Quick Quiz: Population Control and Sex Ratios

If we assume that 50% of births are male, and that no parents have, for some reason, a bias toward having babies of one or the other sex, then this is too easy. Let's add more rules: you can have a third baby, but only if there are at least five left-handed grandmothers living with 72 metres of your home. If your first baby is born on a Tuesday or at night, you can have a second child, but it must be named "Fizzbin" if a boy, "Fizzbina" if a girl.

In the end, we're asking "if 50% of all pregnancies result in a male baby, then what fraction of babies are male?" Still 50%. It doesn't matter how you hand out the pregnancy licenses.

The first round will be 0.5 M and 0.5 F, 1:1. The second round will be 0.25 M and 0.25 F. Looks like the sum stays 1:1. 50% of families with 1 M, 25% with 1 M 1F, 25% with 2 F.

If the correct answer is 50%, and it seems to me almost obvious that it is, what is counter-intuitive about it?

This (50% conclusion) makes some assumptions that I am not sure about:

Does the probability of a male or female child remain constant independent of population density? I seem to recall reading something about this years ago, but can't quickly find anything.

Are all couples equally likely to produce male and female children. In other words, are the genders of the children in two child families independent?

That's cute, you should still have a 1:1 sex ratio.

The one wrinkle I can think of is if some couples are biased to having males while others are biased to females. Then drawing random births you still get 50/50 M:F ratio, but imposing the rule changes things. It becomes a bit more Baysian, and many of the couples who have a girl first could be expected to have a second girl. Then the sex ratio of births will be biased to female.

It does seem pretty clearly fifty percent, for the reasons outlines above. Maybe it's not though- since you promised that a lot of smart people get this wrong, and this seems too obvious. I'm going to have to think it over- this is not my final answer!

The "final" equilibrium ratio is undefined because 0/0 is undefined. Each new generation is only 0.75 times the size of the previous one. But while you're getting there the M/F ratio remains close to 1/1.

Each pregnancy is a distinct event with no influence on the outcome of any other pregnancy. So, neglecting twins and higher N-tuples (as well as selective abortion), the sex distribution will be the same whether or not these rules are imposed.

Proof: Let P be the probability that the pregnancy results in a male child. Thus P is the fraction of couples who have one child who is male. The remaining couples then have a second child, giving P(1-P) as the fraction who have one boy and one girl, and (1-P)^2 as the fraction with two girls. We therefore have P + P(1-P) boys and 2*(1-P)^2 + P(1-P) girls. Expanding the expressions, the ratio of boys to girls is (2P-P^2)/(2-3P+P^2). Both numerator and denominator contain a factor 2-P, which we cancel to arrive at P/(1-P), which is exactly the ratio we would get if there were no restrictions.

Are we to assume that everyone who has a female baby in the first round will have a second child? The higher the percentage of parents choosing to stick with just one girl, the lower the percentage of girls in the overall population.

Does the question assume that all couples will have children until their permission runs out? We know that couples who choose zero offspring do not affect the answer. But what about couples who choose to stop after one girl? Is the problem completely specified?

Ok, the size of each new generation is less than 75% of the size of the previous generation. I think it's proper to re-state the rules: no woman will be permitted to have more than two children and the average number per woman will be less than 1.5. That, it seems, makes it certain that the population will vanish.

@ aaron:

Nope, its still 50/50. Say you have 100 families, the first children will (approximately) 50 boys and 50 girls. Of the 50 families that can have another child, say 40 do. This will result in 20 more boys, and 20 more girls, for a total of 70 boys and 70 girls, a 50/50 split.

Are we to assume that everyone who has a female baby in the first round will have a second child? The higher the percentage of parents choosing to stick with just one girl, the lower the percentage of girls in the overall population.

If the probability of any second pregnancy producing a boy is 50%, it doesn't make any difference. 50% of first pregnancies produce a boy and 50% of second pregnancies produce a boy.

