Why doesn't the solution to the Monty Hall problem make sense?

[r0bHadz]

Homework Statement

You have 3 doors, with 2/3 chance of being wrong.

A host opens a door, and there is no prize.

You are now left with two doors. I would like an explanation why the car is still equally likely to be behind any three doors still after the host opens a door.

I have a hard time that Paul Erdos could not logically come to this conclusion as well. Something is off about this problem

Homework Equations

The Attempt at a Solution

 

[PeroK]

You could write a computer program to simulate it.

Or, you could simulate the experiment using playng cards. Probablity is relative frequency, so if you simulate the experiment a large number of times, you can estimate the probability.

 

[r0bHadz]

You could write a computer program to simulate it.

Or, you could simulate the experiment using playng cards. Probablity is relative frequency, so if you simulate the experiment a large number of times, you can estimate the probability.

Is there any logical way of explaining of why the probability of picking the car doesn't change

 

[PeroK]

Is there any logical way of explaining of why the probability of picking the car doesn't change

That's been discussed on here about a hundred times. The key point is that the host knows where the car is and never accidentally reveals it. The problem is different if one show in three on average Monty ruins things by revealing the car by mistake.

Note that, as a student of mathematical probability, there is nothing funny about this problem. It's only controversial because it's in the public domain and debated by people with no knowledge of how to analyse probabilities. The sort of people who think that if a coin comes up heads 3-4 times in a row it is more and more likely to be heads next time.

It's important, therefore, to analyse both problems:

1) The actual scenario in the show.

2) The scenario where Monty opens a door (without knowing where the car is) and by luck gets an empty door.

In terms of scenario 1, I would imagine there are 100 doors. You pick one. Monty then opens 98 doors, all empty. That leaves two doors. Then I think it's more obvious that there is still only a 1% chance that you picked the right door first time and a 99% chance that the car is behind the one door that Monty didn't choose to open! It's fairly obvious to me where the car is likely to be in that scenario.

 

[r0bHadz]

That's been discussed on here about a hundred times. The key point is that the host knows where the car is and never accidentally reveals it.
1) The actual scenario in the show.In terms of scenario 1, I would imagine there are 100 doors. You pick one. Monty then opens 98 doors, all empty. That leaves two doors. Then I think it's more obvious that there is still only a 1% chance that you picked the right door first time and a 99% chance that the car is behind the one door that Monty didn't choose to open! It's fairly obvious to me where the car is likely to be in that scenario.

Yes, I am questioning the situation in the show, where Monty knows where everything is.

Your post didn't really answer my question though: why wouldn't the probability of picking the car not change?

If he opens 98 doors, leaving two doors, you say there is a 1% chance that he chose the right door from the start. How could it possibly be 99% to 1% if he could have chosen that other door that is left, instead of the one he chose, then it would be 99% to 1% the other way.

 

[PeroK]

Yes, I am questioning the situation in the show, where Monty knows where everything is.

Your post didn't really answer my question though: why wouldn't the probability of picking the car not change?

If he opens 98 doors, leaving two doors, you say there is a 1% chance that he chose the right door from the start. How could it possibly be 99% to 1% if he could have chosen that other door that is left, instead of the one he chose, then it would be 99% to 1% the other way.

That makes no sense.

Simulate the experiment if you don't believe me. Or, bet on it! I'd be happy to take your money!

 

[stockzahn]

You have 3 doors, with 2/3 chance of being wrong.

A host opens a door, and there is no prize.

You are now left with two doors. I would like an explanation why the car is still equally likely to be behind any three doors still after the host opens a door.

I have a hard time that Paul Erdos could not logically come to this conclusion as well. Something is off about this problem

In the scenario you described you can start the game with just two doors left, since the host opens one empty door before you even chose one. He could have done that before you enter the room and it wouldn't change anything, so in your described scenario the chance of getting the car should be 50-50. The original problem is based on the assumption that you have to choose one door before the host opens an empty one:

1. Choose a door
2. The host opens an empty door
3. You have to decide if you switch to third door or stick with your initial choice

In that case you can choose one of three doors: the car door $C$ or one of the empty doors $E1$, $E2$ - each of the choices is made with the same probability:

1 x 1/3) You choose $E1 \rightarrow$ the host opens $E2\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $E2 \rightarrow$ the host opens $E1\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $C \rightarrow$ the host opens an empty door $\rightarrow$ change = lose, stick = win

In two thirds of your initial choices changing leeds two winning the car, in one third you lose. Therefore you should change.

