## Main Question or Discussion Point

Here is a very famous problem:

A random two-child family with at least one boy is chosen. What is the probability that it has a girl? An equivalent and perhaps clearer way of stating the problem is "Excluding the case of two girls, what is the probability that two random children are of different gender?"

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Here is a very famous problem:

A random two-child family with at least one boy is chosen. What is the probability that it has a girl? An equivalent and perhaps clearer way of stating the problem is "Excluding the case of two girls, what is the probability that two random children are of different gender?"
2/3

Options:
Older is a boy, younger is a boy - 1/4
Older is a girl, younger is a girl - 1/4
Older is a boy, younger is a girl - 1/4
Older is a girl, younger is a girl - 1/4

So, removing the 4th case from possibility, the odds of having two different gendered children, without knowing which is older is left as 2 of the 3 remaining possibilities. Hence, since each of the possibilities is equally probable, it's 2/3.

DaveE

CompuChip
Homework Helper
Though this might be counter-intuitive (which is why it is a famous problem) I don't see the paradox.

Though this might be counter-intuitive (which is why it is a famous problem) I don't see the paradox.
I think there is another more-or-less equivalent problem formulated in a more interesting way.

A king of a certain distant country has decided that he wants more men in the population for military purposes. He thus decides of a new law enforcing that a couple is allowed to have only one girl. What is the boy/girl ratio resulting in the population ? (assuming we wait for several generations, after the last person born before the law has deceased, to get a stable asymptotic result for instance).

A family could have say 7 boys, the eighth kid being a girl preventing any further child in the family.

A king of a certain distant country has decided that he wants more men in the population for military purposes. He thus decides of a new law enforcing that a couple is allowed to have only one girl. What is the boy/girl ratio resulting in the population ?
Interesting. I think you're looking for 50%.. But that's making some assumptions.

Technically, I think we need more information regarding how many kids the parents typically want. That is, parents may not WANT (or be able to have!) 6 children. So even though they might theoretically get 5 boys and then 1 girl, they may stop after (say) the 3rd son, and lower the ratio because they didn't keep procreating until they got to their natural limit.

DaveE

But that's making some assumptions.
Good points. So let us assume that they make as many children as necessary for them to reach their first girl. Technically, this is crazy because there is a probability that they have one billion boys and then one girl. But this is a small probability

Technically, I think we need more information regarding how many kids the parents typically want. That is, parents may not WANT (or be able to have!) 6 children. So even though they might theoretically get 5 boys and then 1 girl, they may stop after (say) the 3rd son, and lower the ratio because they didn't keep procreating until they got to their natural limit.
If parents voluntarily stop having children, it would have no effect on the ratio.
If parents who give birth on Tuesdays are not allowed to have more children, it would have no effect on the ratio.
If you pick parents at random and tell them that them must stop having children, it would have no effect on the ratio.
All of these things just reduce the number of people who are allowed to have more children. None of them has any effect on the ratio.

Borek
Mentor
Code:
#include <stdlib.h>
#include <stdio.h>

typedef int BOOL;

int main(int argc, char* argv[])
{
int iTotal = 0, iBoys = 0;

while (1)
{
BOOL bBoy;
do
{
bBoy = rand() % 2;
if (bBoy) iBoys++;
iTotal++;
} while (bBoy);
printf("%lf\n",double(iBoys)/iTotal);
}

return 0;
}
0.5 it is

Borek

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chemisttree
Homework Helper
Gold Member
Here is a very famous problem:

A random two-child family with at least one boy is chosen. What is the probability that it has a girl? An equivalent and perhaps clearer way of stating the problem is "Excluding the case of two girls, what is the probability that two random children are of different gender?"
That would depend on whether the parents have read Dr. Shettles book, "How to Choose the Sex of Your Baby." Success rates vary but it sounds like it is at least a testable theory.

http://www.fertilityfriend.com/Faqs/Gender-Selection-The-Shettles-Method.html

and here.

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CompuChip
Homework Helper
Now let me post my argument (I don't know if it's correct):
The chance of a family having exactly n boys when the chance for a boy or a girl is 1/2 every time (as in, binomially distributed): $$\left( p_\mathrm{boy} \right)^n \cdot p_\mathrm{girl}$$.
So expectation value of the number of boys in the family, assuming every family will have 0, 1, 2, ... boys until they eventually get a girl:
$$E = \sum_{n = 0}^\infty n \left( \frac{1}{2} \right)^{n + 1} = 1$$
So we expect every family to have 1 boy, before they get a girl. That means that 1/2 of all families, and therefore of the population, will be male.

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If parents voluntarily stop having children, it would have no effect on the ratio.
If parents who give birth on Tuesdays are not allowed to have more children, it would have no effect on the ratio.
If you pick parents at random and tell them that them must stop having children, it would have no effect on the ratio.
All of these things just reduce the number of people who are allowed to have more children. None of them has any effect on the ratio.
The easiness with which you answer problems sometimes suggests me that they must really appear trivial to you. In that case, I must say I am not fully satisfied with your answer, which seems to miss the point, although the final conclusion is correct.

