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Monty Hall Problem

hgc said:
Or how about with the 100 door scenario, you pick a door, Monty reveals 98 goats. Do you still think you have a 99% chance by switching? That is the apt analogy.
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It depends on whether Monty knows where the car is, and is purposefully avoiding it - or Monty doesn't know where the car is, and it just so happens that he has avoided revealing it this far. (It also depends on whether Monty must always reveal a door and allow you to change your choice or not, but that adds unneeded complexity.)

If he knows and is avoiding it, your original choice had a 1% chance of being right when you selected it, and it still does... which means that the other door has a 99% chance of having the car.

If he doesn't know and just got lucky, then there is a 50/50 chance.


Here's the analogy that convinced me.
Say you pick one of the three doors.
Instead of showing you that one of the two doors remaining has a goat behind it (which, to be honest, you already knew; you just didn't know which one), Monty gives you the option to select both remaining doors in exchange for the door you just picked.

What would be the odds then?
They would be exactly the same as in the original problem.
 
HarryKeogh said:
imagine there are 100 doors instead of just 3. now imagine Monty asks you after each door is opened do you want to switch. when he gets down to 2 doors do you still think your original door has a 50% chance of being right?

right, it has a 1% chance. The same as when you originally chose.

this analogy helped me understand this puzzle when I first saw it.
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Actually, if he narrowed it down to the door I originally picked and one more door, I do think it is now an 50% chance. How is this not so?
 
Beleth said:
It depends on whether Monty knows where the car is, and is purposefully avoiding it - or Monty doesn't know where the car is, and it just so happens that he has avoided revealing it this far. (It also depends on whether Monty must always reveal a door and allow you to change your choice or not, but that adds unneeded complexity.)

If he knows and is avoiding it, your original choice had a 1% chance of being right when you selected it, and it still does... which means that the other door has a 99% chance of having the car.

If he doesn't know and just got lucky, then there is a 50/50 chance.


Here's the analogy that convinced me.
Say you pick one of the three doors.
Instead of showing you that one of the two doors remaining has a goat behind it (which, to be honest, you already knew; you just didn't know which one), Monty gives you the option to select both remaining doors in exchange for the door you just picked.

What would be the odds then?
They would be exactly the same as in the original problem.
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I guess the big thing I am not getting is why are the two choices not seperated? The original guess is 1/3, yes, and then you get presented with a new choice (yes/no). Your analogy really doesn't make sense to me.
 
DaveW said:
Actually, if he narrowed it down to the door I originally picked and one more door, I do think it is now an 50% chance. How is this not so?
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because he knows which door has the prize behind it. He has to have the prize behind one of those 2 after eliminating 98 other doors.

like Beleth mentioned above, if he didn't know and the prize wasnt revealed it would be a pretty good coincidence and a 50/50 chance

but the important part of the puzzle is that he does know.

I don't know any other way to explain it, hope this helps.
 
HarryKeogh said:
because he knows which door has the prize behind it. He has to have the prize behind one of those 2 after eliminating 98 other doors.

like Beleth mentioned above, if he didn't know and the prize wasnt revealed it would be a pretty good coincidence and a 50/50 chance

but the important part of the puzzle is that he does know.

I don't know any other way to explain it, hope this helps.
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I don't see how that follows. Whether he knows what door has the prize or not, in the end, I am still down to one of two doors. Sure, it'd look suspicious if I originally picked door 1 and he walked down all the other doors from 100 down, skipping just door 57, but, in the end, he could have just picked door 57 at random to not open to leave me to switch to when the prize is behind my door. The last choice I am making in this case is between door 1 and door 57, not door 1 and the other 99 doors (98 of which are open).
 
DaveW said:
Actually, if he narrowed it down to the door I originally picked and one more door, I do think it is now an 50% chance. How is this not so?
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99 times out of a hundred you're going to have the wrong door on your first choice. 99 times out of a hundred that unopened door has the car behind it.

The door you chose and the door that's still closed are not randomly selected.

You make your first choice, picking one door out of a hundred. What are the chances you've got the prize there? 1/100. There is very little chance you got the prize the first time. You KNOW that's not a car behind your door. Why should Monty's shenanigans change that and sudenly make it 50/50 that you got it right the first time? What makes you think there's now a good chance the goat you were certain was there is now a car?
 
Forget the 100 doors explanation. Here's an easier one:

Do you agree that if Monty offered you both of the other doors, you'd switch to them? Of course, since you'd have a 2/3 chance instead of your original 1/3 chance.

Well, that's exactly what Monty did. He silently offered you both doors, then opened the one that doesn't have the car, then asked if you'd like the other one.

~~ Paul
 
rppa said:
99 times out of a hundred you're going to have the wrong door on your first choice. 99 times out of a hundred that unopened door has the car behind it.