 

[PeroK]

... Monty Hall rides again!

 

[r0bHadz]

1 x 1/3) You choose $E1 \rightarrow$ the host opens $E2\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $E2 \rightarrow$ the host opens $E1\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $C \rightarrow$ the host opens an empty door $\rightarrow$ change = lose, stick = win

In two thirds of your initial choices changing leeds two winning the car, in one third you lose. Therefore you should change.

Sorry I meant the original monty hall problem, I didn't feel like it needed to be written out since I thought everyone knows about it.

My question was: Is there any logical way of explaining of why the probability of picking the car doesn't change

and your post doesn't really answer it.

You choose $E2 \rightarrow$ the host opens $E1\rightarrow$ (why doesn't the probability chance to 1/2, 1/2 here, since we only have two doors left?) change = win, stick = lose

 

[PeroK]

My question was: Is there any logical way of explaining of why the probability of picking the car doesn't change

The simplest answer is that the host can always open an empty door. What he does, therefore, does not affect the existing probability that you picked the correct door.

The second event, therefore, carries no information about the door you picked. It carries information only about the doors you did not pick.

 

[stockzahn]

Sorry I meant the original monty hall problem, I didn't feel like it needed to be written out since I thought everyone knows about it.

My question was: Is there any logical way of explaining of why the probability of picking the car doesn't change

and your post doesn't really answer it.

You choose $E2 \rightarrow$ the host opens $E1\rightarrow$ (why doesn't the probability chance to 1/2, 1/2 here, since we only have two doors left?) change = win, stick = lose

It's that: The original problem is based on the assumption that you have to choose one door before the host opens an empty one. But I just can agree with @PeroK. Take three cards and play the game hundred times - you are done within one hour.

EDIT: Following these possible paths seems to me as logical as it can be:

1 x 1/3) You choose $E1 \rightarrow$ the host opens $E2\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $E2 \rightarrow$ the host opens $E1\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $C \rightarrow$ the host opens an empty door $\rightarrow$ change = lose, stick = win

 

[r0bHadz]

The simplest answer is that the host can always open an empty door. What he does, therefore, does not affect the existing probability that you picked the correct door.

The second event, therefore, carries no information about the door you picked. It carries information only about the doors you did not pick.

The host will always open an empty door. If it doesn't change the existing probability that I picked the correct door, why doesn't the probability of the remaining door change?

Mathematically it would be because door 1 is 1/3 then the remaining door would have to be 2/3 to satisfy probability laws.

Which is why I feel as though the probability of the first door changing would have to happen, or the entire model would have to change.

 

[r0bHadz]

It's that: The original problem is based on the assumption that you have to choose one door before the host opens an empty one. But I just can agree with @PeroK. Take three cards and play the game hundred times - you are done within one hour.

EDIT: Following these possible paths seems to me as logic as it can be:

1 x 1/3) You choose $E1 \rightarrow$ the host opens $E2\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $E2 \rightarrow$ the host opens $E1\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $C \rightarrow$ the host opens an empty door $\rightarrow$ change = lose, stick = win

I mean I guess I will have to do it as I've searched all over math stack exchange and there was a question word for word like mine and I haven't found a single answer that satisfied me. I will try out the experiment but it is hardly mathematical and I wouldn't consider it a proof

 

[stockzahn]

I mean I guess I will have to do it as I've searched all over math stack exchange and there was a question word for word like mine and I haven't found a single answer that satisfied me. I will try out the experiment but it is hardly mathematical and I wouldn't consider it a proof

What's mathematically wrong with decision trees?

