Calculating Expected Frequency of Girls per Family in a Biased Society

[mathlete]

Here's the question

"In a faraway land long ago, boys were valued more than girls. So couples kept having babies until they had a boy. The frequency of boys and girls in the populations as a whole remained equal, but what was the expected frequency of girls per family? (Assume that each sex is equally likely)"

I have a hint (see attachment)

I kind of see what they want, but I don't really know how to get it. I tried to find the expected value for a given case (let's say they have n babies), but it doesn't really come out to what they have. I don't really know how the fact that the frequency of boys and girls remains equal fits into the problem, either.

Any help is appreciated.

 

[StatusX]

There's a couple ways to do this. First, you can just find the expected value of the number of girls in a family by <g>=0*p(0)+1*p(1)+..., where p(n) is the probability of a family having n girls. For example, p(0)=1/2 since there is a 50% chance the first child will be a boy and the family will stop after that.

The other way is to note that the expected value of the number of boys per family is <b>=1, since a family won't stop until they have exactly one boy. Then the total number of boys is B=F*<b>=F, where F is the number of families. You also know B=G, ie, the total number of boys equals the total number of girls, (they tell you this, but do you understand why it is?), so what does this give you for <g>?

 

[mathlete]

There's a couple ways to do this. First, you can just find the expected value of the number of girls in a family by <g>=0*p(0)+1*p(1)+..., where p(n) is the probability of a family having n girls. For example, p(0)=1/2 since there is a 50% chance the first child will be a boy and the family will stop after that.

This makes sense... but I don't get the same infinite sum that's given (I get n*x^(n+1) instead of x^(n+1)/(n+1).

The other way is to note that the expected value of the number of boys per family is <b>=1, since a family won't stop until they have exactly one boy. Then the total number of boys is B=F*<b>=F, where F is the number of families. You also know B=G, ie, the total number of boys equals the total number of girls, (they tell you this, but do you understand why it is?), so what does this give you for <g>?

This makes sense too, but wouldn't that just give you 1 for <g>? And sorry, I don't understand why total number of boys has to equal total number of girls

Thanks for taking the time to help, I really appreciate it

 

[0rthodontist]

And sorry, I don't understand why total number of boys has to equal total number of girls

What is the probability that the first child a couple will have is a boy? Assuming the couple makes it to the second child, what is the probability the second child a couple will have is a boy? Assuming the couple makes it to the third child, what is the probability the third child a couple will have is a boy?

 

[StatusX]

This makes sense... but I don't get the same infinite sum that's given (I get n*x^(n+1) instead of x^(n+1)/(n+1).

Yea, I get the same thing as you. I'm not sure what they were going for with that hint, maybe there's another way to do it. But you can evaluate the sum using the same basic method they did, only differentiating where they integrated.

This makes sense too, but wouldn't that just give you 1 for <g>? And sorry, I don't understand why total number of boys has to equal total number of girls

The reason is that the next child who's born (anywhere) is either a boy or a girl with equal probability.

 

[mathlete]

What is the probability that the first child a couple will have is a boy? Assuming the couple makes it to the second child, what is the probability the second child a couple will have is a boy? Assuming the couple makes it to the third child, what is the probability the third child a couple will have is a boy?

1/2 for all of them, correct?

 

[mathlete]

OK, well doing it both ways StatusX described I get 1 as an expected value. This seems to make sense to me, I don't really understand that hint but i'll just ignore it. Thanks for the help everyone!