3 Door Logic Problem

[Mr. Scott]

The fact that the door he opens isn't the one I chose makes me think of it as irrelevant to the outcome.

That's right, but probably in a different way than you are thinking.

Your chance of picking the right door at the beginning is 1/3.

When they open a door showing a goat, and if you resolve to not change your pick, then you still have a 1/3 chance of having picked the right door. Opening a door is indeed irrelevant to the outcome if you resolve not to switch. But switching to the other door does change the probability of a good outcome. It's the two events: the MC hinting at the right door by opening one of the wrong ones, and you switching to the door you didn't pick and the door the MC didn't open that changes the probability of a favorable outcome.

 

[Paul C. Anagnostopoulos]

If, instead of opening a goat door, the host opens one of the two remaining doors at random, how does this affect the benefit of switching?

Monty could end up opening the door with the car.

If, instead of opening a goat door, the host opens one of the two remaining doors at random and we only consider the cases where a goat is revealed, how does this affect the benefit of switching?

If we void the cases where Monty reveals the car, then it doesn't matter whether we stick or switch.

~~ Paul

 

[tkingdoll]

There's a woo aspect to the Monty Hall Problem we haven't touched on yet:

The average person believes he or she has some kind of faint ESP and would make a gut decision on which of the three doors to pick initially. Therefore, when Hall opens a goat door and offers the contestant a chance to change his choice, he's more likely to hold than to change, because changing shows he doesn't trust his gut.

So you have three possible attitudes for that contestant after one door is opened:

1) Stick with the first choice, because I trust my intuition.
2) It's now 50/50 so it doesn't matter.
3) Change my choice, because my intuition is probably wrong.

The woo choice would always be #1, easily proven to be the worst choice.

I'd be interested to see some stats on the actual show - how many people changed versus how many stuck. I want to know the actual ratio.

I did a quick Google and couldn't find anything though

 

[tsg]

If, instead of opening a goat door, the host opens one of the two remaining doors at random, how does this affect the benefit of switching?

If, instead of opening a goat door, the host opens one of the two remaining doors at random and we only consider the cases where a goat is revealed, how does this affect the benefit of switching?

Then the odds of you having chosen the car first go to 1/2. The difference is that the host now has a 1/3 chance of revealing the car.

1/3 of the time you will pick the car.
1/3 of the time the host will pick the car.
1/3 of the time the car will be behind the remaining door.

In the cases where the host reveals the car (eliminating the second outcome), the car is equally likely to be behind your door or the remaining door.

This is why it's important to specify that the host must know where the car is and won't show it to you.

 

[tsg]

There's a woo aspect to the Monty Hall Problem we haven't touched on yet:

The average person believes he or she has some kind of faint ESP and would make a gut decision on which of the three doors to pick initially. Therefore, when Hall opens a goat door and offers the contestant a chance to change his choice, he's more likely to hold than to change, because changing shows he doesn't trust his gut.

So you have three possible attitudes for that contestant after one door is opened:

1) Stick with the first choice, because I trust my intuition.
2) It's now 50/50 so it doesn't matter.
3) Change my choice, because my intuition is probably wrong.

The woo choice would always be #1, easily proven to be the worst choice.

The people I tell this puzzle to who get it wrong almost invariably say "well, it's 50-50 so I'd go with my first choice", in other words, both 1 & 2. There is a definite fear of having been right the first time and changing. Of course it's based on the incorrect assumption that it's 50-50. Most people who can be convinced that it's really 1/3-2/3 have no trouble changing.

 

[monoman]

Okay, I heard it explained 500 times, and I STILL CAN'T WRAP MY HEAD AROUND IT. All I can understand is simply a chart laying out all the options. The conclusion is clear, I just can't wrap my head around WHY it is clear, at all.

And all of you giving examples like "well imagine there are a billion boxes, then it's obvious isn't it?". Nope, that didn't help at all, because all billion of them are no longer a part of the puzzle once they've been opened. No matter how many extra boxes are added and opened, I'm still left with box I picked and box I didn't pick.

Somehow though, the math works out saying I need to switch.

I guess all I can really do is accept it.

This is the way that I think about it – the same as Ashless and Luke T.

You first pick a box.
Monty opens up a remaining empty box.

If you’ve first picked the prize, Monty has a choice of which empty box to show you (it doesn’t matter which).

If you’ve first picked an empty box, Monty has no choice, he just opens up the remaining empty box.

Now, here’s why you should always switch. If you switch then

(1)If you’d previously picked the prize*1, you’ll now switch to an empty box and lose
(2)If you’d previously picked an empty box*2, you’ll now switch to the prize and win

*1 The probability of initially picking the prize is 1/3

*2 The probability of initially picking an empty box is 2/3

Therefore, if you switch, the probability of ending up with the prize is 2/3

Hope this helps!

