Lubos Motl
Dear Steve, of course that the answer to this latest problem will be 43% or so, the program is attached below. But it's because what you're doing is borderline cheating. You want to create the impression that there are 100 years and potentially 400 births in those four families. However, the typical run will only produce 5-10 children. So in every single run, the number of children will fail to be large.

This latest problem has nothing to do with a population. It is a group of 5-10 people that goes extinct within 2 years or so.
Yesterday, 03:40:11 – Flag – Like – Reply

 Lubos Motl
Here is the program, I needed 5 minutes to write it and a few seconds to run it. Input:



sumOfFractions = 0; runs = 1000; years = 100;
For[run = 1, run <= runs, run++,

  stillproducing = {1, 1, 1, 1};
  girlsboys = {{0, 0}, {0, 0}, {0, 0}, {0, 0}};
  (* 0 is boy and 1 is girl *)

  For[year = 1, year <= years, year++,
   For[family = 1, family <= 4, family++,

     randomsex = RandomInteger[];

     girlsboys[[family, randomsex + 1]] =
      girlsboys[[family, randomsex + 1]] + stillproducing[[family]]
     ;
     stillproducing[[family]] = stillproducing[[family]]*randomsex;
     ];
   ]; (* Print[MatrixForm[girlsboys]]; *)
  fraction = Total[girlsboys[[All, 2]]]/Total[Flatten[girlsboys]];
  sumOfFractions = sumOfFractions + fraction;
  ];
overallGirlPercentage = sumOfFractions/runs
N[overallGirlPercentage]


Output:

12803786239/29099070000
0.44000672

This has nothing to do with the original problem because there's no "population" here.
Yesterday, 03:41:28 – Flag – Like – Reply

 Lubos Motl
If you correctly calculated the weighted average over the runs, with weights given by the total number of children in the run, you get 50%:



sumOfFractions = 0; runs = 1000; years = 100;
totalweights = 0;

For[run = 1, run <= runs, run++,

  stillproducing = {1, 1, 1, 1};
  girlsboys = {{0, 0}, {0, 0}, {0, 0}, {0, 0}};
  (* 0 is boy and 1 is girl *)

  For[year = 1, year <= years, year++,
   For[family = 1, family <= 4, family++,

     randomsex = RandomInteger[];

     girlsboys[[family, randomsex + 1]] =
      girlsboys[[family, randomsex + 1]] + stillproducing[[family]]
     ;
     stillproducing[[family]] = stillproducing[[family]]*randomsex;
     ];
   ]; (* Print[MatrixForm[girlsboys]]; *)
  weight = Total[Flatten[girlsboys]];
  fraction = Total[girlsboys[[All, 2]]]/weight*weight;
  sumOfFractions = sumOfFractions + fraction;
  totalweights = totalweights + Total[Flatten[girlsboys]];
  ];
overallGirlPercentage = sumOfFractions/totalweights
N[overallGirlPercentage]


Output:

519/1019 = 0.5093
With a higher number of runs, you get closer to 50%
Yesterday, 03:46:30 – Flag – Like – Reply