The door you chose and the door that's still closed are not randomly selected.

You make your first choice, picking one door out of a hundred. What are the chances you've got the prize there? 1/100. There is very little chance you got the prize the first time. You KNOW that's not a car behind your door. Why should Monty's shenanigans change that and sudenly make it 50/50 that you got it right the first time? What makes you think there's now a good chance the goat you were certain was there is now a car?
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OK, let's start with what little I know of statistics and choices. If a choice is independent, the probability of that choice does not change (think consecutive dice rolls). But, if my choice somehow effects the outcome, the probabilities change. So, I choose 1 door (1/3). Monty shows me one bad door. Now my new chances, if I start all over again with this new knowledge, is 1/2 (I won't choose the obviously bad open door). This is essentially what is happening in this example, as far as I can tell.
 
MEA CULPA

I just rethought the 100 door analogy. It is a 99% chance of getting the car if switching, because it was a 99% chance that the car was in that pool of 99 that you didn't pick, and since Monty was nice enough to reveal 98 goats (and he must, so goes the premise), then it's a 99% chance that the remaining one is the car.

It's 2/3. End of story (until I change my mind again).
 
Paul C. Anagnostopoulos said:
Forget the 100 doors explanation. Here's an easier one:

Do you agree that if Monty offered you both of the other doors, you'd switch to them? Of course, since you'd have a 2/3 chance instead of your original 1/3 chance.

Well, that's exactly what Monty did. He silently offered you both doors, then opened the one that doesn't have the car, then asked if you'd like the other one.

~~ Paul
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Woah! I see it more like: I get a non-sensical first choice. Then he shows me a bad one and I get to choose one out the remaining two. How did I ever get to select 2 doors?
 
Re: MEA CULPA

hgc said:
I just rethought the 100 door analogy. It is a 99% chance of getting the car if switching, because it was a 99% chance that the car was in that pool of 99 that you didn't pick, and since Monty was nice enough to reveal 98 goats (and he must, so goes the premise), then it's a 99% chance that the remaining one is the car.

It's 2/3. End of story (until I change my mind again).
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Gah! Now I'm the only statistics idiot in this thread :p
 
Or here's yet another way to think about it.

You had a 1/3 chance of picking the right door. You have a 2/3 chance of being wrong. When Monty reveals the goat behind one of the other doors, he isn't revealing anything you don't already know. Why? Because given any two doors, one of them MUST have a goat behind it. So there's still a 2/3 chance that the car is behind one of the pair of doors and a 1/3 chance that it's behind the door you picked. The odds don't change because he revealed which of the two doors had the goat.
 
Paul C. Anagnostopoulos said:
But you aren't starting over, because the bad door could have had the car, even though it does not.

~~ Paul
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I feel like I'm not making myself clear... the second choice's probability is dependent on the first choice...just like you seemed to confirm. That is why the statistics should change...
 
Ipecac said:
Or here's yet another way to think about it.

You had a 1/3 chance of picking the right door. You have a 2/3 chance of being wrong. When Monty reveals the goat behind one of the other doors, he isn't revealing anything you don't already know. Why? Because given any two doors, one of them MUST have a goat behind it. So there's still a 2/3 chance that the car is behind one of the pair of doors and a 1/3 chance that it's behind the door you picked. The odds don't change because he revealed which of the two doors had the goat.
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Ah, but he is! He lets me know one of the two doors that was wrong!
 
DaveW said:
Woah! I see it more like: I get a non-sensical first choice. Then he shows me a bad one and I get to choose one out the remaining two. How did I ever get to select 2 doors?
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You get to select two doors by virtue of (a) Monty eliminating one; (b) you getting to choose the other.

Let me restate it:

Do you agree that if Monty offered you both of the other doors, you'd switch to them? Of course, since you'd have a 2/3 chance instead of your original 1/3 chance.

Well, that's effectively what Monty did. He "silently" offered you both doors, then opened the one that doesn't have the car, then asked if you'd like the other one. You are effectively getting to choose both of the other doors.

~~ Paul
 
Just to let everyone know... I'm not trying to bust anyone's chops, I really just don't get it, and I know I don't, but hashing this out like this will hopefully help me learn it. Thanks!
 
Paul C. Anagnostopoulos said:
You get to select two doors by virtue of (a) Monty eliminating one; (b) you getting to choose the other.

Let me restate it:

Do you agree that if Monty offered you both of the other doors, you'd switch to them? Of course, since you'd have a 2/3 chance instead of your original 1/3 chance.

Well, that's effectively what Monty did. He "silently" offered you both doors, then opened the one that doesn't have the car, then asked if you'd like the other one. You are effectively getting to choose both of the other doors.

~~ Paul
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Sure, but I only ever get one door at a time. I never "have" 2 doors.
 
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