Mostly Logical Assumptions:
- Aang and Katara will have children at 20
- All of their children and their children, etc. will have children at 20 because 20 is a nice round number
- Aang will die at 80 years old
- All Avatars will die at 80 years old
- Aang and his descendants will have three children each, assuming three is the average
- All children will live to have children and die at 80 years old
- Bending is inherited, but can also skip generations, therefore all of an airbender's descendants, whether or not they have airbending are able to produce children with airbending, ex: Katara's maternal grandmother was a waterbender (right?). Katara's mother was not a waterbender, but Katara is a waterbender. Therefore, bending can skip generations.
- Because it is very vexing to plot out every possibility of marriage, probabilities will remain constant so there will not be airbenders interbreeding to strength the odds or severe weakening of odds (It's called, I didn't know how to address this problem so I'm throwing it out)
- It is a 50/50 chance for a child to be a bender from two or one bender parents
- Because Aang and Katara are both benders, it is a 1/4 chance for their children to be airbenders
- Therefore, all of the children of Aang's descendants will also have a 1 out of 4 chance of being airbenders
- Despite being able to master all four elements, children of other Avatars will only inherit their "native" element, ex: Avatar Roku had children and there were no indications that they could bend other elements therefore Roku's children that could bend were only firebenders.
- Only whole numbers will be used even though this will cause the rounding errors to become more and more severe with each generation and the margin of error becomes wider
Aang: 20 years old
Aang and Katara have 3 kids, being optimisic, we have one child with no bending, one with water bending and one with air bending.
Running Tally (RT): Aang + 2nd generation = 2 airbenders
Aang: 40 years old
Their 3 kids will have 3 kids each, producing 9 grandchildren.
1:4 will be airbenders therefore 9/4 = roughly 2 airbenders
RT: Aang + 2nd +3rd = 4 airbenders
Aang: 60 years old
9 grandchildren have 27 great-grandchildren. 27/4 = almost 8
RT: 12 airbenders
Aang: 80 years old and dies.
Water Avatar is born.
27 great-grandchildren yield 81 great-great-grandchildren. 81/4 = 20 ish
RT: 12 airbenders + 20 - 1 (Aang died) = 31
Water Avatar: 20 years old
(81)(3) = 243 great3 grandchildren
243/4 = 61 rounding up... ish
RT: 31 +61 -1 (Aang's children die) = 91
Water Avatar: 40 years old
(243)(3) = 729 great4
729/4 = 182 ish
RT: 91 + 182 - 2 = 272 airbenders
Water Avatar: 60 years old
(729)(3) = 2187 great5 grandkids
2187/4 = almost 547
RT: 272 + 547 - 8 = 811 airbenders
Water Avatar is 80 years old and dies. The Earth Avatar is born.
(2187)(3) = 6561
6561/4 = 1640 kind of
RT: 811 + 1640 -20 = 2431
I know there is a way to make this streamed line, but my calculus is really rusty so please enlighten me.