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# One Girl - One Boy

Best Answer Riddari , 13 July 2007 - 07:38 PM

Spoiler for solution
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Posted 11 July 2012 - 04:27 AM

Ummm... I didn't read this entire thread (waste of time), but the issue here is simply a combination/permutation issue.

Since it doesn't matter whether the child stated as the girl is the first born or second born, there isn't as many scenarios as the "answer" claims.

A child is a girl
Possibilities:
Girl-Girl
Girl-Boy } Permutation of
Boy-Girl } the same thing
Boy-Boy

We can rule out the boy-boy field since there is already the claim that one child is a daughter.

What we are left with is
Girl-Girl
Girl-Boy
That is a 50% chance.

If they claimed that the FIRST child was a girl, then you can include the permutations and assume the 1/3 chance of the second child being a girl.

Problem solved.
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Posted 11 July 2012 - 04:55 AM

Hmmm, the more I think about it, the more issues there are with it.

It depends on whether you look at it small scale or large scale.

Small scale it appears to be 50% chance because the birth of one child doesn't affect the other.

However, if you place a limitation on it, and look at it as a large scale model, then the answer is 1/3

For a large scale model you have 100 families.
25 G-G
25 G-B
25 B-G
25 B-B

You know one child is a girl, what are the odds the other child is a girl would be 25/75 (since the B-B are eliminated).
However, this is placing limitations and additional information on the question.

Like was said, small scale and large scale are contradicting each other
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### #333 fabpig

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Posted 11 July 2012 - 10:16 AM

Ummm... I didn't read this entire thread (waste of time), but the issue here is simply a combination/permutation issue.

Since it doesn't matter whether the child stated as the girl is the first born or second born, there isn't as many scenarios as the "answer" claims.

A child is a girl
Possibilities:
Girl-Girl
Girl-Boy } Permutation of
Boy-Girl } the same thing
Boy-Boy

We can rule out the boy-boy field since there is already the claim that one child is a daughter.

What we are left with is
Girl-Girl
Girl-Boy
That is a 50% chance.

If they claimed that the FIRST child was a girl, then you can include the permutations and assume the 1/3 chance of the second child being a girl.

Problem solved.

Hmmm, the more I think about it, the more issues there are with it.

It depends on whether you look at it small scale or large scale.

Small scale it appears to be 50% chance because the birth of one child doesn't affect the other.

However, if you place a limitation on it, and look at it as a large scale model, then the answer is 1/3

For a large scale model you have 100 families.
25 G-G
25 G-B
25 B-G
25 B-B

You know one child is a girl, what are the odds the other child is a girl would be 25/75 (since the B-B are eliminated).
However, this is placing limitations and additional information on the question.

Like was said, small scale and large scale are contradicting each other

1st red. You're missing the fact that there is twice as much chance of a GB combination (as BG also comes into this category)
2nd red. Surely Vice-versa?
3rd red. No. As two - 2 child families have, by definition, 4 children, the chances are that there are going to be 2 of each gender. If family A has (at least) 1 girl, the chances of the other child is a girl is one in 3. ie of the 2 boys and 2 girls, 1 girl is accounted for, leaving 1 girl and 2 boys.
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You must not think me necessarily foolish because I am facetious,  nor will I consider you necessarily wise because you are grave.  Sydney Smith.

### #334 benjer3

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Posted 11 July 2012 - 11:28 AM

Hmm, I think I've had a breakthrough. Let me put it in terms of the OP:

A parent comes up to you and says, "I have two children and one of them is a girl." They then proceed to tell you the gender of the other child. Being the analytic people that we are, we wonder what the odds that the second child was a girl were. While we ponder this, another parent happens to come up and says, "I have two children and one of them is a boy." A hundred parents then proceed to do the same, with the gender of the first child varying. (It also just so happens that, if each parent has at least one girl, they will say, "One of them is a girl.")

Now I ask, can we just discard the cases where the first child was a boy? I propose not, for in doing so we would be selectively taking members out of a sample group that needs to be complete, not to mention skewing the results by doing so selectively. So what are we to do? What if we changed our question, while keeping the same general idea intact: "Given the gender of one in two children, what are the odds that the other is a girl?" This question encapsulates the original question, as well as allowing for other scenarios. But now the answer is obvious, isn't it. In this question, it is obvious that the gender of the "other child" is completely independent of the gender of the first. And thus its answer can be nothing other than 50%.