Well, there is a fourth outcome possible - one girl. Of course, plugging that in leaves the ratio at 1:1, unless I screwed something up.

Showing my work:
First child is either male or female, with probability 0.5 for each.
A boy means no more children, so this outcome is 1M with probability 0.5.
A girl means the parents choose whether or not to have another child.
Choosing to stop at one occurs with probability x, so the outcome 1F occurs 0.5*x of the time.
Choosing to continue occurs with probability 1 - x, so the outcomes 1M + 1F and 2F each occur 0.5*0.5*(1-x) of the time.
So the expected outcome is 0.5*M+0.5*x*F+(0.5*(1-x))*(1.5*F+0.5*M) or (0.75-0.25*x)*(F+M) in a family; because F and M have the same coefficient, it follows that the expected ratio of M:F is 1:1.

I "cheated" and ran a simulation, where each woman has the maximum number of children. The end result is of course that they become extinct, while in the mean time, the sex ratio stays 50/50 (or corrects to this value in one generation if the starting ratio differs).

2/5 M 3/5 F

There's three possible outcomes M, MF or FF each occurring with an equal probability (full compliance). The next generation will simply be MMFFF which gives the fractions above.

Actually it's not true that they occur with equal probability. I can't remember what my justification was for thinking that.

Obviously, there are a number of "real world" things that can mess this up a bit:

1) The ratio of boys:girls isn't 50:50 in the real world. This, however, doesn't matter to the logic problem - the actual answer is "whatever the ratio of boys:girls was before the program began, assuming no human evolution."

2) This assumes no human evolution. :) Since families that have a son first get to have a second child, it seems reasonable to think that the easiest way to have a second child would be to always have boys. Thus, mutations that result in 100% boys would have a really strong tendency to spread through the population. What the final equilibrium is in that case is awfully hard to determine (especially if you allow further mutations of "have a boy first, then girls"). Also, what happens if you have sextuplets in your first pregnancy? Do you have to have an abortion, or kill the subsequent 5? These, too, could have significant selection pressures...

And, yeah, I know this is a physics blog, so we're not allowed to worry about that messy biology stuff...

If the probability of a girl is .62 and if we allow men to be members of more than one family each and if each female born has the maximum number of children in accordance with the rules, then we can have a stable population.

I'm convinced. I guess that makes me one of the smart people who got it wrong.

As usual my readers impress me! The ratio remains 50:50 in the absence of complications. Smart people who aren't used to thinking about independent events often tend to assume that because the system is built to make sure that more families have a boy than don't, the system will thus generate more boys. But each pregnancy is a coin flip, and you'll end up with a 50:50 split no matter what scheme you use to decide who flips what.

I didn't specify the percentage of families who would be satisfied with one girl, but as a few commenters noted it won't make a difference to the ratio.

Of course let the system run long enough and the the ratio will become undefined by virtue of the extinction of the population, since on average each set of parents will produce fewer than the 2 children needed to replace them.

Exctinction? Maybe not. To make the simplest assumptions, couples which have a female first child have twice as many children as couples who have a male child first. Huge difference in fitness, so there will be selection for couples who have a female child first. In the first generation, all the female children and 1/3 the male children originate from couples who have a female child first. So the population should evolve in the direction of having more and more first birth females. Maybe the population reaches 100% female first births and stabilizes. Yes, probably goes extinct.

There was an article in Nature, back in the 90's, about the slight skew to more male births. Turns out this is true of populatins where male children are more valued than female children. In the few populations where females are valued more highly than males, the skew is the other way.

OK so the answer is 1:1 or 50:50. I am not 100% sure yet that I get it. I will think about it but in the meantime:
What is wrong with the following logic?

It looks like there are three possibilities:
i.e if there are an infinite amount of families then each can only experience one the following states:

1. 1 male -> 0 females
2. 1 female-> 1 male
3. 1 female -> 1 female

There are now 2 males for 3 females.
Each family has equal chance of being either in state 1, 2 or 3. Then, if we have an infinite amount of these families, shouldn't the ratio remain 2 males to 3 females?

thanks.

