Probability problem

In the land of Misogynia, the government feels that the proportion of baby girls as opposed to baby boys is too high. The government is pretty authoritarian but draws the line at infanticide, so instead institutes a new policy: once a woman has given birth to a daughter, she is forbidden to have any more children. This law is rigidly adhered to by everyone, but no pregnant woman has an abortion because of the sex of the fetus.

What effect will the decree have on the proportion of baby girls?

Originally posted by Acolyte
What effect will the decree have on the proportion of baby girls?[/b]

None, respectfully submitted without proof for now.

Wow. He's right. That's pretty interesting.

There will be no effect on gender proportions. The gender of any one child is not in any way affected by the gender of the children the mother had in the past.

It's one of the fundamental laws of probability: 'Past incidences cannot have any effect on future outcomes.'

Originally posted by Mr Nice Guy
It's one of the fundamental laws of probability: 'Past incidences cannot have any effect on future outcomes.'

That is why this measure would have the exactly opposite effect. The proportion of baby girls would asymptotically go up to a certain percentage above 50% which can be worked out. I leave it for the smarter guys.

Once a mother has a daughter, no more chances of a boy occur.

I will calculate the number of children a mother can expect to have so that she has only 1 daughter. First for a normal society, then for the one in question:

With Pbbg i am going to mean 'Chance on 2 boys and the third child a girl' and so on.

The expectation is then:

1*Pg + 2*(Pbg+Pgb) + 3*(Pbbg+Pbgb+Pgbb) + ... =
1*1/2 + 2*(1/4 + 1/4) + 3*(1/8 + 1/8 + 1/8) + ... =
1/2 + 1 + 9/8 + 1 + 25/32 + 36/64 + 49/128 + 64/256 + ...

Now for the society in question several chances equal 0:

1*Pg + 2*(Pbg+Pgb) + 3*(Pbbg+Pbgb+Pgbb) + ... =
1*1/2 +2*(1/4 + 0) + 3*(1/8 + 0 + 0) + ... =
1/2 + 1/2 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + 8/256 + ...

Now I cannot conclude what i want, but i can say that the average size of the family would go down.

Another reasoning. Say you have a family of X children. Normally there would be X/2 boys in it on average. Now that would be X-1/2, but now there is a chance of getting that many children!

namely (1/2)^(X-1) (the last can be a boy or girl with equal likelyhood)

So if mother WANTS X children her chances are lessened by a factor of (1/2)^(X-1), but if she DOES manage there will be X-1/2 boys on average in stead of X/2.

Originally posted by ranjan sinha
That is why this measure would have the exactly opposite effect. The proportion of baby girls would asymptotically go up to a certain percentage above 50% which can be worked out. I leave it for the smarter guys.

Incorrect.

Let's assume for simplicity that all babies to be born are born in lockstep on the first day of the year. Let's assume we start with 400 couples and no children, and each viable couple is required to have a child, and let's assume no additional couples come to town.

Year 1: We expect 400/2 = 200 girls and 400/2 = 200 boys to be born.
This leaves us with an expected ratio of 200:200 and 200 remaining viable couples.

Year 2: We expect 200/2 = 100 new girls and 200/2 = 100 new boys to be born.
This leaves us with an expected ratio of 300:300 and 100 remaining viable couples.

Year 3: We expect 100/2 = 50 new girls and 100/2 = 50 new boys to be born.
This leaves us with an expected ratio of 350:350 and 50 remaining viable couples.

And so on throughout the years until no viable couples remain.

It should be clear that at each iteration, the expected ratio of boys to girls is always 1:1. It should also be clear that the assumptions are for simplicity only and could be relaxed without affecting the essence of the above analysis.

Originally posted by DoctorScribbles
None, respectfully submitted without proof for now.

You're right, under a certain homogeneity assumption. But in real life, the decree could have a very slight effect. Why?

Originally posted by Acolyte
You're right, under a certain homogeneity assumption. But in real life, the decree could have a very slight effect. Why?

Hmmm. If the answer lies in the facts of biology, then I fear I am too ignorant to find it.

I make the assumption that after any conception, each resulting child expects to be male with probability .5, and that nothing else influences this. Is this incorrect? Is there some fact like if a couple has had twin boys, there is something about them that would cause their future children to more likely be boys?

Given that my assumption is true, it is not conceivable to me that any practice or rule could influence the proportion of the sexes that are born. Note that I am interpreting the problem statement to mean that couples don't voluntarily abort based on the sex of the fetus, in addition to them not being compelled by law to abort. If couples could voluntarily abort, then a small effect would arise due to new couples that aborted female fetuses in order to be able to have a boy. And there would be no symmetric and opposing influence since new couples can always have a girl even if they have already had a boy.

Dr. S

Originally posted by DoctorScribbles
I make the assumption that after any conception, each resulting child expects to be male with probability .5, and that nothing else influences this. Is this incorrect? Is there some fact like if a couple has had twin boys, there is something about them that would cause their future children to more likely be boys?