 

[Lamuella]

However, I question whether that was ever really established in the first version you heard. Even if it is called the Monty Hall problem, has it ever been established for a fact that he did ALWAYS open a door? I don't know that to be true.

The first time I encountered this problem was in the book "The curious incident of the dog in the night-time" by Mark Haddon. The lead character in that book has asperger's syndrome, and assumes very little. If I remember correctly he spelled the problem out very explicitly.

 

[Lamuella]

Equally, after whichever coin was heads, there's only one possibility for tails left, be it from either coin. There can't be two more tails left in the "deck" after one coin is known to be heads.

So Marilyn isn't removing one of the "cards" from the "deck" or the "tails that can't be"


I think I'm on to something but still confused.

let me explain my reasoning.

All the coins have been flipped at this point, everything is settled. We only need to know the results.

We have more information about person A's coin toss than we do about person B's coin toss. We know for a fact that out of the four equally likely possibilities (HH, HT, TH, TT), two have not come to pass. From the information we have, we know that TH and TT cannot be.

All we know about person B's coin toss is that he has got one or more head. In other words, all we know is that he has not got TT. When you ask "do you have one or more head?" you are in effect asking "Do you have two tails?". Only one of the three equally likely possibilities are ruled out.

Let's do a model of this. The following options are equally likely:

1) A:HH, B:HH
2) A:HH, B:HT
3) A:HH, B:TH
4) A:HH, B:TT
5) A:HT, B:HH
6) A:HT, B:HT
7) A:HT, B:TH
8) A:HT, B:TT
9) A:TH, B:HH
10) A:TH, B:HT
11) A:TH, B:TH
12) A:TH, B:TT
13) A:TT, B:HH
14) A:TT, B:HT
15) A:TT, B:TH
16) A:TT, B:TT

Of the 16 possible options, only 6 (the ones I have emboldened) are possible in this situation with the information we have.

Of those six equally likely options, three show A having two heads. Two show B having two heads.

Thus, the odds of A having two heads are 3/6 or 1/2, whereas the odds of B having two heads are 2/6 or 1/3

 

[tsg]

Okay, I heard it explained 500 times, and I STILL CAN'T WRAP MY HEAD AROUND IT. All I can understand is simply a chart laying out all the options. The conclusion is clear, I just can't wrap my head around WHY it is clear, at all.

The key to understanding is realizing that Monty opening another door doesn't give you any information about the chances you picked the right door. It does give you information about which remaining door is more likely to contain the car. You have a 1/3 chance of picking the right door, and a 2/3 chance that the car is behind the remaining two doors. He's shown you which of the remaining two doors it's not behind. In essence, all he's doing is giving you the choice between the one door you did pick and the two you didn't. The probability of winning by switching is entirely dependent on your odds of having picked the car in the first place.

And all of you giving examples like "well imagine there are a billion boxes, then it's obvious isn't it?". Nope, that didn't help at all, because all billion of them are no longer a part of the puzzle once they've been opened. No matter how many extra boxes are added and opened, I'm still left with box I picked and box I didn't pick.

The problem is the assumption that they are equally likely. After you make your choice, removing a door the car couldn't be behind doesn't change the probability of picking the car any more than if he showed you another 100 doors that the car couldn't be behind.

Another analogy that I have heard that sometimes helps clear it up: I think of a number between one and a million. You guess and I tell you "Okay, it's either the number you picked, or 653,952 (providing you didn't guess that number). Which do you think it is?" One out of a milion tries you will guess right and I'll give you a wrong number to choose from. The rest of the time, you will have guessed wrong and I'll give you the right number. Reduce the problem to a choice of three numbers and you have Monty Hall.

I guess all I can really do is accept it.

This being a skeptic forum, I would like to see you understand it rather than just take my word for it

 

[Beleth]

I'd be interested to see some stats on the actual show - how many people changed versus how many stuck. I want to know the actual ratio.

I don't know the actual ratio, but I do remember watching the show as a child and thinking to myself "gee, they sure do change their minds an awful lot."

 

[Jekyll]

If, instead of opening a goat door, the host opens one of the two remaining doors at random, how does this affect the benefit of switching?

If, instead of opening a goat door, the host opens one of the two remaining doors at random and we only consider the cases where a goat is revealed, how does this affect the benefit of switching?

If he randomly opens any door you didn't pick then the odds aren't changed by switching, in both cases you're just betting that a single door doesn't have a goat behind it which has odds of 1/3.

 

[Trifikas]

If it helps, skip having Monty open a door entirely. Monty isn't showing you anything you don't already know - One of the two doors you didn't pick has a goat behind it. 2 goats, 3 doors, You pick one...yep, at least one goat left.

The problem then boils down to:

Pick a door.
Monty will offer you to keep the door you picked, or to switch to the other two doors.

Is your strategy now to keep the one door you picked, or take the offered two doors?

Him showing you a goat behind one of the other two doors is immaterial - you knew there was a goat behind one of them. So the odds for someone who always switches is 2/3, since they're getting two doors.

