Paradox or did I make a mistake?

[Steven G]

Suppose you visit me in my house, I introduce you to my son and I tell you that I have another child.
Then P( my other child is a girl) = 2/3. (sample space is {bb, bg, gb}

Now suppose you visit me in my house, I introduce you to my son who I tell you is my 1st born and I tell you that I have another child.
Then P( my other child is a girl) = 1/2. (Let x,y mean my 1st child is sex x and my 2nd born is sex y. So sample space is {bb, bg})

Similarly, suppose you visit me in my house, I introduce you to my son who I tell you is my 2nd born and I tell you that I have another child.
Then P( my other child is a girl) = 1/2. (sample space is {gb, bb)

So the last two probabilities are the same.

Now suppose suppose you visit me in my house, I introduce you to my son who I tell you is my ??? born and I tell you that I have another child. The ??? means that you did not hear what I said.
Then P( my other child is a girl) = 1/2 because it does matter what I said.

But wait a minute! If you did not hear me then this is equivalent to the very 1st scenario which has P (other child is a girl) =2/3.

Combining this we conclude that P( other child is a girl) equals both 2/3 and 1/2.

So did I make a mistake or is this some kind of a paradox?

 

[HallsofIvy]

I see no mistake nor paradox. You get different probabilities because you are working with different "equally likely outcomes".

Here is a similar problem- the way I first saw this:

A new family moves into your neighborhood. You have children and are interested in playmates for them so you check up on their children.

1) You find out they have two children and at least one of them is a boy. What is the probability the other is also a boy?
If you only knew they had two children, the "equally likely outcomes" are "BB", "BG", "GB", and "GG" where the first child is the older. Knowing "at least one is a boy" removes "GG" leaving "BB", "BG", and "GB". In one of these three equally likely outcomes, the other child is a boy so the probability is 1/3.

2) You find out they gave two children and the older child is a boy. What is the probability the other is also a boy?
Again, if all we knew was that they had two children the "equally likely outcomes" are "BB", "BG", "GB", and "GG" and again the first child is the older. Knowing that "the older child is a boy removes both "GG" and "GB" leaving only "BB" and "BG". In one of the two equally likely outcomes, the other child is a boy so the probability is 1/2.

It should be no great surprise that the two answers are different because in the second case we are given more information than in the first- "The older child is a boy" is more precise than "at least one child is a boy".

 

[Steven G]

I see no mistake nor paradox. You get different probabilities because you are working with different "equally likely outcomes".

Here is a similar problem- the way I first saw this:

A new family moves into your neighborhood. You have children and are interested in playmates for them so you check up on their children.

1) You find out they have two children and at least one of them is a boy. What is the probability the other is also a boy?
If you only knew they had two children, the "equally likely outcomes" are "BB", "BG", "GB", and "GG" where the first child is the older. Knowing "at least one is a boy" removes "GG" leaving "BB", "BG", and "GB". In one of these three equally likely outcomes, the other child is a boy so the probability is 1/3.

2) You find out they gave two children and the older child is a boy. What is the probability the other is also a boy?
Again, if all we knew was that they had two children the "equally likely outcomes" are "BB", "BG", "GB", and "GG" and again the first child is the older. Knowing that "the older child is a boy removes both "GG" and "GB" leaving only "BB" and "BG". In one of the two equally likely outcomes, the other child is a boy so the probability is 1/2.

It should be no great surprise that the two answers are different because in the second case we are given more information than in the first- "The older child is a boy" is more precise than "at least one child is a boy".

Hi,
I am sorry but I am still confused with this.

You say that you agreed that if you know that I have two children and you met my son that P(other child is a girl)=2/3 (so P(other child is a box) =1/3). This is exactly what you have for 1)

Now you say that you also agree that if you know that my 1st (or 2nd) born is a boy and that I have 2 children that P(other child is a girl)=1/2. So P(other child is a boy) = 1/2. This is basically what you have in 2)

Clearly I got two different answer both of which you agree with. (not saying for the same probability--yet).

You obviously must have a problem with one the following: (*) I claim that if in the above you did not hear whether I said 1st or 2nd born then this is the same as saying this is my son and I have another child a home.

