Poll: A women has two kids, one is a boy, what are the odds the other is also a boy?

[Maze1125]

If you feel there is no ambiguity, perhaps you are not as familiar with the English language as you believe.

And that's the point. This isn't an English question, this is a mathematical question.
And in Maths, the phrase "One is X" has a precise meaning. There is no ambiguity, only apparent ambiguity brought on by the attempt to make the question non-arbitrary, which in turn can confuse people into thinking that ambiguity that exists in English can apply to mathematical questions.

 

[Maze1125]

No, you're just arbitrarily dismissing what I have to say now.

Yes, you are.

Those were just childish responses as, in each case, I justified the comment in the following lines.

The question is simple. We have two variables that could be B or G with equal probability,

No we don't: only one of them can be G. The other must be B. The question is not that simple.

That's why I continued the line.
In the set up of the problem we have two variables that each have 50/50 chance of going each way individually, the problem is then given an extra condition which constrains if further.

Wait, what's SB? That wasn't in any of your responses to me: maybe you're thinking of your response to someone else (totally understandable: this stuff can get confusing).

It was in there the first time I wrote out a proof and although it wasn't directed I you, I suggested that you read.

Anyway.
FB is the event that the first child is a boy.
SB is the event that the second child is a boy.
OB is the event that one of the children is a boy.

If you feel there is no ambiguity, perhaps you are not as familiar with the English language as you believe.

And that's the point. This isn't an English question, this is a mathematical question.

That's ridiculous: this is a WORD PROBLEM: that means it's both an English AND a Mathematical question.

No it isn't, this is a well known probability paradox. It is exclusively in the realm of Mathematics, the words are merely there to convey the mathematical meaning.

 

[Knight Templar]

If you feel there is no ambiguity, perhaps you are not as familiar with the English language as you believe.

And that's the point. This isn't an English question, this is a mathematical question.

That's ridiculous: this is a WORD PROBLEM: that means it's both an English AND a Mathematical question.

No it isn't, this is a well known probability paradox. It is exclusively in the realm of Mathematics, the words are merely there to convey the mathematical meaning.

So it's like a negitive number? Theory demands it exists, but practicality holds that it does not?

 

[DracoSuave]

No it isn't, this is a well known probability paradox. It is exclusively in the realm of Mathematics, the words are merely there to convey the mathematical meaning.

If this statement is true, the problem would have been written using mathematical terms.

It was not.

And 'one is a boy' is not mathematically unambiguous. It could refer to one that is chosen at random and had their sex revealed after selection, or it could refer to one that has been selected by gender. The difference between the two is the difference between 1/2 and 1/3.

If it were mathematically unambiguous, it would have made it clear either through explicit means or by implicit means which sense it means. The only possibility that we leave out is that it does not refer to cardinality of boys.

You can claim otherwise, but prove then that it -cannot refer to one being selected and then sexed.- If you can prove that using only the given problem, then you can prove it is unambiguous. If you cannot, it is already proven that it could be interpreted the other way (An event that has happened can happen) and therefore it is ambiguous.

This word problem lacks the concise mathematical terminology to be clear.

Refusal to understand that point is a refusal to try to understand the otherside of the argument, shouting NO U at it. Given that I have taken the time to read, understand, and acknowledge your own arguments, and have discovered the reason comes down to the ambiguity of the problem at hand, perhaps you should make the attempt to do the same.

But do NOT assume there is no ambiguity just because it results in an answer that you have not previously adopted. That is non-scientific.

 

[Maze1125]

So it's like a negitive number? Theory demands it exists, but practicality holds that it does not?

No, this particular case with boys is unlikely to come up in any practical way, but many equivalent situations can exist with use, and the theory would still hold.

Also, negative numbers are very practical, they're how we can account for debt, or the charge of an electron.

No, you're just arbitrarily dismissing what I have to say now.

Yes, you are.

Those were just childish responses

I don't understand why people act like this just because others disagree with them. This happened in the last thread: I don't see why people have to behave like this.

Ask yourself, you're the one who responded like that.

And as for the rest of your post, it was all points I've already responded to or nit-picking that was irrelevant to the point.

And 'one is a boy' is not mathematically unambiguous.

Yes it is. If you say that phrase to any mathematician, in a probabilistic context, then every single one will understand it to mean that the random variable can only take values in the set (FB union SB).

And, again, prove your claim that the probability can be 1/2.

 

[Maze1125]

No, you're just arbitrarily dismissing what I have to say now.

Yes, you are.

Those were just childish responses

I don't understand why people act like this just because others disagree with them. This happened in the last thread: I don't see why people have to behave like this.

Ask yourself, you're the one who responded like that.

No, at best, you're the one who interpreted it like that. I don't know why people get so sensitive over this problem.