 

[tkingdoll]

I don't know the actual ratio, but I do remember watching the show as a child and thinking to myself "gee, they sure do change their minds an awful lot."

Which would suggest that they also won a lot. Do you have a similar recollection of that happening?

 

[c4ts]

That's right, but probably in a different way than you are thinking.

Your chance of picking the right door at the beginning is 1/3.

When they open a door showing a goat, and if you resolve to not change your pick, then you still have a 1/3 chance of having picked the right door. Opening a door is indeed irrelevant to the outcome if you resolve not to switch. But switching to the other door does change the probability of a good outcome. It's the two events: the MC hinting at the right door by opening one of the wrong ones, and you switching to the door you didn't pick and the door the MC didn't open that changes the probability of a favorable outcome.

So the gameshow is actually detracting from its slight advantage by providing the opportunity to switch? They're now betting on a 1/3 chance that the player was right in the first place by giving him a chance to switch to the wrong door. That's very counterintuitive because usually game shows (or at least the ones I've seen) don't try to help the contestants win. But I guess the idea is some kind of reverse psychology, like what Regis uses when he says "Is that your final answer?" because nobody changes it, adding suspense and pressure on the player without changing the outcome. I guess if the contestants say "no" every time, there is no harm slightly improving the odds.

 

[Beleth]

Which would suggest that they also won a lot. Do you have a similar recollection of that happening?

Unfortunately, no. People won and lost at random so frequently on that show. There was more to LMAD than the three-door Monty. In that regard, it wasn't very much like that suitcase game show at all.

 

[Dustin Kesselberg]

I never leave the house. If I go out side my chances of dying are 50%. I either live or I die. 50/50. The second I make a first step I have a 50% chance of dying. The second step is a 50% chance of dying. My chances of dying after several steps is almost 100%!

 

[Dark Jaguar]

I GET IT!

I can't explain it any simpler than it has already been explained though, so those who don't get it will just have to get it to "click".

Also, the examples of "the billions" of options aren't going to cut it, that only makes perfect sense AFTER you get it anyway, but I GET IT!

Yes, it is clear, rather obvious to me.

By the way, when I said "I'll just accept it" I mean I was accepting the only conclusion the math would allow (listing out every option) as opposed to my intuition, not just "accepting it because you told me to".

Anyway, finally! That thing was starting to bug me. I "got" that he always showed an empty one, but I kept artificially removing the empty one from the puzzle. Took a long time to actually manage to keep the empty one IN the puzzle. The guy opens that empty one, and what are the chances you guessed right? 1/3, meaning the unopened one is going to be the thing you want 2 out of 3 times, BY VIRTUE of the fact that you getting it right first time is only 1/3.

BAM! Feel stupid now, but it's fun reprogramming my intuition.

 

[Dark Jaguar]

I never leave the house. If I go out side my chances of dying are 50%. I either live or I die. 50/50. The second I make a first step I have a 50% chance of dying. The second step is a 50% chance of dying. My chances of dying after several steps is almost 100%!

I hear this from daredevil personalities often enough. I always try to explain "yeah sure I could die going outside, but I'm pretty sure the chances of it are a lot lower than disaster during a mountain climb, and probability is what I care about". And if that doesn't work, it takes an hour long session of slowly explaining to them that my life is not in fact "dull" and "pointless" without that sort of risk taking and I honestly have no desire to do it, at all. It just doesn't interest me. I have no idea why they can't wrap their head around that. Their desire for it must be so ingrained in their psyche that the very idea of someone hiding away from it is like someone refraining from everything they desire out of cowardice, it must mean they have a sad lonely personality defect.

 

[RandFan]

I GET IT!

I can't explain it any simpler than it has already been explained though, so those who don't get it will just have to get it to "click".

Also, the examples of "the billions" of options aren't going to cut it, that only makes perfect sense AFTER you get it anyway, but I GET IT!

That's how I worked out for myself. I wondered if I had a million doors? Then it was so obvious. Of course I would change my mind. I started with a 1 in a million shot and 998,000 were eliminated. There simply is no way that eliminating 998,000 doors would magically improve my odds and that is the only other explanation.

BTW, the problem works best by starting with only the 3 doors. Give someone the problem who has never heard of it but start with one million doors. You'll see most would change doors.

 

[Mr. Scott]

I'd be interested to see some stats on the actual show - how many people changed versus how many stuck. I want to know the actual ratio.

I did a quick Google and couldn't find anything though

As far as I know, the "Monty Hall Puzzle" is often explained in terms of the "Let's Make a Deal" TV show, but the show didn't play out the same exact scenario, but rather an infinite variety of "Deal or No Deal" style sceneria which may or may not have had other doors open. Sometimes the door concealing the car would open up and Monty would offer to exchange that with a newly introduced giant gift box, or cash out of his pocket or...you get the picture. It was more manipulative psychology than statistics.