Cooking For Engineers

[Michael Chu]

A king decides that his country needs more women, so he decrees that families are to be allowed to continue to have children only if they give birth to girls. As soon as a boy is born, the family may not have anymore children. The king says, "This way we'll have families with 2 girls and a boy or 4 girls and a boy - we're sure to have a female heavy population now!" Assuming that the chances of having a girl or a boy are equal, does the king's plan succeed?

[capstinence]

If the odds are constantly equal, the change would be miniscule at best, and could possibly resort in more boys. According to my calculations:

First Births (surveying 100 families):
Girls - 50, Boys - 50

Second Births (surveying the 50 families that had girls):
Girls - 25, Boys - 25

[from here on out, I let any .5 decimals round up for the girls, but it could go either way]

Third Births (surveying the 25 families that had girls):
Girls - 13, Boys - 12

Fourth Births (surveying the 13 families that had girls):
Girls - 7, Boys - 6

Fifth Births (surveying the 7 families that had girls):
Girls - 4, Boys - 3

Sixth Births (surveying the 4 families that had girls):
Girls - 2, Boys - 2

Seventh Births (surveying the 2 families that had girls):
Girls - 1, Boys - 1

After this, I'm assuming that there's at the most one more girl born before a boy is, ending everyone's right to have children. The only reason the girls' side ends up with more is due to the fact that I rounded that way on the decimals. It could obviously be more boys, as well.

[Michael Chu]

yep. yep.

[Cheshire Cat]

... unless, of course, you have the sad situation that exists in some parts of China, where baby girls are 'disposed of' (either through selective abortion or worse) until a boy is produced. They currently have 1.35 boys for every girl...

Back to the problem.

This is just a sum of an infinite series. (Hidden below)

G = 0 x 0.5 + 1 x (0.5)^2 + 2 x (0.5)^3 + ....
= 1/2 x Sum[n=1->inf] ( n/(2^n) )

Steve

[Cheshire Cat]

... of course, I forgot to add...

(Expected number of boys per family)
B = 1

(Expected number of girls per family)
G = 1/2 x Sum[n=1->inf] ( n/(2^n) )

So, if Sum[n=1->inf](n/(2^n))>2 then there are more girls (and it isnt).


However, intuitively, since all births are independent of each other, you know that the King's plan will have no effect. It would be like picking '3' in the lottery because it 'hasn't come up much recently'. Each birth has a 50% chance of a girl no matter what the previous births have been.