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Proportion of boys to girls, if continues to have children until a boy

Posted: September 3rd, 2018, 10:50 pm
by vivekvision
In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the proportion of boys to girls in the country?
Need to solve by forming Probability Density Function and then taking the expectation

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 4th, 2018, 7:12 am
by ppauper
one approach is to find the expected number of girls in a family
Not sure what you're supposed to do here. If you search the net, you can probably find conditional probabilities: if you already have 2 girls, what is the probability of having a 3rd, and so on
If you assume it's a 50-50 chance of boy versus girl at each stage, then
Probability of 0 girls and 1 boy is 1/2
Probability of 1 girl and 1 boy is 1/4
Probability of 2 girls and 1 boy is 1/8
and so on,
so Expected number of girls is [$]\sum_{n=0}^{\infty}\frac{n}{2^{n+1}}=1[$]
so you expect a 1-to-1 ratio

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 5th, 2018, 4:57 pm
by Alan
In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the proportion of boys to girls in the country?
Need to solve by forming Probability Density Function and then taking the expectation
The unstated premise seems to be that families can continue to try to have a boy indefinitely, and they will eventually have one before the "survey" for the statistics is done. While biologically impossible, we can accept the premise mathematically. 

The proportion of boys to girls presumably means the ratio of the number of boys to girls found in the survey, a random variable. 

But this ratio is either undefined or [$]+\infty[$] (depending on how you want to label the case) for families with 0 girls. By ppauper's reasoning, this case has probabilty 1/2. So, if we  follow your instructions and take the expectation of the pdf of this ratio, we either get "undefined" or [$]+\infty[$] for the answer. 

In "real life", dropping the premise, the expectation of this ratio will be "doubly" undefined, since both 1/0 and 0/0 ratios will occur with strictly positive probability.

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 5th, 2018, 6:06 pm
by Paul
There are lots of unstated assumptions. Eg death rates not mentioned. There are several countries where boys are preferred and so this “model” might be a good starting point. But some of these countries tend to give the boys an AK47 as soon as they can walk and send them off to kill each other. To be fair though they do often strap bombs to the girls as well.

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 6th, 2018, 9:53 am
by Cuchulainn
And when you have arranged marriages! 



 

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 6th, 2018, 10:34 am
by Paul
When I was young instead of birthday parties for their friends the parents of the birthday boy or girl would take them all to the movies. If it was a boy then they’d all see a cowboy movie together or something with a space theme. On my birthday once my mother took all my friends to see 7BF7B. Well, I enjoyed it. And bless your beautiful hide, Cuch, for reminding me!

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 6th, 2018, 12:35 pm
by Cuchulainn
You're welcome, Paul

One of my favourites was Oklahoma which I used to sign to large audiences in the 60s. I still remember all the words (in contrast to the Lord's Prayer).

Pity kids are not mote pro-active by acting and signing on stage.

76 Trombones is also cool. Those drums bring tears to my eyes.

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 6th, 2018, 12:55 pm
by Paul
Sign or sing?

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 6th, 2018, 1:13 pm
by ppauper
When I was young instead of birthday parties for their friends the parents of the birthday boy or girl would take them all to the movies. If it was a boy then they’d all see a cowboy movie together or something with a space theme. On my birthday once my mother took all my friends to see 7BF7B. Well, I enjoyed it. And bless your beautiful hide, Cuch, for reminding me!
paint your wagon was another one

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 6th, 2018, 1:21 pm
by Paul
Never really liked Paint Your Wagon, apart from Lee Marvin obvs.

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 6th, 2018, 1:32 pm
by Cuchulainn
Sign or sing?
Most of th signing was done in the Green Room.

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 6th, 2018, 1:38 pm
by ppauper
Pity kids are not mote pro-active by acting and signing on stage.
many of the motes come from diesel autos

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 6th, 2018, 3:55 pm
by Cuchulainn
In the 60s Topol used to sing this great song. 

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 7th, 2018, 1:15 pm
by vivekvision
Still looking for a clue on how to form  Probability Density Function for such problems 

S0 that integration can perform on it calculate the expectation

Re: Proportion of boys to girls, if continues to have children until a boy

Posted: September 7th, 2018, 2:59 pm
by Paul
See the first reply. (There are some assumptions in there but I suspect you don’t need a full list.)