Wonderfully logical illogical probability problem

[Psyk]

Not if the extra information is totally irrelevant.

It makes no difference if the girl is youngest or oldest. It's still just a question of the gender of one child, not it's order or age.

In a question of gender of one child it is one of two outcomes. Any way you cut it, the sibling is either a boy or a girl...ie. 50/50

The truly remarkable thing about the "problem" is that it has managed top confuse so many people. Personally I kept trying to prove a solution that wasn't correct.

Think of the four possibilities:

child 1:boy, child 2: girl
child 1:girl, child 2: boy
child 1:girl, child 2:girl
child 1:boy, child 2:boy

In both situations, the last one is impossible. But the other 3 are equally likely. In two of those cases, the other child is a boy. Hence the possibility of one child being a boy is 2/3. Ok yes it doesn't explicitly say the order matters, but you have to remember there are two ways to end up with a boy and a girl, having a boy first then a girl, or having a girl first then a boy. So the possibility of there being a boy and a girl has to be counted twice.

In the second situation you rule out the second possibility as well because you know the youngest is a girl. So it's back to 1/2.

 

[Jokester]

I'm well aware of what the question is, and I've shown several times in this thread that the answer is 2/3.

You've proved it for a different question though, the two questions have a different sample space.

 

[Ben Cole]

Why is it 50/50 though?
I understand that that the girls sibling is either a boy or a girl, you haven't shown why they are equal probabilities.

He doesn't have to. 1 child, 1 birth always 50/50.

I agree those two statements ask the same question to me.

 

[Jokester]

Think of the four possibilities:

child 1:boy, child 2: girl
child 1:girl, child 2: boy
child 1:girl, child 2:girl
child 1:boy, child 2:boy

In both situations, the last one is impossible. But the other 3 are equally likely. In two of those cases, the other child is a boy. Hence the possibility of one child being a boy is 2/3. Ok yes it doesn't explicitly say the order matters, but you have to remember there are two ways to end up with a boy and a girl, having a boy first then a girl, or having a girl first then a boy. So the possibility of there being a boy and a girl has to be counted twice.

In the second situation you rule out the second possibility as well because you know the youngest is a girl. So it's back to 1/2.

In the first question there is only 2 possibilities as well, as the question is, what is the probability that she has a brother.

The two possibilities are:-

Brother
Sister

Each is equally likely.

 

[Ben Cole]

Think of the four possibilities:

child 1:boy, child 2: girl
child 1:girl, child 2: boy
child 1:girl, child 2:girl
child 1:boy, child 2:boy

In both situations, the last one is impossible. But the other 3 are equally likely. In two of those cases, the other child is a boy. Hence the possibility of one child being a boy is 2/3. Ok yes it doesn't explicitly say the order matters, but you have to remember there are two ways to end up with a boy and a girl, having a boy first then a girl, or having a girl first then a boy. So the possibility of there being a boy and a girl has to be counted twice.

In the second situation you rule out the second possibility as well because you know the youngest is a girl. So it's back to 1/2.

You say there's 4 possibilties. There aren't. Since this is a question about the gender of ONE child, boy then girl and girl then boy are the same.

If it was a question about combinations including order THEN and only then would there be 4 possibilities.

 

[Psyk]

You say there's 4 possibilties. There aren't. Since this is a question about the gender of ONE child, boy then girl and girl then boy are the same.

If it was a question about combinations including order THEN and only then would there be 4 possibilities.

Well I must admit, I'm having a hard time convincing myself I'm right

 

[Ben Cole]

I just think we've all over complicated things. In the end the question is "what is the sex of the sibling"
Answer:Male or female (50/50)

P.S. Never trust Wikipedia.....any old Tom Dick or Harry could nhave written the entry!!!

 

[Psyk]

Actually I'm thinking it is 50:50 now. The problem with my line of thought is that the Girl-Girl possibility should also count twice, since you don't know whether the daughter you've met is the older or younger one.

But, it's 2/3 for the second one because now the order does matter. The 3 possibilities are:
1) She has an older sister
2) She has an older brother
3) She has a younger brother

Each are equally likely.

 

[Zogger]

Here you go. I wrote it for you...
http://www.zogger.co.uk/boygirl.php

(scroll to bottom)

 

[FrostedNipple]

im pretty sure i remember from biology that the chance of having a boy is 25%, and his wording really has no mathematical influence as the probability's are infinite if you try to include it

im saying 25%boy 75%girl

 

[Shoseki]

The fact that he identified her as "his daughter" as opposed to "one of my daughters" means the other child is a boy, or possibly hermaphroditic.

 

[Zogger]

lol, just tried went to take a quick look at that script I wrote from work, and the filter blocks it because it thinks it's pornography (from the url). Oh the things this filter says.

 

[[TR]SweetSpotOz]

But that is not the question that is being asked here, to get 2/3 the question would be:-



What the OP actually asked is



Haircuts example only works for the first case.