There are several ways to answer more or less rigorously as always. A pedestrian way would be the following. The possible family configurations are obviously
(1) G
(2) BG
(3) BBG
(4) BBBG
...
(N) BBBBB...G (N boys)

It is not obviously trivial from this that the ratio 1:2 is conserved. It seems, and that is the initial goal of the king by issuing the law, that there will be more boys. It is of course very easy to count them and find out that it does not work.

The probability for line (N) to occur is (1/2)^N, so the probability weighted total number of individual in this table is
$$T=\sum_{n=1}^{\infty}\frac{n}{2^{n}}$$
while the weighted total number of boys is
$$B=\sum_{n=2}^{\infty}\frac{n-1}{2^{n}}=\frac{1}{2}\sum_{n=1}^{\infty}\frac{n}{2^{n}}$$
from which we find the desired result, that the ratio will indeed be 1:2.

Though this might be counter-intuitive (which is why it is a famous problem) I don't see the paradox.
Because the answer is cleary 1/2, but probability dictates 2/3. Which is it?

Look at it this way. In the first year, every couple has a child. Half the kids are boys, half are girls. Now don't let the parents of girls have any more children. Of those parents that cannot have more children half (0) have boys and half (0) have girls. Of those parents that are allowed to have children half have boys and half have girls. Keep on doing this. Every year, the same number of boys are being born as girls. The only thing that was accomplished is that fewer and fewer people are allowed to have children. How this increases the number of boys is a mystery known only to the king.

Because the answer is cleary 1/2, but probability dictates 2/3. Which is it?
I think this answer belongs in a different thread. In this problem, as in the other one, probability gives the correct answer. Math, it works *itches.

Defennder
Homework Helper
Look at it this way. In the first year, every couple has a child. Half the kids are boys, half are girls. Now don't let the parents of girls have any more children. Of those parents that cannot have more children half (0) have boys and half (0) have girls. Of those parents that are allowed to have children half have boys and half have girls. Keep on doing this. Every year, the same number of boys are being born as girls. The only thing that was accomplished is that fewer and fewer people are allowed to have children. How this increases the number of boys is a mystery known only to the king.
That's a great way to look at it.

Look at it this way. In the first year, every couple has a child. Half the kids are boys, half are girls. Now don't let the parents of girls have any more children. Of those parents that cannot have more children half (0) have boys and half (0) have girls. Of those parents that are allowed to have children half have boys and half have girls. Keep on doing this. Every year, the same number of boys are being born as girls. The only thing that was accomplished is that fewer and fewer people are allowed to have children. How this increases the number of boys is a mystery known only to the king.
This is not the same. Here you are preventing GG while allowing BB. If I called you and said of my two kids one is a girl, are you telling me that you'd need to flip 3 coins to decide what the other one was?

Math works. But are we applying it correctly?

This is not the same. Here you are preventing GG while allowing BB. If I called you and said of my two kids one is a girl, are you telling me that you'd need to flip 3 coins to decide what the other one was?

Math works. But are we applying it correctly?
Sorry, I was answering a different question. I agree, that among families with two children, not both girls, and one of them a boy, 2/3 have a girl. Indeed, the 'one of them a boy' part is meant to confuse you. If they are not both girls, then one of them is a boy. The question I was answering was whether you could get more boys by allowing only certain people to have babies.

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Defennder
Homework Helper
You know, I was thinking about the puzzle of the king's decree that no family can have more than 1 girl and found it quite puzzling that when you look at each family you'll notice that they either have the same number of boys and girls, or more boys than girls, but never more girls than boys. This strongly suggests that if we add up all the families together and if within each discrete family unit, there are never more girls than boys but may well have , shouldn't we expect the ratio to be tilted in favour of males? But of course this isn't the case, as we can see from the above.

Borek
Mentor
but never more girls than boys.
False - exactly 50% families have more girls then boys.

vanesch
Staff Emeritus
Gold Member
False - exactly 50% families have more girls then boys.
Yup, that's the trick! Half of the families started out with a girl, had to stop, and so only had a girl and no boy. Maybe their second would have been a boy, but they weren't allowed to have it.

The king should have allowed all families to procreate indefinitely, but order to kill all new-born girls that weren't the first in the family. Then the production rate would still be 50/50, but as only girls are killed, and not boys, this tilts over the balance.

The essential reason is that each birth is a statistically independent draw from all other outcomes, and each has a probability 50% to be a boy, and 50% to be a girl. So no matter how you organize these drawings (per family, when they can have more or not), their average will always be 50/50.

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Defennder
Homework Helper
Yeah I overlooked those with one female child. A mistake.

seeing as we are looking at families with atleast one boy... the probability is only dependant on the second child. 50%

D H
Staff Emeritus