IF you buy that last line (please entertain me here) then I claim that (**) If you visit me in my house, I introduce you to my son who I tell you is my 2nd born and I tell you that I have another child that is equivalent to If you visit me in my house, I introduce you to my son who I tell you is my 1st born and I tell you that I have another child.
And then 2/3 = 1/2

I know that I should understand what you already replied. I hope that I made it clear where I must be seeing things wrong. Can you please try again?
Jomo

 

[Steven G]

Sure sounds like the Monte Hall classic...

The thing is that the Monte Hall problem has a solution. One might not expect the solution to be what it is but there is a solution. In my problem I am getting two different results and am completely confused about what is going on here.

 

[Ishuda]

...You obviously must have a problem with one the following: (*) I claim that if in the above you did not hear whether I said 1st or 2nd born then this is the same as saying this is my son and I have another child a home.

IF you buy that last line ...

But I, for one, don't buy that last line except under special circumstances which I think you precluded. The fact that you don't know something does not change the existence/non-existence of that something.

To approach this a little differently that HallsofIvy did, you did not hear specifically whether it was the first (or second born) but, you do (1) know or (2) don't know that it was one or the other.

If (1); you do know [and 'the universe' knows] what you missed was either the son was either first born [sample space bb, bg] or second born [sample space gb, bb] then, in either case the probability is 1/2 [assuming equally likely b or g birth which isn't quite true depending on where you are]. Or, paraphrasing HallsofIvy, the additional information changes the odds and you should expect a different answer.

If (2) the special circumstance I was talking about above, you don't know anything of what was said (nor does 'the universe'). In that case, it is the same as saying I have a son and two children, i.e. no further information. So the odds don't change and the sample space is still bb, bg, gb and 2/3=2/3.

 

[Steven G]

But I, for one, don't buy that last line except under special circumstances which I think you precluded. The fact that you don't know something does not change the existence/non-existence of that something.

To approach this a little differently that HallsofIvy did, you did not hear specifically whether it was the first (or second born) but, you do (1) know or (2) don't know that it was one or the other.

If (1); you do know [and 'the universe' knows] what you missed was either the son was either first born [sample space bb, bg] or second born [sample space gb, bb] then, in either case the probability is 1/2 [assuming equally likely b or g birth which isn't quite true depending on where you are]. Or, paraphrasing HallsofIvy, the additional information changes the odds and you should expect a different answer.

If (2) the special circumstance I was talking about above, you don't know anything of what was said (nor does 'the universe'). In that case, it is the same as saying I have a son and two children, i.e. no further information. So the odds don't change and the sample space is still bb, bg, gb and 2/3=2/3.

Thanks for your reply. I am not saying that you are wrong but I do not buy what you are saying..yet. Here is where I am confused. Let's say that I just tell you this is my son and I have another child. Of course this son you just met is either my 1st born or my 2nd born. So why am I wrong if I say that you did say which position this son is in and I did not hear you when in fact you never said anything about the position of this son. This is precisely where I am getting lost.
If in fact this is a paradox then I will never answer another probability problem again since it may be a paradox that is almost impossible to see unless you are looking for one from every possible angle.



Here IS a paradox, but is easy to see that this is a paradox. If I have two envelopes marked A and B and I tell you that one has twice as much money as the other. I ask you to pick an envelope. Then I ask you if you want to switch envelope with me. Should you switch? Suppose the envelope that you pick has $2x. If you switch and pick the higher valued envelope (it has $4x) then you increased your winnings by $2x. If on the other hand you switch to the lower valued envelope (it has $x) then you only decreased your winnings by $x. So you should switch as you will increase by more than you would decrease. But wait a minute, because if you were now asked again if you wanted to switch the answer would be yes again. Then you have your original envelope. This is a paradox and is easily seen. I actually did see this immediately as a student even before my Professor asked if one should switch back if allowed. It was the 1st and last time I impressed my probability teacher.

 

[HallsofIvy]

The thing is that the Monte Hall problem has a solution. One might not expect the solution to be what it is but there is a solution. In my problem I am getting two different results and am completely confused about what is going on here.

As I said before, you are getting two different results because these are two different problems and have two different solutions!

