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

[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.

 

[StoneTemplePython]

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.

do you want symbol manipulation or a heuristic / intuitive argument? My feeling is that page 1 has more than enough for intuition. Note: Erdos was finally convinced by a simulation -- there's no shame in getting your intuition from simulations.

As far as symbol manipulation goes, I've given the most accessible one I can think of...
- - - -
perhaps for intuition:
(i) either just accept it is the fact of how complementary events work (not totally satisfying).
(ii) alternatively, and very roughly, you could think of it as 1/3 of the time you are right to begin with so Monty's maneuvers are irrelevant... but 2/3 of the time you have chose wrongly and Monty is 'merging' all of the other options into one door for you. I.e. 2/3 of the time Monty is doing you a huge favor by narrowing the options down to one door that is correct, and 1/3 of the time what Monty is doing is irrelevant. (Ok this is really just a rehash of (i), I suppose).

From a symbol manipulation standpoint, you could try applying Bayes Rule to some door you didn't select, but it'll be a bit messy. There's something of a reference frame problem underneath this -- the frame of reference is the door you selected and its hard to disentangle from that.

I suspect this is what is inhibiting intuition here but I won't dwell on this. My view is sometimes you need to find (at least) 2 approaches to a problem -- one to get the proof and another complementary one (ha) to reinforce your intuition.

 

[PeroK]

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?

Your question has been answered at least a dozen times. Why you can't accept these answers is, frankly, beyond me.

 

[mfb]

1/3 of those 2/3 are gone now because it was revealed and the prize is not there.

The show host will never reveal a prize. The host will actively avoid revealing a prize and will always open an empty door. There is nothing that goes away - apart from the risk to switch from one wrong door to another, something you can't do.

You would be right if the show host wouldn't know where the prize is. In that case you can lose the option to switch. But this is not the scenario considered.

 

[bhobba]

Its counter-intuitive - simple as that. When you see the answer you go - dah - is it that easy. Yes it is. The answer is opening the door does not change the chances of winning - its an independent action. The chance is 1/3 you are correct. Opening the door doesn't change that. But you now only have two doors so the chance of it being behind the other door is 2/3. A lot of problems in probability are like that, which is why solving such requires well developed intuition - it's the reason the actuarial probability exam is generally considered so hard. Some people have that intuition (or developed it), others do not. I am in that do not group. When I saw the solution to the Montey Hall problem - I went - dah.

Thanks
Bill

 

[FactChecker]

The show host will never reveal a prize. The host will actively avoid revealing a prize and will always open an empty door.

That is the critical point. Monte Hall has intentionally eliminated a door without the prize from the set of doors that you have not chosen. That improves the odds of the other door you have not chosen but does not improve the odds of your door.

This insight can help one to intuitively understand the answer. But a methodical application of Baye's rule should always give the correct answer to deceptive problems like this -- intuitive or not.

 

[Nugatory]

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.

Here's a more extreme example: I toss a fair coin. You cannot see how the coin landed, but you can safely say that when you do look the probability that you will see it heads-up is 50%. Then I look at the coin and tell you truthfully that it landed heads-up - and your 50% probability changes to 100%.

So you shouldn't be surprised that the probabilities change when new information is revealed. The tricky part is correctly calculating how they change, and that's what the other posts in thread are explaining.

 

[FactChecker]

A more extreme case is intuitively obvious. Suppose there were a thousand doors and you picked one. You know that you have almost no chance of having the prize door. Now suppose Monte one-by-one opens the other doors till there is only one other door remaining that you did not pick. Obviously, he has avoided that door because it is the prize door. If you do not switch to that door, you are making a big mistake.

 

[PeroK]

A more extreme case is intuitively obvious. Suppose there were a thousand dors and you picked one. You know that you have almost no chance of having the prize door. Now suppose Monte one-by-one opens the other doors till there is only one other door remaining that you did not pick. Obviously, he has avoided that door because it is the prize door. If you do not switch to that door, you are making a big mistake.

It's even better if Monty goes through the doors in order. Let's say you picked door 1.

He opens doors 2-175, then coughing and shuffling he says "let's miss out door 176", then he opens doors 177-1000.

I wonder why he missed out door 176?

