Boys and girls in the village

[bonanova]

In one village boys are desired, to work the land. And so couples are told to be sure they have a boy and then stop having children.

In another village girls are desired, to increase the population. In that village, couples must bear a girl and then stop having children.

The villages are of equal size, and heterosexual monogamy is practiced.

By symmetry, there will a girl in each village for every boy in the other village.

But marriages are permitted only within one's own village.

What percentage of children in each village therefore can be expected not to find a mate?

[harey]

If you take n=8 families:

4 families have b => 4b

2 families have gb => 2g 2b

1 family has ggb => 2g 1b

1 family has gggb => 3g 1b (or gggg, but does not matter much)

Total: 7g 8b

For a large n, g -> b.

Limits: A woman cannot have an infinity of children and twins may not obey to the rule.

[bonanova]

If you take n=8 families:

4 families have b => 4b

2 families have gb => 2g 2b

1 family has ggb => 2g 1b

1 family has gggb => 3g 1b (or gggg, but does not matter much)

Total: 7g 8b

For a large n, g -> b.

Limits: A woman cannot have an infinity of children and twins may not obey to the rule.

I'll mark this solved.

Is there a more succinct solution?

[harey]

Is there a more succinct solution?

There is: just the formula giving the number of boys. But it is quite unreadable and to understand it, you have more or less to go thru what I have written.