Because saying things like that is pointless and childish.
It makes it sound like you don't want to debate the point but just want an argument for the sake of it.

And as for the rest of your post, it was all points I've already responded to or nit-picking that was irrelevant to the point.

Like I said, you're just arbitrarily dismissing what I have to say now by doing things like calling it 'nit-picking that is irrelevant to the point'.

It's not arbitrary if it's true. I was making points, you criticised most of those points on an issue that wasn't relevant to the point itself. So, either you didn't understand the point I was making in the first place, or you were just nitpicking.

 

[Maze1125]

No, you're just arbitrarily dismissing what I have to say now.

Yes, you are.

Those were just childish responses

I don't understand why people act like this just because others disagree with them. This happened in the last thread: I don't see why people have to behave like this.

Ask yourself, you're the one who responded like that.

No, at best, you're the one who interpreted it like that. I don't know why people get so sensitive over this problem.

Because saying things like that is pointless and childish.
It makes it sound like you don't want to debate the point but just want an argument for the sake of it.

No, it's because for some reason you don't want to debate the point past a certain level and consider any further discussion to be argument for the sake of it. That's your fault, not mine.

And as for the rest of your post, it was all points I've already responded to or nit-picking that was irrelevant to the point.

Like I said, you're just arbitrarily dismissing what I have to say now by doing things like calling it 'nit-picking that is irrelevant to the point'.

It's not arbitrary if it's true.

I agree. However, the question then becomes whether it IS true.

I was making points, you criticised most of those points on an issue that wasn't relevant to the point itself.

I disagree. I think my criticisms were fully relevant.

So, either you didn't understand the point I was making in the first place, or you were just nitpicking.

False dichotomy: the third option is you didn't understand my criticisms, so rather than get me to clarify what I was saying, you just started calling what I had to say "arbitrary" and "nit picking."

The fourth option is you don't want to deal with my questions because you can't answer them, so you are PRETENDING that I am "nitpicking"

The fact we're even having this argument shows that you don't care about the topic and just care about having an argument. So I see no point in continuing this discussion with you.

 

[Maze1125]

No, you're just arbitrarily dismissing what I have to say now.

Yes, you are.

Those were just childish responses

I don't understand why people act like this just because others disagree with them. This happened in the last thread: I don't see why people have to behave like this.

Ask yourself, you're the one who responded like that.

No, at best, you're the one who interpreted it like that. I don't know why people get so sensitive over this problem.

Because saying things like that is pointless and childish.
It makes it sound like you don't want to debate the point but just want an argument for the sake of it.

No, it's because for some reason you don't want to debate the point past a certain level and consider any further discussion to be argument for the sake of it. That's your fault, not mine.

And as for the rest of your post, it was all points I've already responded to or nit-picking that was irrelevant to the point.

Like I said, you're just arbitrarily dismissing what I have to say now by doing things like calling it 'nit-picking that is irrelevant to the point'.

It's not arbitrary if it's true.

I agree. However, the question then becomes whether it IS true.

I was making points, you criticised most of those points on an issue that wasn't relevant to the point itself.

I disagree. I think my criticisms were fully relevant.

So, either you didn't understand the point I was making in the first place, or you were just nitpicking.

False dichotomy: the third option is you didn't understand my criticisms, so rather than get me to clarify what I was saying, you just started calling what I had to say "arbitrary" and "nit picking."

The fourth option is you don't want to deal with my questions because you can't answer them, so you are PRETENDING that I am "nitpicking"

The fact we're even having this argument shows that you don't care about the topic and just care about having an argument. So I see no point in continuing this discussion with you.

So the fact that *we're* having an argument shows us something negative about *me* but nothing about you?

I make a criticism of what you have to say, you dismiss my criticism as "arbitrary" and "nit picking," and now you say the fact that I dispute your claim that my criticism was not relevant to the topic...proves I don't care about the topic?

That's rather convenient for you, isn't it? Quite a Catch-22 you're putting me in there: my act of disputing your accusation is proof that your accusation is true?

Not really, you could have easily proved the relevance of your points by explaining them. That would have both proved me wrong and continued the original discussion.
Equally, right now you have have proved me wrong again by reverting back to the original topic, but instead you choose to continue the extremely tangential line by talking about catch-22 issues that don't even exist.

When you first started making comments it seemed like we could have a productive discussion, but all your comments on this last page have seem to be simply childish, nit-picky or completely irrelevant, and you have given me no reason to believe that things are anything other than how they seem.

So, I apologise, but this will have to be my last response to you in this topic.

 

[DracoSuave]

And 'one is a boy' is not mathematically unambiguous.

Yes it is. If you say that phrase to any mathematician, in a probabilistic context, then every single one will understand it to mean that the random variable can only take values in the set (FB union SB).