 

[FactChecker]

It's even better if Monty goes through the doors in order. Let's say you picked door 1.

He opens doors 2-175, then coughing and shuffling he says "let's miss out door 176", then he opens doors 177-1000.

I wonder why he missed out door 176?

I think we have a winner! This is the example I will use in the future.

One aspect this brings up is elementary game theory. If he is allowed to use game strategy and knows that you have picked the prize door, he might be tricking you into trading it away.

 

[JeffJo]

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

Pardon me for making one small change to what you asked. I did it because "can't" is usually used in explanations. And there isn't a reason why it can't change, because it can. (Don't hit the "reply" button yet.)

Under the assumptions that you should make about how the host chooses a door to open, it won't change. But how this comes about is seldom explained. And I believe that the reason the problem keeps coming back, is because intelligent puzzle-solvers, who are inexperienced in probability, can sense that the assertion "it can't change" hasn't been supported.

Go back to the start of the thread, where you said that your original choice had a 1/3 chance of having the prize. This is a step everybody takes for granted, but you may not know why:

• The reason is that you have no information that can make your assessment of door #1's chances different than door #2, or door #3.
• In such cases, you can only treat them as if they are all equally likely.
• This is called the Principle of Indifference. When there are N options that you cannot distinguish from each other, except by name, then each has a 1/N probability of being the actual case. That way they add up to 1.
  
• This remains true even if there is information, unknown to you, that can make them different. If I ask you to call a coin flip, you have a 50% chance to get it right even if I know that it favor one side. Say the bias is B. Since you don't know which result is favored, there is a B/2 chance that you will pick the favored side and win, and a (1-B)/2 chance that you will pick the unfavored side and win. Since B/2+(1-B)/2=1/2, your chances of winning are 50%.

The reason people think that the two remaining doors should each have a 50% probability, is because they apply the Principle of Indifference to them. But the set of information you have about each is different:

Set 1: There is one door that the host could have opened, and did.
Set 2: There is one door that the host could have opened, but did not.

Case A: If it has the prize, he could not have opened it.
Case B: If it does not, that means your door has it. But in this case, the host had to choose between the two doors you didn't pick.​

Set 3: There is one door that the host not could have opened; the door you choose.

Only the door that Set 3 applies to is unaffected by what you learned when the host opened a door. The door that Set 1 applies to is eliminated. The door that Set 2 applies to is different, and that difference is why the probabilities for Cases A and B are different. Specifically, the probability for Case B needs to be reduced to account for the chance that the host would have opened a different door. Since you can only assume (remember the Principle of Indifference?) that the host would have chosen the other door half of the time, Case B is not half as likely as Case A. To make them add up to 100%, you make the probability for Case A 2/3, and the probability for case B 1/3.

+++++

This kind of problem fools many people, including many who should know better. Paul Erdos was one, but he recognized and admitted his error. Here is another, that I will probably get arguments about:

Q1: I have two children. What is the probability that both have the same gender?

There are four possible gender combinations, which I ordered by alphabetizing their first names: BB, BG, GB, and GG. Each is equally likely (remember the Principle of Indifference?), so the probability of (BB or GG) is 1/4+1/4=1/2.

Q2: At least one of them is a boy. Now what is the probability that both have the same gender?

Most teachers will remove the GG possibility, and apply the Principle of Indifference to BB, BG, and GB to get an answer of 1/3. This is the exact same solution that makes people say the answer in the Monty Hall Problem is 1/2.

The PoI does not apply anymore, for the same reason it doesn't in the Monty Hall Problem. There is one combination that I can't have (GG) since I would have had to tell you about a girl, two where I could have told you about a boy or a girl (BG and GB), and one where I could only tell you about a boy (BB). The same reasoning as above applies, but now there are two cases that are reduced by half. The probability I have BB is 1/2, that I have BG is 1/4, and that I have GB is 1/4.

And if you don't believe me, consider how you would have answered:

Q2.1: At least one of them is a girl. Now what is the probability that both have the same gender?
Q2.2: I wrote the gender of at least one on a notepad in front of me. Now what is the probability that both have the same gender?