 

[PeroK]

I mean I guess I will have to do it as I've searched all over math stack exchange and there was a question word for word like mine and I haven't found a single answer that satisfied me. I will try out the experiment but it is hardly mathematical and I wouldn't consider it a proof

This is what I warned you about: people who say "it's more likely to come up heads next time". Then, when you do an experiment with a real coin and it's shown that it's 50-50 every time, they say "it's mathematically more than 50%, but in an experiment it's 50%".

What you're saying is that "mathematically the probabilty is 50-50". It's just that reality differs from your mathematics. In which case your mathematics is wrong!

 

[r0bHadz]

What's mathematically wrong with decision trees?

I was talking about the act of carrying an experiment out. I could carry out experiments until I die and it would not constitute a proof under "mathematics" am i right?

This is what I warned you about: people who say "it's more likely to come up heads next time". Then, when you do an experiment with a real coin and it's shown that it's 50-50 every time, they say "it's mathematically more than 50%, but in an experiment it's 50%".

What you're saying is that "mathematically the probabilty is 50-50". It's just that reality differs from your mathematics. In which case your mathematics is wrong!

I completely agree, but this is a mathematical problem. It's in a textbook, not the real world. So it only feels like it should be able to be worked out through logic and proven.

I will have to go talk to my prof in a couple of hours here since I don't feel like I convert my doubts and my understanding from reading text is limited I guess but hopefully I can move along soon. I have wasted a lot of time on this already -_-

 

[stockzahn]

I was talking about the act of carrying an experiment out. I could carry out experiments until I die and it would not constitute a proof under "mathematics" am i right?

You are right. Then differently: Why don't you accept the decision tree as mathematical method?

 

[r0bHadz]

You are right. Then differently: Why don't you accept the decision tree as mathematical method?

Are you talking about this:

1 x 1/3) You choose $E1 \rightarrow$ the host opens $E2\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $E2 \rightarrow$ the host opens $E1\rightarrow$ change = win, stick = lose
1 x 1/3) You choose $C \rightarrow$ the host opens an empty door $\rightarrow$ change = lose, stick = win

If so, the problem to me doesn't seem like "if you choose E1," "if you choose E2" ... and so on (if we look at this problem with the 100 door example)

It's: "You've chosen a door. A door with no prize is eliminated. There are now two doors, you can have the same choice or change it"

I'm looking for something that satisfies the question "why does the probability not change"

We can look at this problem after a trail of 100, even 1 million, but this problem is mathematically a trail of just 1. (Actually, on a game show it is too. Youre probably not going to get invited again.)

 

[PeroK]

I was talking about the act of carrying an experiment out. I could carry out experiments until I die and it would not constitute a proof under "mathematics" am i right?
-

That's not true of applied mathematics and especially probability theory and computer science.

It's not necessarily about proving something rigorously but about informing your logic.

 

[stockzahn]

It's: "You've chosen a door. A door with no prize is eliminated. There are now two doors, you can have the same choice or change it"

It's: You have three choices of equal probability (paths). Two of the three paths lead to the car, if you change. The third path leads to the car, if you don't change. Therefore in two out of three choices, changing is the winning strategy.

I'm looking for something that satisfies the question "why does the probability not change"

Because you made your first door choice before the host opens one of the empty doors. The choice was made at 1/3 probability - that doesn't change even another door is opened afterwards (consider the 100 doors example of @PeroK).

 

[mfb]

If it doesn't change the existing probability that I picked the correct door, why doesn't the probability of the remaining door change?

A door you didn't pick can get opened, a door you picked cannot.

A very simple way to consider the question:
If you keep your door, you win if and only if you pick the right door initially.
If you switch your door, you win if and only if you picked one of the two wrong doors initially.
Which case is more likely?