And, again, prove your claim that the probability can be 1/2.

When you say 'One is a ______, what is/can be/qualities may be had by the other?' it could be taken to mean, in a mathematical sense, to mean that one specific entity is called on, and that the other one is the unit in question.

IF you are taking a specific child and -then- revealing their gender, the selection process becomes the FB is a defined event, and the SB is a non-defined event. The question becomes an isomorphism of 'I have flipped a coin and it came up heads. What are the odds of the next flip?' 50%. This is -trivial-.

IF you are taking a random child, and -then- revealing their gender, there are three randomized elements: The child selected, and the gender of each child respectively. Each has a probability of 50% for each outcome; You could select the first child, or the second child, and said child could be male, or female, and the other child could be male or female.

This leads to EIGHT different outcomes: (FC-SC-Selected child)

1-) B-B-FC
2-) B-B-SC
3-) B-G-FC
4-) B-G-SC
5-) G-B-FC
6-) G-B-SC
7-) G-G-FC
G-G-SC

Then, we eliminate the outcomes that this case could not have evoked, 4, 5, 7, and 8. This leaves:

1-) B-B-FC
2-) B-B-SC
3-) B-G-FC
6-) G-B-SC

Using Draco's Law of Counting the damn outcomes, I see 4 outcomes. 2 of them have the selected child have a brother, 2 of them have the selected child have a sister.

50%.

So, -if- the possibility exists that the mother -could have- revealed a girl in the method of her selection, then that does, AS PROVEN ABOVE, directly affect the probability, in the -exact same manner- that in the Monty Haul problem, if the host -could have revealed a car-, that changes the probability of switching being advantageous. This does not care if she picked the first child, or a child at random.

It -does- become 33%, however, if the woman selected by the quality "boy" rather than "child". In -that- case the woman is screening your outcomes, and this bias sets the probability away from the 50%.

Therefore, the probability is affected by the probability chance this woman is selecting by boy rather than by child regardless of gender.

Selecting by boy might include answering the question 'Do you have a boy?' She, however, has not answered 'I have a boy.' She has told you 'One of them is a boy.' You are not certain the manner of selection she used, nor -her- criteria, but that directly affects the results.

 

[Knight Templar]

So it's like a negitive number? Theory demands it exists, but practicality holds that it does not?

No, this particular case with boys is unlikely to come up in any practical way, but many equivalent situations can exist with use, and the theory would still hold.

Also, negative numbers are very practical, they're how we can account for debt, or the charge of an electron.

What I meant was negitive numbers don't really exist, you can't point to something tangeable that is a negitive as in the real world you can't have less than nothing. I may have used the wrong word with "practicality".

Cheers for the explination.

 

[DracoSuave]

So it's like a negitive number? Theory demands it exists, but practicality holds that it does not?

No, this particular case with boys is unlikely to come up in any practical way, but many equivalent situations can exist with use, and the theory would still hold.

Also, negative numbers are very practical, they're how we can account for debt, or the charge of an electron.

What I meant was negitive numbers don't really exist, you can't point to something tangeable that is a negitive as in the real world you can't have less than nothing. I may have used the wrong word with "practicality".

Cheers for the explination.

Negative numbers do exist in the real world. Any time you have an arbitrary zero point that is not at the lowest extreme of a measurement, negative numbers can and will occur.

Examples: Temperatures not measured in Kelvins
Electric Charge

As well, any time you have vectors in a one-dimensional space, or a zero-sum system:

Debits/Credits in accounting.
Profit/loss in trading

Saying negative numbers do not exist in the real world is naive. Numbers and quantifiable values have more uses in the real world than counting.

 

[Knight Templar]

snip

Again I used the wrong word for what I meant, tangeable is more accurate.

 

[Maze1125]

And 'one is a boy' is not mathematically unambiguous.

Yes it is. If you say that phrase to any mathematician, in a probabilistic context, then every single one will understand it to mean that the random variable can only take values in the set (FB union SB).

And, again, prove your claim that the probability can be 1/2.

When you say 'One is a ______, what is/can be/qualities may be had by the other?' it could be taken to mean, in a mathematical sense, to mean that one specific entity is called on, and that the other one is the unit in question.

IF you are taking a specific child and -then- revealing their gender, the selection process becomes the FB is a defined event, and the SB is a non-defined event. The question becomes an isomorphism of 'I have flipped a coin and it came up heads. What are the odds of the next flip?' 50%. This is -trivial-.

IF you are taking a random child, and -then- revealing their gender, there are three randomized elements: The child selected, and the gender of each child respectively. Each has a probability of 50% for each outcome; You could select the first child, or the second child, and said child could be male, or female, and the other child could be male or female.