Q2.1 and Q2.2 must have the same answer as Q2. But it would be a paradox if the information in Q2.2 makes its answer different is than the answer to Q1, since you have no information that is different. This paradox actually has a name: Bertrand's Box Paradox (no, Bertrand didn't refer to the problem as a "paradox,", he used this paradox to explain why the answer to all of the Q2's must be the same as Q1).

 

[sysprog]

This short book on the subject is entertaining and many would find it clarifying: http://faculty.winthrop.edu/abernathyk/Monty Hall Problem.pdf

 

[JeffJo]

This short book on the subject is entertaining and many would find it clarifying: http://faculty.winthrop.edu/abernathyk/Monty Hall Problem.pdf

Read it. In fact, what I said about Bertrand's Box Paradox comes from it, starting ion page 24. You should even be able to see how I constructed my explanation of the Two Child Problem to be parallel to Jason Rosenhouse's explanation of Bertrand's problem.

In particular, note:

Bertrand intended this as a cautionary tale of what happens when you are too cavalier in assigning equal probabilities to events. Lest you find this point too trivial to bother with, I assure you that some very competent mathematicians throughout history have managed to bungle it.

 

[PeroK]

Q1: I have two children. What is the probability that both have the same gender?

There are four possible gender combinations, which I ordered by alphabetizing their first names: BB, BG, GB, and GG. Each is equally likely (remember the Principle of Indifference?), so the probability of (BB or GG) is 1/4+1/4=1/2.

Q2: At least one of them is a boy. Now what is the probability that both have the same gender?

The issue with these questions, especially Q2, is there is no logical way to decide why someone has decided to ask you these questions. These problems all work better if information is given in response to direct questions:

How many children do you have?
Do you have at least one boy?

But, if someone simply produces this question out of thin air, what sort of logical reasoning can you apply?

Perhaps a good example might be if someone has a five-card hand of cards. If you ask them: do you have the Ace of Hearts, then you can work with the answer. And you might calculate the probability they also have the Ace of Spades. But, if they unilaterally provide you with this information: "I have the Ace of Hearts", then it's not clear on what basis you can start to calculate probabilities.

You could of course assume that they simply picked a card uniformly at random and told you what it was. But, you have no concrete basis for that assumption: unless you told them to do that; and, even then, the fact they picked a "top" card suggests the process may have been far from a uniform selection process.

In particular, it's difficult to deal with the case where someone has both the Ace of Hearts and the Ace of Spades and draw any conclusion about how likely it is that they chose to tell you about the Ace of Hearts.

If we go back to Q2. If someone accosts you in the street and says:

I have two children. At least one of them is a boy. What is the probability that both have the same gender?

Then, what are your assumptions? I might argue that someone who has only one boy would say so. Perhaps not 100% of the time, but most of the time. In fact, most people would tell you both genders: I have two boys or a boy and a girl. So, I might lean towards an answer that they probably have two boys; or, I might lean towards them having only one boy. But, there is no way to know without a statistical and/or psychological study!

In summary, I believe these problems are ill-posed as problems in probability theory. They require an element of statistical analysis to support any assumptions on why someone produces such a question.

On the other hand, information gained from direct questions can be input into straight probability calculations without any such concerns.

 

[PeroK]

Q1: I have two children. What is the probability that both have the same gender?

Q2: At least one of them is a boy. Now what is the probability that both have the same gender?

In fact, I would like to propose an answer to Q2.

Assumption 1: no one in the history of mankind has ever (or will ever) pose that particular question unless they have heard about this riddle and are asking it for that purpose.

Assumption 2: the riddle is almost always given in terms of "boy". Now, it's not impossible that someone might change it to "girl", but I'll assume that the default is to ask the question in terms of boys. Every time I've heard it that has been the case.

Assumption 3: The only (likely) circumstances in which the question gets changed to "at least one girl" is in the case where the questioner has two girls.

Given that this is a riddle, I'm not convinced that the questioner definitely has two children. They may have any number of children and are asking it as a pure riddle. So, I'll have a final assumption:

Assumption 4: the questioner definitely has two children, and is providing genuine information.

With these assumptions, then I can conclude that the questioner has two boys with probability 1/3.

Corollary:

If the questioner asks the question in terms of "at least one girl", then the probability they have two girls is close to 1.

