Probability Question (fairly simple)

[pico]

suppose a country as adopted a new child quota system where each couple can only have 1 male kid.

if the first kid is female, you can have as many female kids as you want until you get your first male kid.

if the first kid is male, you can't have any more kids.

What is the male-female ratio for the next generation in that country?

[pico]

after you think about this question, start thinking about if-bets.

[Ganchrow]

suppose a country as adopted a new child quota system where each couple can only have 1 male kid.

if the first kid is female, you can have as many female kids as you want until you get your first male kid.

if the first kid is male, you can't have any more kids.

What is the male-female ratio for the next generation in that country?

Assuming the probability of having a boy were constant at 50%, then the answer would be 1:1.

There's no child-birthing strategy to be dreamt up (short of gender-determined abortion) that could ever alter that ratio.

[wannabet]

couples who have a male first would have a high % of having a male next. couples who have a female first would have a higher % of having another female. end up with more females.

[pico]

couples who have a male first would have a high % of having a male next. couples who have a female first would have a higher % of having another female. end up with more females.

mutation in male sperm yielding less y chromosome?

[wannabet]

I don't know if the biology is wrong. Was wondering more about the mathematical element when I read the question. yes, children evolve to produce less y chromosome.

[wannabet]

next generation, is the kids of the kids, or just the kids?

[pico]

you're thinking too much. think of it as one million couples's kids.

ganch already figure out the answer. i just thought this is sort of related to if-bets.

[wannabet]

so, just the kids. then, I don't know the answer, too complicated. Don't know what if-best are.

[eidolon]

how would it not be 2:1?
would that only be the maximum case, in a million years?

***
scratch that...since all families will have at least 1 male, it will make it an even ratio at 1:1.

[Ganchrow]

since all families will have at least 1 male, it will make it an even ratio at 1:1.

Not necessarily.

A couple could choose to stop having children before they have any males.

Nevertheless, the answer remains 1:1.

[reno cool]

couple that has boy first will have 1 more boy than girl.1-0
those that have girl first will avg 1 more girl than boy.

[Data]

Ganchrow has already given mathematically correct answer. The easiest way to understand this is not to think about born babies but rather realize that any given couple that is going to have a baby will have either a boy or a girl at 50/50.

Having said that, I cannot help but notice that the real probability of having a boy is not 0.5 but rather 0.517. See sex ratios for different age groups here:

https://www.cia.gov/library/publications/the-world-factbook/geos/xx.html#People

[Ganchrow]

Having said that, I cannot help but notice that the real probability of having a boy is not 0.5 but rather 0.517. See sex ratios for different age groups here:
https://www.cia.gov/library/publicat...xx.html#People

Insofar as this is the case the ratio of boys to girls would then be 0.517 (1-0.517) ≈ 1.07.

EDIT: Although clicking on the above link it looks like you had derived the results in reverse:

1.07/(1.00+1.07) ≈ 51.7%

Too clever, Data.

[curious]

Not necessarily.

A couple could choose to stop having children before they have any males.

Nevertheless, the answer remains 1:1.

I don't see the logic of this. Or the probabilities, or the statistics.

I don't know what the mechanism is that will keep a couple from having any more children after they have 1 male child. Let's say it's something benign. A shot that you get and you only have to get it once and it works 100% for the rest of your life. I'll leave out the complication that this couple may divorce....etc, etc. Let's say divorce is illegal. Hey if they can keep you from having children they can keep you from getting divorced.

So, given a large enough sample size to have meaning, say 10 million couples.

The odds of any of the couples having a male child first is 50/50. So, half the couples will have males, half will have females. Now you have:
5 million couples - 1 male
5 million couples - 1 female

Now the 5 million couples try for a male, again at 50/50 odds
2.5 million couples 1 female, 1 male
2.5 million couples 2 females

Now the 2.5 million couples that have 2 females try for a male
1.25 million couples 2 females, 1 male
1.25 million couples 3 females

This will keep on going, the female to male ratio in this country is going to be astronomical

I want to move to this country!

[Ganchrow]

I don't see the logic of this. Or the probabilities, or the statistics!

-snip-

the female to male ratio in this country is going to be astronomical

A simple way to see otherwise mathematically would be as follows:

let q = probability of having a girl
let n = maximum number of children a couple can physically have (where 1 ≤ n ≤ ∞)

By induction:

Prob of exactly 0 girls = (1-q) for all n
Prob of exactly 1 girl = (1-q)*q for n ≥ 2
Prob of exactly 2 girls = (1-q)*q² for n ≥ 3
Prob of exactly i girls = (1-q)*qⁱ for n ≥ i+1

Prob of exactly 1 girl = q for n = 1
Prob of exactly 2 girls = q² for n = 2
Prob of exactly i girls = qⁱ for n = i

And for boys:

Prob of exactly 0 boys = qⁿ
Prob of exactly 1 boy = 1 - qⁿ

Hence:^*

[nbtable][tr][td]E(# of girls) = (1-q)*[/td] [td] [/td] [td]( i*qⁱ ) + n*qⁿ [/td] [/tr] [/nbtable]
= (1-q) * ( q*(1-q^n-1)/(1-q)² - (n-1)*qⁿ/(1-q) ) + n*qⁿ
= (1-qⁿ) * q / (1-q)

And:

E(# of boys) = 0 * qⁿ + 1 * (1 - qⁿ)
= 1-qⁿ

So the ratio of boys to girls would be:

1-qⁿ / ( (1-qⁿ) * q / (1-q) )
= (1-q)/q

So for q = Pr(girl) = Pr(boy) = 50%, the male/female ratio would be (1-50%)/50% = 1:1.