This leads to EIGHT different outcomes: (FC-SC-Selected child)

1-) B-B-FC
2-) B-B-SC
3-) B-G-FC
4-) B-G-SC
5-) G-B-FC
6-) G-B-SC
7-) G-G-FC
G-G-SC

Then, we eliminate the outcomes that this case could not have evoked, 4, 5, 7, and 8. This leaves:

1-) B-B-FC
2-) B-B-SC
3-) B-G-FC
6-) G-B-SC

Using Draco's Law of Counting the damn outcomes, I see 4 outcomes. 2 of them have the selected child have a brother, 2 of them have the selected child have a sister.

50%.

So, -if- the possibility exists that the mother -could have- revealed a girl in the method of her selection, then that does, AS PROVEN ABOVE, directly affect the probability, in the -exact same manner- that in the Monty Haul problem, if the host -could have revealed a car-, that changes the probability of switching being advantageous. This does not care if she picked the first child, or a child at random.

It -does- become 33%, however, if the woman selected by the quality "boy" rather than "child". In -that- case the woman is screening your outcomes, and this bias sets the probability away from the 50%.

Therefore, the probability is affected by the probability chance this woman is selecting by boy rather than by child regardless of gender.

Selecting by boy might include answering the question 'Do you have a boy?' She, however, has not answered 'I have a boy.' She has told you 'One of them is a boy.' You are not certain the manner of selection she used, nor -her- criteria, but that directly affects the results.

You are adding information that is not anywhere in the question.
The mother is irrelevant, she did not pick the child, she has absolutely nothing to do with the problem. The only information we have is "One is a boy." that is simply a piece of information with a precise meaning. How it was obtained it irrelevant.

If the child was selected, that would actually give the problem more information than it has and therefore it would be a different problem.

The only issue here is that fact that it the problem has been attempted to be formulated in a "realistic" way. This is causing confusion such as the idea the mother has anything to do with it.

Therefore I suggest we instead consider the purely probabilistic problem that is "There are two identical variables that can each take state A with probability 1/2 and state B with probability 1/2. One of the variables (remembering that this has a precise meaning) is in state A, what is the probability they are both in state A?"

 

[DracoSuave]

You are adding information that is not anywhere in the question.

Actually, I am stating that there is information -missing- and accounting for that missing information. 'There is missing information' is not the same as 'adding information.'

That's what -ambiguity- means.

The mother is irrelevant, she did not pick the child, she has absolutely nothing to do with the problem. The only information we have is "One is a boy." that is simply a piece of information with a precise meaning. How it was obtained it irrelevant.

The method of obtaining that information is the difference between 1/2 and 1/3. That means that the probability she is asking you for is dependant on the manner of obtaining that information.

This must be some new meaning of the word 'irrelevant' that I am not familiar with.

I think the word you are looking for is 'relevant.'

If the child was selected, that would actually give the problem more information than it has and therefore it would be a different problem.

I agree it would give the problem more information. In fact, it would give you enough information to make the problem -solveable- with a real number answer rather than a polynomial.

The only issue here is that fact that it the problem has been attempted to be formulated in a "realistic" way. This is causing confusion such as the idea the mother has anything to do with it.

No, the problem is that you feel there -must be- a definative real-number answer. When presented with direct evidence that the manner in which the information was provided has a -direct result- on the mathematical results, your response is to bury your head in the sand and pretend that it doesn't.

The -scientific- and -mathematical- approach is to account for it in your calculations.

n/3 + (1-n)/2 does that. Your answer does not.

Therefore I suggest we instead consider the purely probabilistic problem that is "There are two identical variables that can each take state A with probability 1/2 and state B with probability 1/2. One of the variables (remembering that this has a precise meaning) is in state A, what is the probability they are both in state A?"

That would be great, except that is not what we are presented with. You have said 'Instead of presenting us this problem that we have, consider this other problem which gives me the answer I desire.'

That, sir, is chicanery.

It has been proven that the method by which the information is obtained -DIRECTLY- affects the result and the correct answer of the problem.

That -mathematically proves- the question is ambiguous.

Does the method of obtaining the information affect the result? Yes. So how can you, with any mathematical honesty, make -any- claim that it is irrelevant? Or that the question is ambiguous.

I'm sorry, you're too wrapped up in your confirmation bias to actually discuss this problem on a meaningful mathematical level. You've assumed it means something that it does not state that it means--You've claimed -in the face of evidence to the contrary--that it cannot be ambiguous.

But the fact remains that it is, the method of information gathering is -vital- to the solution of the problem, and therefore, if a non-variable answer is what you seek, then more information is required.

If this concept proves problematic that you can end up with a variable answer to a mathematical problem when faced with incomplete knowledge of all relevant information, you need to return to high school algebra.