 

[JeffJo]

The issue with these questions, especially Q2, is there is no logical way to decide why someone has decided to ask you these questions.

Yet, when the differences you talk about are not demonstrated as I did, many people will answer them as if they had asked me "Do you have any boys?" See https://en.wikipedia.org/wiki/Boy_or_Girl_paradox.

But I disagree that it is a problem:

• Q1 IS answerable. The answer is 1/2.
• Q2 has the same information content as Q1, and so IS answerable. The answer is 1/2.
  
• Q2 and Q2.1 are just as answerable as Q2.2
• Q2 and Q2.1 can't have a different answer than Q2.2

Probability is not an absolute measure. It measures your uncertainty about a situation. So the uncertain parameters you describe can be handled. Example: I have in front of me a biased coin. It lands on one side 75% of the time, and on the other 25%. If I flip it, what probability should you assign to the proposition that it will land "Heads"? Answer: 1/2. I can assign a different probability only because I know which side is favored.

But, if someone simply produces this question out of thin air, what sort of logical reasoning can you apply?

The genders of a person's children can't affect the form of the information he states, but it can affect the values he states in that form. So, a parent of BB is just as likely to state "I have two, and at least one is a <insert gender here>" as is a parent of BG, GB, or GG. The parents of BB and GG have only one gender they can insert, but the others have two and are equally likely to insert either.

Result: the answer to each question I asked is 1/2.

Perhaps a good example might be if someone has a five-card hand of cards. If you ask them: ...

But I was trying to illustrate how to solve a problem when no question is asked of the informant.

In particular, it's difficult to deal with the case where someone has both the Ace of Hearts and the Ace of Spades and draw any conclusion about how likely it is that they chose to tell you about the Ace of Hearts.

See Principle of Indifference.

 

[sysprog]

This seems to me like it might make the MH game problem reasonings clearer for some people:

After a person mis-evaluates, and says either that the chances after the goat is revealed go to 50-50, or that both of the doors retain their original 1/3 chance, or that, for whatever reason, he doesn't improve his chances by switching, present a modified version of the problem as follows:

The contestant picks a door. 1/3 chance per door of the car being behind it.
Monty says: I'm in an especially generous mood today, so I'm going to make you an offer: you can have both of the other two doors in exchange for your chosen door.

Should the contestant switch?

I think most people will recognize that the 2 doors between them have a 2/3 chance compared to the original 1/3 chance, so most people will say the contestant should switch.

So let's say in this version of the game that the contestant trades his 1 door for the 2 doors.

Monty then says: I'm going to open one of your 2 doors first. If neither door has the car, I don't care which one of them I open, but if one of them has the car, I'm going to open first the only one that doesn't. Either way, I'm going to open one of your doors that doesn't have the prize, whether it's the only one available for that or not, and I'm not going to open your originally chosen door right now.

The contestant isn't too worried about this, because he knows that only one of the doors could have the car anyway. He's anxious to see whether his 2/3 chance will pay off, but that's all that worries him in the game at that moment.

So Monty opens one of the contestant's 2 doors, and reveals a goat. No-one, including the contestant, is surprised, but a murmer comes from the audience.

Monty says: You traded in your 1/3 chance for what was then, at least then, a 2/3 chance. Now there are only 2 doors remaining. If you want to, I'll let you trade back for your original door."

The contestant becomes flustered, because for a moment, the audience murmers more loudly.

Should he switch back?

Monty then offers him 10% of the car's value to switch. If he's thinking his chances are "1 door 1/3" as at the start, or that they went up to 2/3 when he was allowed to switch for 2 doors, but now that there are only 2 unopened doors, his chances are now 50-50, the 10% cash should tip the scales for him, so he might think he should switch back.

Should he switch back?

I think most people will realize that, given what Monty said, the opening of a non-winning door of the 2 doors switched for won't diminish the 2/3 chance. The contestant knew when he switched for the 2 doors that at least one had to be a non-winner, because there's only 1 car in the game.

If a question respondent who thought in the original problem that there was no advantage to switching, thinks that in this version there's an advantage to not switching back, ask him to reconsider, after closer examination, whether in the original problem switching doors after the reveal is equivalent to not switching back after the reveal in the second version.