Krist, half of families will have a male first and fall into your first state. The other half will have a female and fall into either the second or third state, with half of them in each. So states 2 and 3 each make up 25% of the population while state 1 makes up 50%. Adjust by multiplying the first state by 2, since twice as many families fall into it as opposed to either state 2 or 3. Now you have 3 males and 3 females.

The error that crops up in this problem is the most common probability error, that of assuming that all events have equal probability. It's the most common simplifying assumption in the basic study of probability, and we use it so much it's easy to forget that it is an assumption.

Trey- wonderful explanation! I get it now. I was thinking that each state has a 1/3 probability of occurring since there are three possibilities and each is just as likely to occur as the others. But actually state 1 is twice the other two! Makes sense now. I was thinking too globally about each family unit rather than each individual unit-the 50% chance of male and 50% female.

@Krist: what Trey described is correct. it is known as the density of states, a subject Matt has mentioned recently in the Bose-Einstein posts. (number 2 i believe)

I assumed that it was going to be a complex question, and so I drew a flow chart type diagram with the various possible outcomes of each outcomes as I read it, and when done, had to ask myself if it was a trick question because it was blindingly obvious that it was always 50/50 no matter what, but that could be because the diagram made forced me to stay within the declared parameters and not make any assumptions or intuitive leaps of logic.

Please forgive my composition typo of "outcomes of each outcomes " in place of "of outcomes of each possibility".

It seemed pretty obvious to me that the correct answer (assuming no complicating issues such as unequal gender probabilities) is that the population will become 50% male and 50% female. I neglected to notice that the population would become extinct. But thinking on it, it occurs to me that the reason I may have found the "correct" answer intuitive is because I am familiar with the "Monte Hall" problem. That wiki thing has an interesting discussion on the Monte Hall problem and other "paradoxes" similar to it.

I'm surprised nobody has mentioned it up to this point, but the policy Matt described is not theoretical. It's the actual population control policy in place in rural areas of China. (In urban areas the policy is even stricter: maximum one child even if it's a girl.) I assume they don't intend to leave it in place indefinitely, but for now a smaller population in China would be a good thing overall. One side effect will be that India, which has no similar policy, is expected to overtake China in population by 2050.

@hedberg: The general case isn't that hard; I solved it upthread. It doesn't matter what the expected gender ratio would be without the policy; you get the same gender ratio with the policy in place.

"Exctinction? Maybe not. To make the simplest assumptions, couples which have a female first child have twice as many children as couples who have a male child first. Huge difference in fitness, so there will be selection for couples who have a female child first. In the first generation, all the female children and 1/3 the male children originate from couples who have a female child first. So the population should evolve in the direction of having more and more first birth females. Maybe the population reaches 100% female first births and stabilizes. Yes, probably goes extinct."

Well you gotta consider that once there is a big majority of females in the population a male child can generate a lot more grandchildren than a female can. I suspect we would see some sort of equilibrium with a skewed sex ratio but not 100% females.

"There was an article in Nature, back in the 90's, about the slight skew to more male births. Turns out this is true of populatins where male children are more valued than female children. In the few populations where females are valued more highly than males, the skew is the other way."

Surely that is because of selective abortion or some similar feature and not some inborn feature of the human race?

Natural selection does not care about what social value the genders are ascribed. If one sex is less represented it will have more reproductive success on average causing the balance to swing back towards equality. That should suffice to make balanced gender ratios the most favorable under most circumstances. Fishers classical explanation for balanced sex ratios should work regardless of how patriarchal or matriarchal a culture is.

Johan: selection pressure can only draw from the phase space of extant entities. One cannot assume that there is some heritable trait present in the population that skews the birth rate. I suppose that an X-linked fatal recessive would do the job, a female with that trait would, for every four live births, produce 1 normal male (males cannot be carriers), two carrier females, and one normal female. They would also have reduced fertility (0.75 of normal), so this trait should be selected out over the long term in the absence of manipulations such as we are positing here.