 

[Steven G]

As I said before, you are getting two different results because these are two different problems and have two different solutions!

I apologize for not getting this. I really do not see why these are two different problems.

1) If one is never told anything about which position the son is in.
2) One is told which position the son is in but did not hear clearly what was said.

Why are these two different?

Please explain this as I must settle this in my head.

Thanks,
Jomo

 

[Ishuda]

I apologize for not getting this. I really do not see why these are two different problems.

1) If one is never told anything about which position the son is in.
2) One is told which position the son is in but did not hear clearly what was said.

Why are these two different?

Please explain this as I must settle this in my head.

Thanks,
Jomo

In order to answer your question, please answer this question for 2). Irregardless of the fact that you did not hear clearly, did you ('the universe') gain any information? For example, I'm not sure of exactly what was said but it was either first born or second born.

 

[Steven G]

In order to answer your question, please answer this question for 2). Irregardless of the fact that you did not hear clearly, did you ('the universe') gain any information? For example, I'm not sure of exactly what was said but it was either first born or second born.

You made a good point but I still do not buy it (sorry). In order to help me consider this:

1)you tell me 1st or 2nd born but I do not hear you clearly.

2) You never mention 1st or 2nd but I reasonably assume that the son I met was either your 1st or 2nd born. I do not see why it is unreasonable for me to assume this.

Thanks,
Jomo

 

[Ishuda]

You made a good point but I still do not buy it (sorry). In order to help me consider this:

1)you tell me 1st or 2nd born but I do not hear you clearly.

2) You never mention 1st or 2nd but I reasonably assume that the son I met was either your 1st or 2nd born. I do not see why it is unreasonable for me to assume this.

Thanks,
Jomo

Will you answer the question please. Irregardless of the fact that you did not hear clearly, did you ('the universe') gain any information? For example, I'm not sure of exactly what was said but it was either first born or second born. If you don't answer the question then it really looks to me like what you are putting forth is something like there are 50 bananas available on Friday. Who ate the orange? That is, it is not a matter of misunderstanding, it is a matter of deliberately withholding information about the conditions of the problem and expecting the correct answer.

 

[Steven G]

Will you answer the question please. Irregardless of the fact that you did not hear clearly, did you ('the universe') gain any information?I do not think that I gained any information. The reason is that I was told that this person has two children and I met the son. So I knew that this son whom I met was the 1st or 2nd born For example, I'm not sure of exactly what was said but it was either first born or second born. If you don't answer the question then it really looks to me like what you are putting forth is something like there are 50 bananas available on Friday. Who ate the orange? That is, it is not a matter of misunderstanding, it is a matter of deliberately withholding information about the conditions of the problem and expecting the correct answer.

My answers to your questions are above. I guess that I did not do that before, sorry about that. Please do not be insulted by what I am about to say and understand that this is a forum for learning. When I was in college I had a fellow student inform me after I used the word irregardless that this word does not exist. I was told to simply use regardless.
Thanks,
Jomo

 

[pka]

My answers to your questions are above. I guess that I did not do that before, sorry about that. Please do not be insulted by what I am about to say and understand that this is a forum for learning. When I was in college I had a fellow student inform me after I used the word irregardless that this word does not exist. I was told to simply use regardless.

Although you are correct about irregardless. But how is that germane to this thread? Therefore, one might indeed find it insulting. I have been very reluctant to join in the fun, but this is going nowhere. Your restatement of the second question using \(\displaystyle ???\) really confused the issue. It was only after I realized you were using \(\displaystyle ???\) as a variable did I understand your confusion. Probability question of this sort do not contain unbound variables. So that wording makes two questions, depending upon the value of \(\displaystyle ???\). Therefore, there are different answers each of two different questions.
There are just two different questions in this whole discussion.
1) If I have two children and one is male, then what is the probability both are?
2) If I have two children and the second-born is male, then what is the probability both are?

The first has (2/3) as an answer, (1/2) is the answer to the second.

 

[Steven G]

Although you are correct about irregardless. But how is that germane to this thread? Therefore, one might indeed find it insulting.