If he doesn't think so, ask him to imagine the game played both ways with the car behind the same door, and the same door originally chosen in each game. Why should the game-1 contestant not switch, if the game-2 contestant was right to switch before a door was opened and would be mistaken to switch back afterward?

 

[PeroK]

The genders of a person's children can't affect the form of the information he states,

For example, if you were in a society (and there are plenty of these still alive and kicking on Earth even in the 21st Century) where men generally are ashamed to have daughters and want only sons, then this would affect the information they give you. In such a society they would talk only about their sons and not their daughters.

Therefore, when the information is unrequested, the problem becomes one of statistics - or data gathering.

I agree that it's not so clear cut with the children question, but for other types of questions it might be much stronger (*). So, it's a bad policy to assume that information given unsolicited is the same as information given to direct questions.

In any case, almost all of the errors caused in these problems are as a result of the information being given unsolicited. If you stick to the information being obtained by direct question, then there are no such issues.

(*) To give an example. Suppose you ask Ms Jones:

Are you a mountaineer? Yes
Have you climbed the Matterhorn? Yes.

Then, let's assume that 1% of climbers who have climbed the Matterhorn have also climbed Everest. Then, there is a 1% chance that Ms Jones has climbed Everest.

But, if you are talking to Ms Jones and she says:

I'm a mountaineer and I've climbed the Matterhorn.

Then, there is strong probability that if she had climbed Everest she would have told you this instead of telling you about the Matterhorn.

In this case, the probability that Ms Jones has also climbed Everest is potentially much lower than 1%. And, it cannot be determined by probability theory directly, as it involves assumptions about psychology.

 

[cosmik debris]

I still think drawing a truth table, as in Wikipedia, is the easiest way to view the MH problem.

https://en.wikipedia.org/wiki/Monty_Hall_problem

Cheers

 

[JeffJo]

Therefore, when the information is unrequested, the problem becomes one of statistics - or data gathering.

Why? Any biases you find using statistics will apply only to the society you used to gather those statistics. I can postulate another society with the opposite biases. And since I didn't say what society this parent belongs to, you can't claim that the one you use is a better choice.

I asked a hypothetical question. Not one about a society you will choose. And before you say there is no such thing as a purely hypothetical question, you have to assume one to have the gender probabilities be equal and independent. Because neither are true if you resort to statistics.

I agree that it's not so clear cut with the children question, but for other types of questions it might be much stronger (*). So, it's a bad policy to assume that information given unsolicited is the same as information given to direct questions.

Which is why I didn't make any such assumption. Yes, I got the same answer as someone who did, but that does not mean I made that assumption. In fact, the only assumption I made was that there was no pre-determined form to the information.

And this has a direct application to the Monty Hall Problem. We don't know that Monty doesn't favor the door he opened. We also don't know that he doesn't favor the other door, and only opened the one he did because he had to. So we can only model his choice as equiprobable. Which is not the same thing as saying that we know he chose with equal probability.

And one reason this is important, is because this is not an equivalent problem:

So let's say in this version of the game that the contestant trades his 1 door for the 2 doors.

Monty then says: I'm going to open one of your 2 doors first. If neither door has the car, I don't care which one of them I open, but if one of them has the car, I'm going to open first the only one that doesn't. Either way, I'm going to open one of your doors that doesn't have the prize, whether it's the only one available for that or not, and I'm not going to open your originally chosen door right now.

The probability that this contestant gets the car is 2/3, regardless of how Monty Hall chooses a door in the case he says he doesn't care about. Because it is determined before the reveal. But if Monty Hall opens a door before the switch is offered, and we model his choice of the door he actually opened as a probability of Q, then the probability the contestant wins after switching is 1/(1+Q).

The reason the MHP continues to baffle people, even after explanations like sysprog's, is because that explanation is incorrect. It gets the right answer, because we must model Q=1/2, but it is not based on a mathematically-correct solution method. And it teaches an incorrect method, that you can ignore how information is acquired.

 

[PeroK]

Why? Any biases you find using statistics will apply only to the society you used to gather those statistics. I can postulate another society with the opposite biases. And since I didn't say what society this parent belongs to, you can't claim that the one you use is a better choice.