So think what you will. I did change the question, but as I said, I believe the question is a perfect substitute for the original as it asks the exact same thing but allows for a complete sample group.
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### #335 fabpig

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Posted 11 July 2012 - 11:34 AM

Apart from 1 thing...........If one of the 2 children is a boy, the chance of the other being a girl is 2/3
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### #336 benjer3

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Posted 11 July 2012 - 12:08 PM

On the contrary. Look at this program (and let me know if it's wrong ):

```class Program
{
static void Main(string[] args)
{
Random random = new Random();
int numberOfFamilies = 1000;

int numberSecondGirl = 0;
int numberGirlGirl = 0;
int numberBoyGirl = 0;

for (int count = 0; count < numberOfFamilies; ++count)
{
Family family = new Family();

int childIndex = random.Next(2);

bool firstChild = family.Children[childIndex];
bool otherChild = family.Children[(childIndex == 0) ? 1 : 0];

if (otherChild)
{
++numberSecondGirl;
if (firstChild)
{
++numberGirlGirl;
}
else
{
++numberBoyGirl;
}
}
}

Console.WriteLine("Out of {0} families, {1} of the \"other children\" were girls. Out of those, {2} of the first children were girls ({3}%), while {4} of them were boys ({5}%)",
numberOfFamilies, numberSecondGirl, numberGirlGirl, ((double)numberGirlGirl / (double)numberSecondGirl) * 100D,
numberBoyGirl, ((double)numberBoyGirl / (double)numberSecondGirl) * 100D);

}
}

public class Family
{
private static readonly Random random = new Random();

public bool[] Children = new bool[2];

public Family()
{
for (int index = 0; index < Children.Length; ++index)
{
Children[index] = random.Next(2) == 1;
}
}
}```

For one run, this resulted in "Out of 1000 families, 515 of the 'other children' were girls. Out of those, 271 of the first children were girls (52.6%), while 244 of them were boys (47.4%)." The results are kinda backwards from what you stated, but I think you get the idea.
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### #337 fabpig

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Posted 11 July 2012 - 01:46 PM

so what were the other 485? If you're claiming that they're boys, then that makes 485*2 boys = 970 + (244 out of your 515 families) which leaves 786 children of whom 271 are girls.and the other 515 are ???.

Or are you taking it that the girl you've been introduced to is the eldest child?
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You must not think me necessarily foolish because I am facetious,  nor will I consider you necessarily wise because you are grave.  Sydney Smith.

### #338 benjer3

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Posted 12 July 2012 - 02:39 AM

so what were the other 485? If you're claiming that they're boys, then that makes 485*2 boys = 970 + (244 out of your 515 families) which leaves 786 children of whom 271 are girls.and the other 515 are ???.

Or are you taking it that the girl you've been introduced to is the eldest child?

In the other 485, all of the "other children" were boys, and the first children could be boys or girls. I could modify the program to show that the first children were equally distributed among boys and girls in this group as well. That leaves 515 "other" girls, 485 "other" boys, (271 + ~0.5 * 485) "first" girls, and (244 + ~0.5 * 485) "first" boys, giving a near perfect distribution. And if you look at the program, I randomly select from the first and second child of the family (though it wouldn't make a difference either way; I just did that to make it more realistic), so I'm not assuming anything about the children I've been introduced to.
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### #339 benjer3

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Posted 12 July 2012 - 02:51 AM

Spoiler for Code

This yields "Out of 1000 families, there were 239 girl-girls (23.9%), 247 girl-boys (24.7%), 280 boy-girls (28%), and 234 boy-boys (23.4%)." Near-perfect distribution.
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### #340 syonidv

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Posted 16 July 2012 - 02:38 PM

If you have 2 children, both girls, and 1 of them is a girl, then the 'other' child is also a girl. I really don't see your point.

I have 2 daughters. If someone says to me "Do you have at least 1 daughter?", I say "Yes". If they then ask what the gender of my 'other' child is, I don't say "what other child?"

Indeed, sir, but to the question "What is the gender and age of your other child?", the parents would answer, "What 'other' child?"

The problem is unique in that even if there is no specific 'other' child, we have the liberty of assuming 'female' since this is the only possibility. My point is that the problem statement specifically asks for the gender of the 'other child', indicating that the information we have pertains to one child, and not to two collectively. To wit, we are asked for the gender of the 'other child', not "What is the probability both children will be girls?"

I also hold fast to the 1/2-ist view and the hard semantics of 'other' (rather than Mr. Ben's reasonable "both can be correct" view) because applying conditionals to groups (e.g. "at least one of X or Y is") can lead to inconsistencies for even slight variations of the representative simulation.

Consider a dentist's office that serves 500 2-child families. Each child has a file. Suppose the dentist asks the receptionist to place a star on the file of any child whose family has at least one girl. Later, the dentist asks the receptionist to look through all the files with stars on them and for each one write down the gender of the patient's sibling.

In this case, the list of siblings would be roughly 2/3 female and 1/3 male.
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