SERVING THE QUANTITATIVE FINANCE COMMUNITY

• 1
• 2

vivekvision
Topic Author
Posts: 9
Joined: March 22nd, 2017, 9:45 am
Contact:

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

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

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

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

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

Alan
Posts: 10026
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

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

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.

Paul
Posts: 10418
Joined: July 20th, 2001, 3:28 pm

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

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.

Cuchulainn
Posts: 61518
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

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

And when you have arranged marriages!

http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

Paul
Posts: 10418
Joined: July 20th, 2001, 3:28 pm

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

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!

Cuchulainn
Posts: 61518
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

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

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.
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

Paul
Posts: 10418
Joined: July 20th, 2001, 3:28 pm

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

Sign or sing?

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

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

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

Paul
Posts: 10418
Joined: July 20th, 2001, 3:28 pm

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

Never really liked Paint Your Wagon, apart from Lee Marvin obvs.

Cuchulainn
Posts: 61518
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

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

Sign or sing?
Most of th signing was done in the Green Room.
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

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

Pity kids are not mote pro-active by acting and signing on stage.
many of the motes come from diesel autos

Cuchulainn
Posts: 61518
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

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

In the 60s Topol used to sing this great song.
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
Jean Piaget

vivekvision
Topic Author
Posts: 9
Joined: March 22nd, 2017, 9:45 am
Contact:

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

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

Paul
Posts: 10418
Joined: July 20th, 2001, 3:28 pm

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

See the first reply. (There are some assumptions in there but I suspect you don’t need a full list.)

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...

 JOBS BOARD

Looking for a quant job, risk, algo trading,...? Browse jobs here...

GZIP: On