Your asssertion that the problem is well-posed demands that there cannot be any significant factors that are unknown. You say that the answer is 1/2 and I say the problem is ill-posed. And, in general, unsolicited information is different (whether you like it or not) from information obtained from a direct question. That's a basic fact. There can be no argument about that.

I looked at the Wikipedia link and it provided the problem in a different format. It says you are simply told:

Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

Your view if I understand it, is that the answer is unequivocally 1/2.

Whereas, my answer is that it depends how this question was arrived at.

a) If Mr Smith would have been rejected if he had two girls, hence the question always relates to boys, then the answer is 2/3.

b) If the question could equally well have been "... at least one of them is a girl ...", then the answer is 1/2.

This is critical and, ironically, is analogous to the difference between Monty Hall and the alternative. The Monty Hall problem depends not just on the evidence presented to your eyes but on knowing what options Monty has.

Likewise, the two-child problem depends on how the knowledge that Mr Smith has at least one boy was arrived at.

Your problem therefore, is that IF the "Mr Smith" question was generated by process a), then your answer of 1/2 is wrong. It's only correct IF the question was generated by process b). And, since the Wikipedia page doesn't specify how the question was arrived at, there is no definite answer and the problem is ill-posed.

 

[PeroK]

PS In fact, the Wikipedia article says exactly what I am saying:

"The Boy or Girl paradox surrounds a set of questions in probability theory which are also known as The Two Child Problem,[1] Mr. Smith's Children[2] and the Mrs. Smith Problem. The initial formulation of the question dates back to at least 1959, when Martin Gardner published one of the earliest variants of the paradox in Scientific American. Titled The Two Children Problem, he phrased the paradox as follows:

• Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
• Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

Gardner initially gave the answers 1/2 and 1/3, respectively, but later acknowledged that the second question was ambiguous.[3] Its answer could be 1/2, depending on what more information is available beyond that you found out just that one child was a boy. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk,[4] and Nickerson.[5]"

...

This question is identical to question one, except that instead of specifying that the older child is a boy, it is specified that at least one of them is a boy. In response to reader criticism of the question posed in 1959, Gardner agreed that a precise formulation of the question is critical to getting different answers for question 1 and 2. Specifically, Gardner argued that a "failure to specify the randomizing procedure" could lead readers to interpret the question in two distinct ways:

• From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.
• From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of 1/2.[4][5]

Grinstead and Snell argue that the question is ambiguous in much the same way Gardner did.[12]

 

[JeffJo]

Your asssertion that the problem is well-posed demands that there cannot be any significant factors that are unknown.

Where did I say "well posed?" I don't even think you can define what that means in general. But my point is quite the opposite of what you just said. The entire point of Probability is to model the unknown aspects of a problem. So the presence of unknown aspects is not an obstacle. And if there are no such unknown aspects, you have a deterministic problem.

The issue here is that you have decided that some categories of unknown information are unacceptable for treatment with probability. What I and trying to say is that a question is solvable if you can reasonably model the ambiguous aspects of it with probability.

You say that the answer is 1/2 and I say the problem is ill-posed.

I say that any answer except 1/2 is unacceptable, because it produces a variation of Bertrand's Box Paradox (not the Problem, the Paradox that Bertrand identified and that Jason Rosenhouse described in his book). And I go on to demonstrate a reasonable probability model that shows the answer can be 1/2. I do this by modeling the part that you consider to be ill-posed with probability.

And, in general, unsolicited information is different (whether you like it or not) from information obtained from a direct question. That's a basic fact.

It is also the point that I am trying to make. You can't assume that unsolicited information is an answer to any specific question, but you can treat the information content in the unsolicited statement as an element of an event space.

Your view if I understand it, is that the answer is unequivocally 1/2.

My point is that when the information is unsolicited, or stated without an indication of how or if it was solicited, then any answer other than 1/2 leads to a logical paradox and so is unacceptable.

My point is that I can justify 1/2 as an answer, by modeling what information was given with probability. Not that I know why the information was given.