Jokester, how you can not think they are the same thing is causing me absolute bewilderment.

 

[Jokester]

Jokester, how you can not think they are the same thing is causing me absolute bewilderment.

I should probably point out I've got an SYS in Statistics and Probability .

The two questions the OP has asked are exactly the same from the point of view of probability, and are not the same as the question that yields 2/3.

 

[D.P.]

I had to work this one out this morning, i first thought wtf!? Then i worked it out, see what you think:

You meet your new tutor in town accompanied by a young girl. He says to you "I have two children, this is my daughter Lisa". What is the probability that his other child is a boy. Would it make a difference if he had said "I have two children, this is my youngest child, Lisa"? (You may assume a 50% chance of any one birth being a boy or a girl.


Have fun!

OSB

100% chance of being a boy. If he had 2 daughters then he would say "This is my youngest/eldest daughter".

Took about 0.2 seconds, maybe less.

However, this assumes an agent that tries to maximise information gain in communication. When dealing with humans one shouldn't assume a rational agent.

 

[[TR]SweetSpotOz]

.

 

[OSB]

I should probably point out I've got an SYS in Statistics and Probability .

The two questions the OP has asked are exactly the same from the point of view of probability, and are not the same as the question that yields 2/3.

I'd like to point out that doesn't mean you're views are correct. The two cases ARE different. Clarification:

When having two children the following combinations are possible (note order is important):

BB
BG
GB
GG

Where B=Boy and G=Girl.


Now in the first case the information you are given is that he has two children and one is a girl (ignoring any linguistical nuances where he might phrase it this way or that way because of whatever). So this rules out the BB option, leaving only BG, GB and GG. Out of these 3 remaining possibilities 2 contain a boy, therefore the probability his other child is a boy is 2/3.

In the second case the information we have is not only that one child is a girl, but that his second child (the youngest) is a girl. Now the only remaining possibilities are BG and GG. Here out f these two options only 1 contains a boy, therefore the probability his other child is a boy in this case (given this information) is 1/2.


Hope that is clear enough for everyone?

 

[Jokester]

I'd like to point out that doesn't mean you're views are correct. The two cases ARE different. Clarification:

When having two children the following combinations are possible (note order is important):

BB
BG
GB
GG

Where B=Boy and G=Girl.


Now in the first case the information you are given is that he has two children and one is a girl (ignoring any linguistical nuances where he might phrase it this way or that way because of whatever). So this rules out the BB option, leaving only BG, GB and GG. Out of these 3 remaining possibilities 2 contain a boy, therefore the probability his other child is a boy is 2/3.

In the second case the information we have is not only that one child is a girl, but that his second child (the youngest) is a girl. Now the only remaining possibilities are BG and GG. Here out f these two options only 1 contains a boy, therefore the probability his other child is a boy in this case (given this information) is 1/2.


Hope that is clear enough for everyone?

Seriously, this is WRONG! for the questions posted by the OP.

The 2 cases from the OP are both differently worded versions of case 1 from this page here:-

http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

For the 2/3 case (case 2), the question can't state that the first or second child is definitely female/male because it reduces the probabilites to what is the probability of that other child being male or female which is always 50/50.

It makes a big difference in terms of probability saying this is my daughter, what is the probability my other child is a boy, to I have a two child family, at least one is female, what is probability I also have a boy, because the answer also relies on whether it's the first child or the second that is female, whilst in the first case we have stated with certainty that the first child is female.

 

[OSB]

Seriously, this is WRONG! for the questions posted by the OP.

The 2 cases from the OP are both differently worded versions of case 1 from this page here:-

No they're not. I'm the OP, the first case i stated is analogous to case two on your link (in that it is only the gender of one child is stated, not whether that is the older or younger child). Where as the second case in the OP both the gender and whether that child is the older or the younger is also stated, analogous to case 1 on that link. I don't see how you think they're both case 1 from your link?!

 

[Jokester]

No they're not. I'm the OP, the first case i stated is analogous to case two on your link (in that it is only the gender of one child is stated, not whether that is the older or younger child). Where as the second case in the OP both the gender and whether that child is the older or the younger is also stated, analogous to case 1 on that link. I don't see how you think they're both case 1 from your link?!

Ah sorry didn't realise you were the OP, no they are both case 1, because you have stated "this is my daughter" in both cases, the age of the child is irrelevant to the probability. By stating "this is my daughter" removes the opposite case of being possible and it's irrelevant what one was born first. If you KNOW (ie probability = 1, not 0.5) which child is female there are only two options possible, BG, or GG. the alternative boy girl combination of GB is NOT possible. Younger or older makes no difference as all it does is change GB to BG and BG to GB, it's restricted the sample space by removing the third option that gives the 2/3.

The correct question you need to ask is "I have two children, if at least one is female (this allows for both combinations of BG and GB) what is the probability that the other is male"