Wonderfully logical illogical probability problem

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markyp23 said:
Ahhhh...

The question is written fairly badly! I'm still not convinced either way. What is wrong about Haircut's explanation, then?
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In the case that he has two children, one is definitely female

There are only 3 options, because he hasn't stated that one is younger than the other etc

BB
BG (order is irrelevant since he hasn't stated younger/older)
GG

Because we know with certainty that one is female, BB = 0, BG = 1.B and GG = 1.G, since we know that the probability of any one child being male is 50% and the same for female, this again reduces to 50/50.
 
Jokester said:
In the case that he has two children, one is definitely female

There are only 3 options, because he hasn't stated that one is younger than the other etc

BB
BG (order is irrelevant since he hasn't stated younger/older)
GG

Because we know with certainty that one is female, BB = 0, BG = 1.B and GG = 1.G, since we know that the probability of any one child being male is 50% and the same for female, this again reduces to 50/50.
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You can't just say that though as if you're going to say the order is irrevelant you need to consider that BG is twice as likely as BB and GG.
This gives the 2/3 figure.
 
From my thinking there seems to be a misunderstanding about numbers of outcomes.

There are 4 combinations of children BB,BG,GB,GG.

However we are talking about the gender of the child of which there are only three outcomes. Boys, Girls, Mixed.

In the question we eliminate one outcome which means we are left with two. ie. 50/50

It seems some people were over analysing and considering BG and GB as two different outcomes when given the question about gender they are the same.

Or am I wrong? ;)
 
Haircut said:
You can't just say that though as if you're going to say the order is irrevelant you need to consider that BG is twice as likely as BB and GG.
This gives the 2/3 figure.
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But the problem is in the BG case we know with 100% certainty that one is female, so the probability that any one child is male is always 50%.
 
Jokester said:
But the problem is in the BG case we know with 100% certainty that one is female, so the probability that any one child is male is always 50%.
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No no no no, that doesn't work.
I'll try and think of a way to explain it better, I'll be back in a bit!
 
And just to demonstrate it further

what is the probability that two random children are of different gender
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Is the question that is asked to give the 2/3 answer.

This is not this case as they are not random as we have been told one child is most definitely female.
 
The possibility in both instances is 50%. No matter how badly worded the statements may be to try to persuade you to think one way or the other, there is still in both instances a 50% possibility that the other child is a boy. (Edit: as the person above me has just demonstrated, perfectly)
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agreed
 
Haircut said:
you need to consider that BG is twice as likely as BB and GG.
This gives the 2/3 figure.
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This is only valid if you are asking about two children.

ie. Is it more likely that a 2 child family has mixed gender children or two girls.

To which the answer is, it is twice as likely. (your 2/3 answer)


Since we know there is a girl (gender irrelevant it just makes it a one child question) the question is reduced to "does she have a brother or sister"

50/50 since it is a single event with two possible outcomes of equal liklihood.
 
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Sorry Jokester....I'm posting the same as you a little bit later and in a slightly different way. :o

You're definitely on the same wavelength as me. :cool:
 
OK, hopefully this will convince the doubters.
Let me know which points you disagree with

There are two children in the family, let's call them Child A and Child B.

All the possible ways of arranging this are shown below and they all have probability of 0.25


Child A = Girl
Child B = Girl

Child A = Girl
Child B = Boy

Child A = Boy
Child B = Girl

Child A = Boy
Child B = Boy

Now the crucial point here is that you don't know whether you have just met child A or child B, only that you have met a girl, so this only eliminates the last possibility (where both A and B are boys)

So you are left with

Child A = Girl
Child B = Girl

Child A = Girl
Child B = Boy

Child A = Boy
Child B = Girl

You know you have met the girl.
As we can see from the above given that you have met a girl the other child will also be a girl in only one case.
The other child will be a boy in two cases, thus the probability of being a boy is twice as high.

So, this probability is 2/3

If anyone can find a valid flaw in this then let me know and I'll go and give back my maths degree as I'm clearly not worthy of it :p
 
markyp23 said:
You missed out a massive part of that question though:

"Excluding the case of two girls, what is the probability that two random children are of different gender?"


Replace that with:
Excluding the case of two boys, what is the probability that two random children are of different gender?

Then you have a 2/3 answer.
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But that is not the question that is being asked here, to get 2/3 the question would be:-

A random two-child family with at least one girl is chosen. What is the probability that it has a boy?
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What the OP actually asked is

A random two child family with as least one girl is chosen. What is the probability that the girl has a brother?
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Which is 50%

Haircuts example only works for the first case.
 
The most intuitive way to see there's a difference is to look at the information you are being given. In the first situation you are only being told one of the children is a girl. In the second situation you are being told specifically that the youngest child is a girl. You are being told an extra piece of information, so it makes sense that the probabilities will change.
 
Psyk said:
The most intuitive way to see there's a difference is to look at the information you are being given. In the first situation you are only being told one of the children is a girl. In the second situation you are being told specifically that the youngest child is a girl. You are being told an extra piece of information, so it makes sense that the probabilities will change.
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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.
 
Haircut said:
Surely if there is a boy in the family then the girl has a brother :confused:
The two are logically equivalent.
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They aren't though, one is asking it from the point of view of the daughter where there are two possibilties (B or G or 50/50), the other is asking what the possible combinations of two siblings are (BG GG or GB) and out of them what the probability that one contains a boy (2/3).
 
Jokester said:
They aren't though, one is asking it from the point of view of the daughter where there are two possibilties (B or G or 50/50), the other is asking what the possible combinations of two siblings are (BG GG or GB) and out of them what the probability that one contains a boy (2/3).
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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.

And also, the two statements below are equivalent unless you can show me a two child family where one is a boy but yet the girl doesn't have a brother.

A random two-child family with at least one girl is chosen. What is the probability that it has a boy?

A random two child family with as least one girl is chosen. What is the probability that the girl has a brother?
 
Haircut said:
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.

And also, the two statements below are equivalent unless you can show me a two child family where one is a boy but yet the girl doesn't have a brother.

A random two-child family with at least one girl is chosen. What is the probability that it has a boy?

A random two child family with as least one girl is chosen. What is the probability that the girl has a brother?
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Because ultimately the question is:-
What is the probability that his other child is a boy?
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And it becomes a simple 50/50 probability, the use of other, implies it's a simple single event.
 
I'm well aware of what the question is, and I've shown several times in this thread that the answer is 2/3.

Anyway, I'm off to bed.
This is as fruitless as trying to convince people that 0.999... = 1
 
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