My point is that the technique I use to do this is exactly what is needed to correctly solve the MHP. (As opposed to asserting "the other two doors start with 2/3 probability, so the chance that switching wins after the reveal must also be 2/3" which is incorrect math.)

Whereas, my answer is that it depends how this question was arrived at.

And the answer to "Will this coin land on Heads or Tails?" depends on how much torque you applied when you flipped it, among other factors. If you know these factors, the coin flip becomes deterministic and you can calculate the result. If you don't know these factors, you use probability.

If you don't know why the information was given in the TCP, you can use probability the same way.

PS In fact, the Wikipedia article says exactly what I am saying:

"The Boy or Girl paradox surrounds a set of questions in probability theory which are also known as The Two Child Problem,[1] Mr. Smith's Children[2] and the Mrs. Smith Problem. The initial formulation of the question dates back to at least 1959, when Martin Gardner published one of the earliest variants of the paradox in Scientific American. Titled The Two Children Problem, he phrased the paradox as follows:

• Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
• Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

Gardner initially gave the answers 1/2 and 1/3, respectively, but later acknowledged that the second question was ambiguous.[3] Its answer could be 1/2, depending on what more information is available beyond that you found out just that one child was a boy. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk,[4] and Nickerson.[5]"

Since I wrote much of that, I suggest to you that it is not contradicting what I am trying to tell you now. The reason I didn't continue, in Wikipedia, with the more detailed explanation I provide here, is because many people do not want to accept it. But there are references I could have used.

Yes, the problem is ambiguous. So is "If I flip a coin, will it land on Heads, or on Tails?" The entire purpose of probability is to handle such ambiguous situations.

With the coin, the answer isn't "Heads" or "Tails," it is the ambiguous statement "50% chance for Heads, and 50% chance for Tails." That is also the answer if we are told the coin is unfair, but not told how unfair or in which direction. This is because probability is not a property of the system, as you are trying to make it out to be. The system itself contains enough information to be deterministic. Probability is a property of what we know - or more specifically, what we don't know - about the deterministic parts of that system.

With the ambiguous question "Why do we know Mr. Smith has a boy?", the answer can only be "we can't state the absolute probability, but if it is P for a man with BB, then we must use P/2 for a man with BG or GB, P/2 for knowing he has a girl if he has BG or GB, and P for knowing he has a girl if he has GG.

I DON'T KNOW THAT THOSE PROBABILITIES ARE CORRECT IN THE ACTUAL SYSTEM. But I can't assume anything else. And regardless of what I assume, I can show that any answer other than the one I get when I use those probabilities is incorrect.

 

[PeroK]

"Why do we know Mr. Smith has a boy?",

You could have asked him. In fact, you could have asked him if he had two girls and he could have said "no".

There's a similar issue if three coins are tossed and you are told that the first and the third are the same. The probability that the middle coin is the same then depends on the process by which that information was obtained. And, crucially, whether the second coin has already been looked at.

In this situation, you cannot make any meaningful progress until you resolve how the available information was generated.

Anyway, the Wikipedia article explains clearly that everyone involved in the research accepts the fundamental ambiguity in the question and the two possible interpretations.

 

[stevendaryl]

The issue here is that you have decided that some categories of unknown information are unacceptable for treatment with probability. What I and trying to say is that a question is solvable if you can reasonably model the ambiguous aspects of it with probability.

I suppose you can model anything unknown using probabilities. But in general, you can't get a unique answer for some uncertainties.

Suppose a man comes up to you and tells you: "I have two children. At least one of them is a boy."

You can model it using probabilities this way:

• A = He tells you that he has two children, at least one of which is a boy.
• B = He has two boys
  
• C = He has one boy and one girl.
• D = He has two children, at least one of which is a boy = B or C.

So you're trying to figure out P(B|A), the probability of B being true given that A is true. Note that A and D are different events: It's possible for D to be true while A is not true.