That's not quite necessary - all you need to assume is that the probability of a child being male is independent of the child's mother. In fact, this is not quite true, and under certain circumstances a woman is signficantly more likely to give birth to girls than boys. For example, a woman can miscarry an embryo (usually before she knows she's pregnant) because her immune system rejects it, and this is more likely to happen to a male than a female fetus. So a woman with an oversensitive immune system is more likely to have daughters than sons, whereas for a woman without this problem boys are slightly more likely than girls.

The result of this is that Misogynia's decree, taken as an average effect on the population, restricts the fecundity of women who would tend to have a greater proportion of daughters slightly more (whether because of their or their sexual partner's biology) than it does average women. I don't know the exact numbers, but I would imagine this might become a detectable effect on the scale of millions of births - not a particularly effective way of reducing the proportion of girls by any means! The government could probably get better results by improving the nutrition of its populace, as male fetuses are a bit more vulnerable to the effects of poor maternal diet than female ones.

Originally posted by Acolyte
That's not quite necessary - all you need to assume is that the probability of a child being male is independent of the child's mother. In fact, this is not quite true, and under certain circumstances a woman is signficantly more likely to give birth to girls than boys. For example, a woman can miscarry an embryo (usually before she knows she's pregnant ...[text shortened]... as male fetuses are a bit more vulnerable to the effects of poor maternal diet than female ones.

These other factors are not part of the conditions specified in the problem. If we are to strictly follow the analysis of ThMasters37 , as indicated in his post above The average no of children of a couple i.e. in a family works out to just 2. And the no. of girl children in a family is equal to 1.


Where does that leave us?

Originally posted by cheskmate
These other factors are not part of the conditions specified in the problem. If we are to strictly follow the analysis of ThMasters37 , as indicated in his post above The average no of children of a couple i.e. in a family works out to just 2. And the no. of girl children in a family is equal to 1.


Where does that leave us?

I've heard this reasoning and it's amazingly simple:

The second society has about the same rules as the following. Let all families have as many children as they want, but kill all the people born after a girl in a family.

The ratio of the killed people is 1:1, you have as much chance to kill a boy as you have chance of killing a girl. This means the killing does not affect the ratio of boys and girls.

The endresult is the same as in the second society.

Originally posted by Acolyte
In the land of Misogynia, the government feels that the proportion of baby girls as opposed to baby boys is too high. The government is pretty authoritarian but draws the line at infanticide, so instead institutes a new policy: once a woman has given birth to a daughter, she is forbidden to have any more children. This law is rigidly adhered to by everyone ...[text shortened]... e of the sex of the fetus.

What effect will the decree have on the proportion of baby girls?

The answer is impossible to determine due to undefined variables.

For the countrys intent to come to fruition, there must exist a genetic variable that acts as a protagonsit for the acceptance of the XY chromosome during the formation of gender. Also for the countries intent to be realized, the populace with the genetic disposition to birth males must procreate multiple times. In this scenario, the end result will be conducive to the intent.

However, if you remove any of the above factors, I can see no other outcome other than the converse of the intended result (i.e. an increase in the percentile of the females in the populace).

Thats my analysis atleast.

Best Regards,
Omnislash

Originally posted by Omnislash
I can see no other outcome other than the converse of the intended result (i.e. an increase in the percentile of the females in the populace).

I'll admit that this was my first intution and hypothesis, but then my senses got the better of me. Think of the couples producing babies as nothing more than people flipping coins. Banishing people from the city for flipping Heads won't do a damn thing to cause the heads to tails ratio to deviate from 1:1.

Originally posted by DoctorScribbles
I'll admit that this was my first intution and hypothesis, but then my senses got the better of me. Think of the couples producing babies as nothing more than people flipping coins. Banishing people from the city for flipping Heads won't do a damn thing to cause the heads to tails ratio to deviate from 1:1.

With a completely systematic 50/50 ratio I agree that it would do nothing. Like I said, my concern is the genetic variants that could produce something other than a 50/50 ratio. The data within the question suggests that such a genetic disposition already exists which favors female births (i.e. a genetic variant which is not conducive to the acceptance of the male gender producing XY chromosome). If such a genetic trait exists within the populace, then absence of a counter balancing trait which more readily accepts the XY chromosome would indeed produce atleast a small increase in the fameal populaces birth ratio to males.

However, if we simply take the data given and do not consider any of these genetic possibilities, then you are quite correct that it will have no effect whatsover and the ration remains 1:1. What I intend to suggest is that if we consider the possible genetic and behavioral variables which are the root of this allegedly systemic unbalance and evaluate how the proposed action would effect these variables we can reasonably predict either a slight increase in the female populaces percentile or a slight increase in the male populaces percentile in relation to the populace birthed, dependant upon which genetic variables exist and what the behavioral patterns are. This is what I propose to be the major determining factors atleast.

I suppose we could go so far as to debate the effects of nutrition and other variants relevant to the problem, but this would be going even farther away from topic into unknown variables. Being that this is a question of probability, the data given limits us to conclude the 1:1 ratio, and hence that the actions would have no effect. I find this hard to swallow though, as this would also suggest such demographics as each couple having only one child. Obviously, this is a fallacious assumption for a sustained populace.

Anyhow, I suppose I've ranted on long enough about the question itself. 1:1 is the ratio if the question is taken at face value and you are correct.

Best Regards,
Omnislash