 

[DrClaude]

If Monty Hall always choses a door where the car is not, offering to change after opening the door is the same as offering you the chance between

• keeping your original choice
• choosing all the other doors together

 

[OmneBonum]

The OP's original question was 'Why doesn't the solution to the Monty Hall problem make sense?'
The answer to that question isn't strictly mathematical. It has to do with the human brain, and how it evolved to do the kinds of things that it needs to do on a regular basis to ensure the survival of the species.
I remember when I first was presented with the problem, on an episode of MythBusters. I actually thought to myself, "If this is true, if switching beats staying pat, then I am ready to believe it is a magical phenomenon!" But then, I had a good think on the throne and I thought to myself, "Oh, wait. Monty knows where the prize is. Information is entering the system!" I eventually worked it out, but I will never forget, there was that very weird moment when my mind just could not fathom it! There's a nice, quick article on this in WIRED which I think the OP might have read or even linked to previously: https://www.wired.com/2014/11/monty-hall-erdos-limited-minds/

 

[StoneTemplePython]

That's been discussed on here about a hundred times. The key point is that the host knows where the car is and never accidentally reveals it. The problem is different if one show in three on average Monty ruins things by revealing the car by mistake.

Note that, as a student of mathematical probability, there is nothing funny about this problem. It's only controversial because it's in the public domain and debated by people with no knowledge of how to analyse probabilities...

The only thing I'd add, is a lot of people who should know better have messed this problem up. I think the bulk of it comes from people who don't understand basic probability. But a fair amount of mathematicians have messed it up (most famously Erdos) because there's something of a linguistic sleight of hand in how the problem is typically posed (e.g. by vos Savant).

If the problem is posed and the question asker addresses underlined part explicitly, then I think people who should know better tend to do much better.

- - - -
as for mathy-er solutions:
my vote goes to a Bayes formulation, or posing it as a renewal rewards problem.

 

[Ray Vickson]

Sorry I meant the original monty hall problem, I didn't feel like it needed to be written out since I thought everyone knows about it.

My question was: Is there any logical way of explaining of why the probability of picking the car doesn't change

and your post doesn't really answer it.

You choose $E2 \rightarrow$ the host opens $E1\rightarrow$ (why doesn't the probability chance to 1/2, 1/2 here, since we only have two doors left?) change = win, stick = lose

Look at the "do not switch" strategy. Initially (before any doors are opened) you have P(win) = 1/3, P(lose) = 2/3. If the car is behind your chosen door, Monty opens one of the other doors, so you still win. However, if the car is not behind your door, Monty opens the other door (because you have chosen one door already, and Monty avoids opening the car-door). So, in any case you lose, and the chance of that is 2/3. So, basically, by not switching it does not matter whether Monty opens a door or not; you still have P(win) = 1/3, P(lose) = 2/3. In this case, your statement that the probability of picking the car does not change is correct.

Now look at the "always switch" strategy. In 1/3 of the cases the car is behind your chosen door, so switching will cause you to lose. However, in 2/3 of the cases the car is not behind either your original door or the door opened by Monty, so switching will cause you to win---every single time. So, switching leads to P(win) = 2/3 and P(lose) = 1/3. In this case, by switching, you do, in fact, change the probabilities of winning the car

 

[r0bHadz]

A door you didn't pick can get opened, a door you picked cannot.

A very simple way to consider the question:
If you keep your door, you win if and only if you pick the right door initially.
If you switch your door, you win if and only if you picked one of the two wrong doors initially.
Which case is more likely?

They're both equally likely, because before you switch your door, a door that has nothing behind it is revealed.

"A door you didn't pick can get opened, a door you picked cannot."

Right, and the other door I didn't pick can't get opened either.

I will look into what StoneTemplePython is saying because I need this answered mathematically: "why does the probability not change."

 

[r0bHadz]

Look at the "do not switch" strategy. Initially (before any doors are opened) you have P(win) = 1/3, P(lose) = 2/3. If the car is behind your chosen door, Monty opens one of the other doors, so you still win. However, if the car is not behind your door, Monty opens the other door (because you have chosen one door already, and Monty avoids opening the car-door). So, in any case you lose, and the chance of that is 2/3. So, basically, by not switching it does not matter whether Monty opens a door or not; you still have P(win) = 1/3, P(lose) = 2/3. In this case, your statement that the probability of picking the car does not change is correct.

Now look at the "always switch" strategy. In 1/3 of the cases the car is behind your chosen door, so switching will cause you to lose. However, in 2/3 of the cases the car is not behind either your original door or the door opened by Monty, so switching will cause you to win---every single time. So, switching leads to P(win) = 2/3 and P(lose) = 1/3. In this case, by switching, you do, in fact, change the probabilities of winning the car

Great explanation, and I do seem to understand it. My original question of: why does the probability not change, remains to be unanswered though. I would like an explanation on why after a door is exposed do we have a probability of 2/3d on one door.