Consider another case. In the United States there is a tradition of shooting at deer at some times of the year. This produces a strong selection pressure toward bulletproof, or at least bullet-resistant, deer. Still no clear success on that evolutionary front after many generations.

I am not sure against which of my statements you are arguing against but probably the first one. My point was mostly to dispute a previous argument about the effects of natural selection not to make any definitive predictions myself.

Obviously whether sex-skewing mutations would arise depends on the time-scales involved before the population went extinct.

Johan makes some interesting points. If there is no genetic mechanism regulating the sex of first-birthed children then differences in fitness are meaningless in this sitution. If we postulate such mechanisms, then we might predict an eventual stable population with 100% female first births, and 100% male second births. If there is no such genetic mechanism, the population will simply shrink by 25% each generation.

There is a slightly skewed natural propensity for boy babies, but boy babies have a higher mortality rate than girl babies, so it sort of balances out.

Marc: I think your analysis could work, but it depends on whether the babies are bosons or fermions.

Is the Proposed Trans Global Highway a solution for population concerns and global warming?
One tremendous solution to future population concerns as well as alleviating many of the effects of potential global warming is the proposal for the construction of the "Trans Global Highway". The proposed Trans Global Highway would create a world wide network of standardized roads, railroads, water pipe lines, oil and gas pipelines, electrical and communication cables. The result of this remarkable, far sighted project will be global unity through far better distribution of resources, including including heretofore difficult to obtain or unaccessible raw materials, fresh water, finished products and vastly lower global transportation costs.
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WORLD POPULATION DAY

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NOW ANYBODY CAN DARE TO CHALLENGE THE POVERTY.

Poverty is not at all any curse. It is totally the lack of one's awareness. Plenty is everybody's birth right. Any sort of help from any corner, never makes the poor plenty. Mere ever long dreaming for better future will never become a reality. It is only the right Knowledge of awareness along with total determination, to overcome one's poverty, will only make the poor, plenty. Here is an assured hope for the poor to make their lifelong dream, easily, early, a reality. Now anybody can dare to challenge the poverty.

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âKRISHNAMURTIâS FORMULAâ AWARENESS WAR ON POVERTY

Poverty and hunger are the two faces of the same coin. What is poverty? Poverty is the lack of income / resource to meet the minimum requirements like food, clothing and other common needs of the family members. Poverty is the prime reason for Hunger. Hunger is lack of minimum notorious food at least twice a day to the family members. Hunger is nothing but Hell on the earth. POVERTY is the abbreviation of P â Pity, O â Oppression, V â Violence, E â Extortion, R â Regret, T â Torture, and Y â Yell. HUNGER is abbreviation of H â Humility U â Unbearable, N â Nasty, G â Gruesome, E â Extortion, and R â Rage.

Hunger is the ancient enemy of the humanity. It is the most hidden self secret of the mankind as its revelation will bring them shame and humiliation. 50,000 people die every day due to poverty related causes. More than 854 million go to bed hungry every day. 1 billion canât read and write. 1.2 billion People have no access to safe drinking water. Almost half the world lives with less than $2 a day. 35% of worldâs poor live in India. 40% of the worldâs underweight children under the age of 5 live in India. As the food crisis is being compounded by ever multiplying population, extreme weather conditions, ecological imbalance and reduction in land for agriculture due to occupation of lands under various developmental activities. Every year4 million people join the worldâs shameful club of poverty. Hunger related disease has got numerous disadvantages such as lack of self esteem, fear, lack of competence, confidence, inferiority complex, laziness, lethargy, negligence, negligence of hygiene, ill health, poor body immunity, lack of creativity, missing opportunities, social status, illiteracy and respect.
There are numerous International and National NGOâs are working to eradicate the largest enemy of the world HUNGER. Even though their services are highly appreciable, they bring slow and steady reforms. However to certain extent their support will make the hungry people steadily dependable because of lack of self awareness rather than trying to be self-reliant. It is not possible for any government or organization to pump huge funds constantly to eradicate ever increasing poverty unless the poor population becomes self reliant ultimately. Only such people will swim out of the dark sea of self poverty forever. To fully support such challenging task of âPOVERTY FREE STRUGGLEâ here are Sri Missionâs amazing 20awareness tips to âROOT OUT HUNGERâ forever for generations.