Does it really matter what the subject at hand is when it comes to learning? I understand that some may think it matters and others will not. I just remember how glad I was for my classmate pointing this error out to me and I just wanted to do the same. Sure this is an open forum but I do not think that anyone would look down on someone for not knowing not to use a word, especially when it is a common mistake. To be honest, I decided to make the statement because I felt in the end that it would be best for the person to know.
I know that I am new to this forum but I want to be part of this family and help where I can.
Jomo

 

[Ishuda]

My answers to your questions are above. I guess that I did not do that before, sorry about that. Please do not be insulted by what I am about to say and understand that this is a forum for learning. When I was in college I had a fellow student inform me after I used the word irregardless that this word does not exist. I was told to simply use regardless.
Thanks,
Jomo

Since this is a forum for learning, maybe you will be pleased to know that, according to wikipedia and others, "In the last twenty-five years, irregardless has become a common entry in dictionaries and usage reference books, although commonly marked as substandard or dialect." So the use would be perfectly acceptable (as far as anything can be perfect, that is) unless, of course, you are the type of person who always insists on the 'speaker' you are 'listening to' be limited to your vocabulary and language. In that case, it would only be unacceptable to you (and possibly others) but not the world at large.

Do you honestly think that the sample space [bb bg gb gg], i.e. knowing a person has two children and thus, because human women are considered to be unable to birth two children at once, one must have been born first and the other must have been born second is the same as the sample space of either [bb bg] or [bb gb] but not both, i.e. one or the other was said, not both was said? If so, stick around this forum for learning and maybe you will learn something if you pay attention to what is said rather than ignoring what you don't want to 'hear'.

 

[Steven G]

Do you honestly think that the sample space [bb bg gb gg], i.e. knowing a person has two children and thus, because human women are considered to be unable to birth two children at once, one must have been born first and the other must have been born second is the same as the sample space of either [bb bg] or [bb gb] but not both, i.e. one or the other was said, not both was said? If so, stick around this forum for learning and maybe you will learn something if you pay attention to what is said rather than ignoring what you don't want to 'hear'.

Trust me that I want to hear the solution more than you might expect. Sometimes it is hard to get through to me. I almost bought what you said about giving birth to two babies at once until I realized that one baby comes out first.
I am just trying to settle this in my mind.
I truly can't think of why if I am not told that this son that I met comes from a 2 children family then this son was 1st born or 2nd born. How can that assumption be wrong?
What am I missing?

 

[Deleted member 4993]

.... When I was in college I had a fellow student inform me after I used the word irregardless that this word does not exist. I was told to simply use regardless.
Thanks,
Jomo

That fellow student was wrong.

From http://www.merriam-webster.com/dictionary/irregardless

[h=2]Usage Discussion of IRREGARDLESS[/h]Irregardless originated in dialectal American speech in the early 20th century. Its fairly widespread use in speech called it to the attention of usage commentators as early as 1927. The most frequently repeated remark about it is that “there is no such word.” There is such a word, however. It is still used primarily in speech, although it can be found from time to time in edited prose. Its reputation has not risen over the years, and it is still a long way from general acceptance. Use regardless instead.


[h=2]Examples of IRREGARDLESS[/h]

• I told them that irregardless of what you read in books, they's some members of the theatrical profession that occasionally visits the place where they sleep. —Ring Lardner, The Big Town, 1921

[h=2]Origin of IRREGARDLESS[/h]probably blend of irrespective and regardlessFirst Known Use: circa 1912

 

[pka]

That fellow student was wrong.

I can give examples where the use of that word in a job interview lost the candidate the job. All of the material you posted means little to leadership at certain colleges and universities.

 

[Ishuda]

I can give examples where the use of that word in a job interview lost the candidate the job. All of the material you posted means little to leadership at certain colleges and universities.

I don't go to job interviews - if they want to hire me and I find it interesting enough I just might take the job.

 

[Deleted member 4993]

Subhotosh, what's an IRREGARD anyway

This thread has turned out to be the weirdest one I've seen in quite a while,
regardless of irregardless, or irregardless of regardless...

"Give My Irregards to Broadway"

irregards,
Denis

When you don't have more of those - you will be called irregardless. Like hair-less or heart-less...