 

[mfb]

They're both equally likely

"You initially pick the right door" and "you initially pick a wrong door" are equally likely? Do you really think this?

Ignore what happens afterwards for a moment. Just focus on the initial selection. Do you really think these two options are equally likely?

What about a lottery? "You initially pick all the right numbers" vs. "you initially don't pick all the right numbers"?

 

[r0bHadz]

"You initially pick the right door" and "you initially pick a wrong door" are equally likely? Do you really think this?

Ignore what happens afterwards for a moment. Just focus on the initial selection. Do you really think these two options are equally likely?

What about a lottery? "You initially pick all the right numbers" vs. "you initially don't pick all the right numbers"?

No of course not. Before a door is revealed you have a 1/3 chance picking any of the three doors.

 

[mfb]

Good. Then what is unclear? If you decide to not switch (you can make this decision long in advance, doesn't matter) you are less likely to be successful (1/3). If you decide to switch then you are more likely to be successful (2/3).

 

[StoneTemplePython]

I will look into what StoneTemplePython is saying because I need this answered mathematically: "why does the probability not change."

No of course not. Before a door is revealed you have a 1/3 chance picking any of the three doors.

ok, so if you want to see the symbol manipulation spelled out, here's the Bayes Formulation:

Define $A$ as the event that your initial selection has the prize and $B$ as the event that Monty opens a door, after you've made your selection, and that door does not have a prize behind it.

Now think a carefully about the event B in this problem: Monty opens a door that you haven't selected and that door doesn't have a prize behind it. As I underlined in my earlier post (and a few others have said), this is the key insight to the puzzle. He does this no matter what-- when you have selected the correct door he does this with probability 1 and when you have not selected the correct door he does this with probability 1. That is the event $B$ has a raw probability of 1 -- i.e. $P\big(B\big) =1$, and if we condition on your selection (or anything really) it still has probability of 1.

Plugging this into Bayes' Rule, you have

$P\big(A \vert B\big) = \frac{P\big(A\big)\cdot P\big(B\vert A\big) }{P\big(B\big)} = \frac{\frac{1}{3} \cdot 1 }{1}=\frac{1}{3}$ hence the probability of winning given that you stick with your initial selection is equivalent to the probability that your initial selection is right given event $B$ and that is $\frac{1}{3}$.

 

[r0bHadz]

Good. Then what is unclear? If you decide to not switch (you can make this decision long in advance, doesn't matter) you are less likely to be successful (1/3). If you decide to switch then you are more likely to be successful (2/3).

Why would you have a 2/3 success at switching if one of the doors were already revealed? 1/3 of those 2/3 are gone now because it was revealed and the prize is not there.

 

[r0bHadz]

StoneTemplePython, I appreciate the post but that doesn't really say anything about my question: why, after monty opens a door, does one of the two remaining doors have a probability of 2/3 that the prize is behind it, while the one you chose has only 1/3?

 

[StoneTemplePython]

StoneTemplePython, I appreciate the post but that doesn't really say anything about my question: why, after monty opens a door, does one of the two remaining doors have a probability of 2/3 that the prize is behind it, while the one you chose has only 1/3?

Actually it does answer exactly this.

I need this answered mathematically: "why does the probability not change."

And again, it answers exactly this in as direct a mathematical way as I could think of.
- - - -
Evidently, the problem is that you don't understand Bayes Rule and/or events.

 

[r0bHadz]

Actually it does answer exactly this. And again, it answers exactly this in as direct a mathematical way as I could think of.
- - - -
Evidently, the problem is that you don't understand Bayes Rule and/or events.

I understand bayes rules and I understand why you have a 1/3 chance if you remain with your door. And I understand that there has to be another 2/3 somewhere to satisfy 1/3 + x = 1. What I don't understand is how does monty revealing one of the doors give the remaining door that you have the option to switch to, give it a probability of 2/3, since it started off with a probability of 1/3 before he revealed anything.