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D&D 3E
Mathematician help 
2nd-May-2008 04:23 pm
I need some math help. specifically in statistics. I have the population of one particular continent of my world. The population is currently 2000. I want to calculate how many people I would have in 100 and 200 years given a lifespan of 50 years and an average birthrate of 2 children per family. Anyone good at this kind of thing?

edit: if any other information would be helpful please let me know. I would like the most realistic results possible.
2nd-May-2008 09:07 pm (UTC)
Er, roughly? 2000.

Your birth rate is static. Now, as there will be a slight decline, as fewer people make it to breeding age each generation, even if they still average 2 children per family. And some will never get married/have a spouse of the same gender, etc. But for a simplified model? the answer is 2000.

It also isn't a sustainable community without high levels of technology. There's a reason people have large families -- free labour -- in agrarian societies.
2nd-May-2008 09:08 pm (UTC)
First, an average birthrate of 2 children per family is considered below replacement rate so the population will decline. Even if everyone survived to adulthood, married, then had an average of 2 children, the population in 200 years would still be 2000.

But in general, population growth is like compound interest. This formula considers each family to continue to bear children forever.
BR = average children born per family
Gen = Time per generation, i.e. average age of a family to have children
Final Population / 2 = ((Initial Population) / 2) * ((1 + BR/2) ^ (Years / Gen))

But this doesn't account for the death rate. If we instead think of the population replacing itself every generation, then after each generation the couple is replaced by a certain number of child bearing pairs.

FP/2 = (IP/2) * ((BR/2) ^ (Years / Gen))

This would be an idealized approximation of the number of families over time. Making realistic calculations becomes incredibly hard (as we account for mortality rate, marriage rate, etc). If I had to run the numbers, I'd write a program to simulate this.
2nd-May-2008 10:17 pm (UTC)
Final Population / 2 = ((Initial Population) / 2) * ((1 + BR/2) ^ (Years / Gen))

Final Population / 2 = ((2000) / 2) * ((1 + 2/2) ^ (200 / 20))

Final Population / 2 = (1000) * ((1.5) ^ (10))

Final Population / 2 = 1000 * .15

Final Population / 2 = 1500

Final Population = 750?
2nd-May-2008 10:42 pm (UTC)
(1.5) ^ (10) = 1.5 * 1.5 * 1.5 * ... 1.5 = 57.6 ...

And FP / 2 = 1500 -> FP = 3000

But, that formula was faulty.

Use the other one.
2nd-May-2008 10:59 pm (UTC)
FP/2 = (IP/2) * ((BR/2) ^ (Years / Gen))

so given the above numbers...

fp/2 = (2000/2) * ((1.5 to the 10th)

fp/2 = 1000 * 58

fp/2 = 5800

fp = 11,600

That's not bad using a birth rate of 3, with a generation every 20 years.

2nd-May-2008 10:35 pm (UTC)
Is anyone going to take into account the suicide rate?! LOL

Who the fuck wants to go on living when you're only going to live up to 50?! LULWOT
2nd-May-2008 10:48 pm (UTC)
Oh wait... I'm on the d&de board... Fantasy games aren't supposed to make sense. 50 years old is normal since kids can have babies and old people can be grandparents younger than Real Life... :P
2nd-May-2008 11:03 pm (UTC)
Well, you have to take into account murder, death by wild animals and bandits, stillborns, warfare, plagues, illness and the like.

Take these 10 people...
person one is stillborn = 0
Person two dies of Pneumonia at 2
Person three is killed by a grizzly at 5
Person four is murdered at 20
Person five falls off of a cliff and dies at 30
Person six lives to 30 due to bad genes
Person seven lives to 60
Person eight lives to 61
Person nine lives to 70
Person ten lives to 72

That's an average of 35. If we don't count the stillborn, that's still 38.88~
If we add in someone who lived to 100, it brings it up to 40.9 or 45 if we don't count the stillborn.

So yes, a medieval world is perilous.
2nd-May-2008 11:20 pm (UTC)
Oh wait... I'm on the d&de board... Fantasy games aren't supposed to make sense. 50 years old is normal since kids can have babies and old people can be grandparents younger than Real Life... :P

I'm 28. If I were a woman, it'd be moderately likely that I'd be a grandmother by now, if I lived in ye olden times upon which fanatsy medivalism is based.

As I'm a man and not a woman, I'd still be on my first generation, with my eldest, assuming he survived, being about 10. In another eight years, I might become a grandfather. And eighteen years after that, well, odds are looking pretty good I'd be dead already.

There's no medical reason, and in some states no legal barrier, to becoming a parent as young as 14. Round it up to fifteen for the sake of of math, and by the time you reach your life expectant age of 75 or so, you'll be on your fifth generation of descendants. That is, you'll be a great-great-great grandfather.

If you think it doesn't make sense, you don't know much, do you?

(And "assume 2 kids per family" isn't that unrealistic, given the high mortality rate. If my wife and I have ten kids, half of whom die before they become adults, and another half of whom are eaten by trolls or something, we only added 2.5 adults to the population.)
3rd-May-2008 12:07 am (UTC)
Um... I was just saying it was "normal for a fantasy game". I'm fully aware of age expectancy in the dark ages.