K: KEY IS TO IMPROVE YOUR JOB SKILLS: By attaining job skills you will be competitive, confident, creative and efficient. You can do your job to the optimum satisfaction of your superior or customer with least wastage in shortest time. Your earnings and opportunities, by way of good customer satisfaction will improve apart from your job and self confidence that enhances your hope of achieving whatever you wanted.

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away from all sorts of bad companions, so as to keep away from all sorts of bad habits.

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T: TACKLE LAZINESS: Laziness leads to lethargy, procrastination, negligence, take it easy attitude, loss in earnings, loss of many opportunities, loss of faith in employer, colleagues, friends, relatives, neighbors, family members, erodes yourself confidence, cuts your own self esteem, weakens you financially, makes you unfit physically because of over rest, develops cracks in self confidence, job skills, makes you financially dependable, makes friends away as you often seek help from them, encourages to adopt bad habits, ultimately all the way to ruin. Keep away from laziness to achieve the prime path of prosperity.

I: IMPROVE SAVINGS: Your savings will improve your self confidence, courage to face any eventuality, secure future, hope of better tomorrow and enhance love & faith in your family members. Think of ways and means to save, purchase, economically, quality, quantity items at competitive rates. Savings proves to be super power.

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O: ONLY HOPE-THINK POSITIVE: Nobody ever achieved anything in the world without positive thoughts in them. Positive thinking has got tremendous invisible power which can turn impossible things t o become possible, unbelievable things believable, miracles to happen practically. It enhances your strength, self c confidence, enormously. Think positive and move confidently towards your goal.

R: REFUSE RISKY LOANS: Life is full of twists and calamities. Some unavoidable incidences like accidents, deaths, operations, court cases, disputes, settlements etc. may force you to go in for debt traps. You have to use your presence of mind, tact and ensure how best you can avoid such risky loans, which will really push you to the debt traps future.

M: MAKE MULTIPLE EARNINGS: You have to search various ways, means to improve your income. House wives can also take up some full time/part time job to enhance family income. Serving people can do overtime, part time job or work from home etc. Wisely invest your savings in safe schemes, mutual funds for risk free, safe, regular return of income.

U: UTILISE INSURANCE COVER: Even though nobody can stop the mishaps, there is insurance which will come to your rescue in case of calamities like death, theft, accident, crop damage etc. Something is always better than nothing. So try to cover your valuables, lives, cattle, crops, with insurance cover. Few banks offer even free insure ance cover for saving bank account holders. Various private insurance companies offer budgetary attractive schemes at competitive premiums. Beware to utilize various free travel and such other schemes to secure your lives and properties.

L: LOVE PEOPLE: The greater secret for success is investment on people. That is not spending money, but just love. It will get you unbelievable lifelong unlimited returns. Nobody knows who will be helpful in a miracle way. Sincerely caring and loving, everybody will prove definitely a wonder.

A: *ADOPT DISCIPLINE: Discipline is like back bone for life because self discipline enhances your determination, it gives tremendous self confidence, implements personal punctuality, builds leadership qualities, enhances confidence in your near and dear. Without self discipline any achievement becomes difficult. Discipline ensures easy success.