QED

As per the second entry of "Finite sum" under Wikipedia's list of power series:

[nbtable][tr][td] [/td] [td] (i*xⁱ) = x * (1-xⁿ)/(1-x)² - n*xⁿ⁺¹/(1-x)[/td][/tr] [/nbtable]

[Ganchrow]

The odds of any of the couples having a male child first is 50/50. So, half the couples will have males, half will have females. Now you have:
5 million couples - 1 male
5 million couples - 1 female

Now the 5 million couples try for a male, again at 50/50 odds
2.5 million couples 1 female, 1 male
2.5 million couples 2 females

Now the 2.5 million couples that have 2 females try for a male
1.25 million couples 2 females, 1 male
1.25 million couples 3 females

Just to demonstrate the error with this particular mode of thought, let's just say, for the sake of argument, that couples will have no more than 3 children each.

As you've demonstrated, out of 10 million couples, 5 million will have exactly 1 boy, 2.5 million will have 1 boy and 1 girl, 1.25 million will have 1 boy and 2 girls, and 1.25 million will have 3 girls.

So that's 5,000,000 + 2,500,000 + 1,250,000 = 8,750,000 boys and 2,500,000 * 1 + 1,250,000 * 2 + 1,250,000 * 3 = 8,750,000 girls for a ratio of 1:1.

In general, if couples will have no more than N children each, then with 10,000,000 couples we'd expect to see 10,000,000 * (1-0.5^N) each of both boys and girls. (Were there no limit to the total number of possible children per couple, than this would converge to an expectation of both 10,000,000 boys and 10,000,000 girls -- so on average we'd expect each couple to have one child of each sex.)

[curious]

Just to demonstrate the error with this particular mode of thought, let's just say, for the sake of argument, that couples will have no more than 3 children each.

As you've demonstrated, out of 10 million couples, 5 million will have exactly 1 boy, 2.5 million will have 1 boy and 1 girl, 1.25 million will have 1 boy and 2 girls, and 1.25 million will have 3 girls.

So that's 5,000,000 + 2,500,000 + 1,250,000 = 8,750,000 boys and 2,500,000 * 1 + 1,250,000 * 2 + 1,250,000 * 3 = 8,750,000 girls for a ratio of 1:1.

In general, if couples will have no more than N children each, then with 10,000,000 couples we'd expect to see 10,000,000 * (1-0.5^N) each of both boys and girls. (Were there no limit to the total number of possible children per couple, than this would converge to an expectation of both 10,000,000 boys and 10,000,000 girls -- so on average we'd expect each couple to have one child of each sex.)

What if all couples will keep having babies until they have 1 boy, no matter how large N has become?

The end result would be 10,000,000 boys of course, but wouldn't the number of girls have to be larger than that because 5 million couples would have 1 boy and 5 million couples would have 1 boy and between 1 and N girls?

That was the assumption I was working under.

The total number of boys has to be 10 million. The total number of girls is found by:
5 million *1 +
2.5 million *2 +
1.25 million *3 +
.625 million *4 +
.3125 million *5 +
.15625 million * 6 +
.078125 million * 7 +
....

I'm not sure how large N will become here. The number of couples will continue to be divided by 2 until the number of couples = 1 or 1 and some fraction. Obviously the number of couples cannot be less than 1.

Now, I am assuming that the odds of having a boy are always 50/50 and is not dependent on the previous trials of having girl babies. This may be a false assumption. A couple conceiving a girl baby may affect the probability of having a male baby the next time they conceive. I'm not sure about that.

[reno cool]

damn curious, how'd you play professional blackjack if you can't figure this out?

5mil have 1 boy

5mil have 1 boy and avg 2 girls

[Peep]

That was fun Ganchrow, thanks. "Things are not always what the at first appear to be".

[Ganchrow]

The total number of girls is found by:
5 million *1 +
2.5 million *2 +
1.25 million *3 +
.625 million *4 +
.3125 million *5 +
.15625 million * 6 +
.078125 million * 7 +
....

Remember that half the couples (that's 5,000,000) will have 1 boy and 0 girls.

So the expected # of girls would be:

5 million * 0 +
2.5 million * 1 +
1.25 million * 2 +
.625 million * 3 +
.3125 million *4 +
.15625 million * 5 +
.078125 million * 6 +
...

As an infinite series:^*

[nbtable][tr][td] E(# of girls) = 5,000,000 * [/td][td] [/td] [td] (i*0.5ⁱ) = 5,000,000 * 0.5 / (1-0.5)² = 10,000,000 [/td][/tr] [/nbtable]

As per the second entry of "Infinite sum" under Wikipedia's list of power series:

[nbtable][tr][td] [/td] [td] (i*xⁱ) = x / (1-x)² [/td][/tr] [/nbtable]

[Wea]

One of the scientific theories for the birth of more boys than girls is the fact that the male sperms weigh less due to the smaller Y-chromosone (compaired to the X) and are therefore faster swimmers!