As far as modern day life goes, age expectancy of 50 is not normal. It's why I said "I just realised this was the d&d3e board". Because that's literally what I meant. I commented on this from my Friends Page without realising it was the d&d3e board...

Also my statement of a fantasy game not making sense, still holds. Dragons and trolls and whatnot, you'd have to be insane to believe such things ever existed. Dragons are clearly superior beings, amongst many other fantastical creatures from d&de (mind flayers, etc)... If such creatures ever did exist, they'd probably have killed us all by now.
3rd-May-2008 12:34 am (UTC)
Also my statement of a fantasy game not making sense, still holds. Dragons and trolls and whatnot, you'd have to be insane to believe such things ever existed. Dragons are clearly superior beings, amongst many other fantastical creatures from d&de (mind flayers, etc)... If such creatures ever did exist, they'd probably have killed us all by now.

Nope, sorry. Wrong on all counts.

I'll start with the offensive one -- there's a huge difference between having an unusual, but unfalsifiable, belief, and being mentally ill. The former might think that aliens exist or there is a secret society controlling the world. The latter mispercieves those "maybes" as facts they have experienced. (I don't know if that's more offensive to the mentally ill, or the religious.)

Moving backwards, a fantasy game--or, indeed, any setting or human intellectual construction -- needs to make sense. If it's not internally consistent, it's impossible to advance within. I don't know a DM who doesn't want their setting to follow from logical conclusions after the supernatural facts are established. ("If dragons exist, they must eat...")

Lastly -- if you think dragons or mind flayers are less capable than human beings, you haven't been paying attention. Both creatures have impressive physical might, but they are individualistic and short-sighted, both of which don't tend well to defeating foes that are not.
3rd-May-2008 02:42 am (UTC)
most fantasy games are based on the medieval era or the dark ages when the average life expectancy in europe was something like 40. that's when they came up with the idea that you ought to respect your elders. anyone who made it past 50 had to be pretty on the ball.

but they kept the saying "respect your elders" going even until now, when the oldest among us are slower and dimmer than the youngest among us.
3rd-May-2008 12:20 am (UTC)

let's take some further assumptions.

In your population of 2000, roughly 2% (40) die each year. Since you don't state what kills them, we'll assume random chaos -- and give no special protection to the young.

Since you decreed that a family has an average of 2 children, we only care about the number of children that don't grow up to form a family. We'll assume that the actual birth rate is high enough that, for any given 5 of newlywed couples, 10 children will be born by the time all the newlyweds die.

If you pigenhole the date of motherhood at 15 -- a compromise between Juliet's age and the modern age of consent -- you'll lose 30% of the possible mothers (and father's) before they get married in the first place.

This will have serous consequences, as each generation will be 30% smaller than the previous one. Every fifty years, your popluation loses 30% of its population.

After 100 years, those 2,000 settlers will have 1,400 descendants. At 200, they'll be down to 980.

If that's not what you want, the average family size needs to get bigger, or you need to open up the immigration rate.
3rd-May-2008 03:01 am (UTC)
Two surviving offspring in a pre-industrial family is very low, I would think (I am not an Anthropologist, so whatta I know).

But more to the point. Shouldn't population statistics be secondary to your narrative? I mean the important thing is what kind of story are you trying to tell. Are these people dieing out from famine, plague and depredations from Orcs and Ogres, just looking for Heroes? Or are they a happy, peaceful prosperous people who like the status quo and look upon strange interlopers (your players) with suspicion?
3rd-May-2008 05:44 am (UTC)
I see where you are going, but I'm slowly building a world, game by game. In each game I increase time incrementally for "the boring stuff" to happen. Call it times of peace. I've decided where towns will be, and what groups will leave to settle them. I'm just trying to get an idea of how to describe those towns without the wittier of my players saying "there can't possibly be 4000 people in this town... there were only 2000 people in the whole world 200 years ago...
4th-May-2008 11:15 pm (UTC)
Eh, just tell them somebody cast "Plentiful Children" on the town's wombs or something. Yes, fantasy worlds do still have to make logical sense, but never forget that magic breaks everything.

Or maybe there was a religious revival (Yondalla?) that emphasized proper prenatal care so more children were born alive and more survived to adulthood?

Just make up a couple of witty comebacks like that and then if they get brought up incorporate them into the world retroactively. Or something. ;-)
4th-May-2008 12:37 pm (UTC)
This "Population simulator for SF writers" may or may not be helpful. I like the idea that you can alter the values of the variables year on year if you like, so for example they might have had a really bad harvest one year which saw the death rates sky rocket etc.

While there are loads of simplifications made to reach the end results, I think the key is to have something that's justifiable and believable enough to be able to silence your "wittier players" :)

Edited at 2008-05-04 12:39 pm (UTC)
22nd-May-2008 03:38 pm (UTC)
I would think first: what has happened in the last 200 years?
Have there been advances in technology? society? magic? medicine?
Has there been a plague? a war? a mass extinction? starvation? war? genocide?
Is there some new incentive or disincentive to having children?

Unless something happened that would either positively or negatively affect the population, I would keep the population the same. Then again, that isn't very dramatic.
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