These 20 awareness tips will surely, easily make you rich sooner than you think. Every point mentioned above is important and will make your dream a reality faster. These points not only eradicate self poverty, even they make the rich ever richer. Wish you to walk soon the path of prosperity by easily becoming rich. Srimissionâs effort in eradicating poverty of each individual in t he world is a mission. Will you not be a part of this mission to spread invaluable message to entirely eradicate word âpovertyâ itself, from the world fast and to make it as a past? Come and join this gigantic darkness eradicating mission to lit bright light of smiles on each and every poorâs face and forever generations to come.

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ART OF AWARENESS LIVING.

Awareness is assured all-round advance asset. Awareness is infinite treasure. It is the real; unseperatable ever preventive and ever protective mobile wealth. Awareness is required in every step of life. To spread such precious awareness easily and effectively to the entire mankind, Srimission, Awareness foundation for the world peace, is spreading ART OF AWARENESS LIVING in very simple ways. One among such popular methods is PERIODICAL AWARENESS LECTURES (PALs).
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Periodical Awareness Lectures are very attractive. In fact, in other words the programme is known as âUnique Unlimited Universityâ. Because, from an university one gets a degree certificate in one subject, in which he is qualified, whereas at Periodical Awareness Lectures, every interesting individual gets reasonable enough knowledge in the subject conducted every time, lifelong. Lectures will be conducted on all common manâs needy subjects in routine life, strictly excluding religion and political subjects.
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Periodical Awareness lectures are delivered by well qualified, experienced professionals, on their respective subjects in whom they have full command. Any Doctors, Lawyers, Professors, Professionals, scholars, scientists, personality/management development trainers, councilors, not one, anybody who can add value to enrich lives with their knowledge treasure can present their lectures. In addition to that, persons who want to develop their public speaking skills will be given training and encouraged to develop their skills by presenting lectures on PALs platform. Anybody, who will think that he has subject to deliver a lecture on PALs platform, can register as a resource person.
WHERE PERIODICAL AWARENESS LECTURES WILL BE CONDUCTED?
Usually PALs will be conducted at Metropolitan Cities, District headquarters/heavily populated towns. PALs will be conducted at Higher secondary Schools, PU Colleges, Tutorial Colleges, Libraries, Associations, Clubs, such any other place where interesting listeners group is available, provided they have a classroom type accommodation for audience to sit is available. Such accommodation will be utilized periodically, maximum 2 hours, once a month, on the local weekly holiday. Interesting such accommodation owners can register with Srimission for the consideration of their venue to conduct PALs.
WHO WILL CONDUCT PALs?
PALs will be conducted by Local organizer (LO).Any person who has experience in public speaking ability, to coordinate resource persons as well as, attract audience and confident that he can raise all the credentials of PALs to further highs, can register as an LO. It is an added advantage, if the LO is well versed with English, apart from local language.
WHEN PERIODICAL AWARENESS LECTURES WILL BE CONDUCTED?
PALs will be conduct minimum quarterly once, initially. Then on the improvement of audience strength, the frequency of PALs will be increased steadily to bimonthly and later to monthly. Quarterly PALs are conducted in the months of January, April, July and October. In case of Bi monthly, PALs are conducted in the six even months of the English calendar.
WHAT ARE THE TIMINGS OF THE PERIODICAL AWARENESS LECTURES?
Generally PALs are conducted on the 4th Sunday (or local 4th weekly holiday) of any calendar month, from 5-00 to 7-00 p.m at the local venue. Even though day and time of conducting PALs is left to the local PALs team, it should be bear in mind that ultimately maximum audience to attend the PALs, to make them all-round success.
Who are all voluntarily interesting to offer their valuable services as Local Organizers, Resource Persons/Speakers, Venue Providers and support Art Of Awareness Livingâs PALs program me, they are all requested to register with Srimission with their Name, postal address, mobile as well as land line phone numbers, email id etc. to email to srimission@gmail.com, or by phone to ph nos:91 80 23641839/919900004200.
For further Art of Awareness living various activities; all-round, easy simple, single window awareness solutions visit website: www.srimission.org.