false, and to do this we have to consider variants. The consideration of variants was brought in to show that assumptions about Monty's knowledge and the rules he follows are indeed relevant features of the original game, not just irrelevant details like doors vs. curtains or the host's name.



Or, a simpler answer: do we need a separate thread for each minor variant of the MH problem? No *********** way. So it's happening here.

 

[GlennB]

Discussing variants

Or, a simpler answer: do we need a separate thread for each minor variant of the MH problem?

No we don't, but if variants are discussed with little or no clue that is a variant under discussion then it can easily look like a misunderstanding of the basic game.

Just above, while composing this post, there was a box called "Title" where I put the subject of this post. If people would use it (requires 'Go advanced' sometimes as I recall) then there would be less dispute.

Now get off my lawn

 

[Dr. Keith]

If that were a vital concern, the thread shouldn't be more than half a page long.

Well, there was at least one successful troll in there somewhere. That'll make a simple thread long.

 

[jiggeryqua]

Well, there was at least one successful troll in there somewhere. That'll make a simple thread long.

I didn't spot him.