Using Bayesian probabilities, we can calculate:

$P(B|A) = \dfrac{P(A|B) P(B | B or C)}{P(A|B) P(B | B or C) + P(A|C) P(C | B or C)}$

To solve the problem, you need to know the conditional probabilities:

1. P(B | B or C)
2. P(C | B or C)
3. P(A | B)
4. P(A | C)

I would say that you don't know any of those conditional probabilities. You might assume that each time someone has a baby, the baby is equally likely to be a boy or a girl. But that doesn't imply that all four of these possibilities are equally likely: (1) Two boys, (2) two girls, (3) oldest is a boy, youngest is a girl, (4) oldest is a girl, youngest is a boy. They may not be equally likely, because maybe the decision to have a second child depends on the sex of the first child. Maybe a couple wants a boy, and so if they have a boy first try, they stop with one child. If their first baby is a girl, they try a second time. If that's their plan, then two boys would have probability 0.

You also don't know the conditional probabilities 3 or 4.

Even if we know that the man is truthful, we haven't specified the rules for how he reports on his children. Maybe he follows this rule:

If he has two boys, he would say "I have two children, both boys".
If he has two girls, he would say "I have two children, both girls".
If he has one of each, he would say "I have two children, at least one of them is a boy"

If he follows that rule, then you know exactly that he has one boy and one girl. So P(A|B) = 0 and P(A|C) = 1.

So that gets to the question of "ill-posed" problems. A problem is ill-posed (I think this is what it means) if it is impossible to solve without making additional assumptions that were not implied by the original statement of the problem.

 

[JeffJo]

"Why do we know Mr. Smith has a boy?"

You could have asked him. In fact, you could have asked him if he had two girls and he could have said "no".

Or you could just know he participates in Boy Scout events. There are lots of ways, many of which don't involve asking him vague questions. The point you are (purposely?) ignoring is that we don't know how. So speculation is irrelevant.

What you are also ignoring, is that we don't need to know , in order to assess a probability for the general case where we have this information. We do need to know it for a specific situation, but we don't have that information.

I suppose you can model anything unknown using probabilities. But in general, you can't get a unique answer for some uncertainties.

And my point is that, with the information we have, any answer other than 1/2 creates a paradox that makes that answer unacceptable. While a simple, and reasonable, application of the Principle of Indifference makes the answer 1/2.

Suppose a man comes up to you and tells you: "I have two children. At least one of them is a boy."

You can model it using probabilities this way:

• A = He tells you that he has two children, at least one of which is a boy.
• B = He has two boys
  
• C = He has one boy and one girl.
• D = He has two children, at least one of which is a boy = B or C.

...

Note that A and D are different events: It's possible for D to be true while A is not true.

Indeed. That's part of my point. But I prefer:

• B2 = He has two boys.
  
• B1G1 = He has one boy and one girl
• G2 = He has two girls.
• ALOB = He tells you that he has at least one boy.
• ALOG = He tells you that he has at least one girl.
• SE = He tells you something else (including nothing) about genders.

Can we agree that {B2,B1G1,G2}x{ALOB,ALOG,SE} is a nine-element partition of the sample space, given that he has two children?

Then, since we don't know why he would say this, the only things we can assume about probabilities are:

• Pr(B2) = P2(G2) = 1/4.
• Pr(B1G1) = 1/2.
  
• Pr(ALOB|B2) = Pr(ALOB or ALOG|B1G1) = Pr(ALOG|GG). Call this value Q.
  
• Pr(ALOB|B1G1) = Pr(ALOG|B1G1) = Q/2.
• Pr(ALOB|G2) = Pr(ALOG|B2) = 0.

From this, we can deduce Pr(B2|ALOB)=1/2.

I would say that you don't know any of those conditional probabilities.

And there is a difference between:

1. Knowing a situation-specific conditional probability, that applies to one instance of a random procedure, and...
2. Assigning a conditional probability to the general case of that random procedure.

You are saying we don't know #1. I agree. I'm solving a question about #2. I never said I know the conditional probabilities that apply specifically to your Mr. Smith. I said the Principle of Indifference applies to the population of all Mr. Smiths who end up in a similar situation.

Similar example: Some studies claim to show a correlation between how you hold a fair coin before flipping it, and the result of the flip. If true, that still doesn't change the answer to "What is the probability that flipping this fair coin will result in Heads?" The answer, IN GENERAL, is 1/2, You don't know how I intend to hold it, and you can only assume there is a 50% chance either way. The fact that the information exists cannot affect the probability you use IF YOU DON'T KNOW IT.