How Many Girls for Every Boy?

skeptic2

A society has the rule that if a couple gives birth to a boy, they may not have any more children. If they have a girl they may continue having girls until a boy is born. The sex of the babies is purely random and overall the chances of having a boy or a girl are exactly equal. What is the percentage of girls in the population assuming both sexes live to the same age?

humanino

Half and half. Once I asked this here, and jimmysnyder found it completely obvious.

Jimmy Snyder

To put it in even simpler terms, the ratio of boys to girls is determined by the people who have children, not by the people who don't. The people who do will produce them in the ratio 50:50.

Borek

http://www.physicsforums.com/showthread.php?t=233255

markmcdo

I can see how all 1 and 2 child families are 50/50, but wouldn't 3+ child families have more girls? I know I'm missing something but don't know what.

davee123

Yes, all the 12-child families would have FAR more girls. But there aren't many of them. By contrast, there are a LOT of families with only 1 boy that were forced to stop early, which balances it out.

Look at it this way:

A given set of N families are trying to have 4 children.

A) 1/2 of them have 1 boy, and are forced to stop
B) 1/4 of them have 1 girl, then 1 boy, and are forced to stop
C) 1/8 of them have 2 girls, then 1 boy, and are forced to stop
D) 1/16 of them have 3 girls, then 1 boy, and stop because they have their desired 4 children
E) 1/16 of them have 4 girls, and stop because they have their desired 4 children

A) produces N/2 boys
B) produces N/4 boys and N/4 girls
C) produces N/8 boys and 2*N/8 girls
D) produces N/16 boys and 3*N/16 girls
E) produces 4*N/16 girls

Total boys:
= N/2 + N/4 + N/8 + N/16
= (8N + 4N + 2N + 1N)/16
= 15/16 N

Total girls:
= N/4 + 2*N/8 + 3*N/16 + 4*N/16
= (4N + 4N + 3N + 4N)/16
= 15/16 N

In fact, the same holds true of families trying to have ANY number of children-- the total number of boys actually produced is the same as the total number of girls actually produced.

The only way it would affect things is if certain families were actually more likely to produce girls than boys, or visa versa. That is, if it weren't always a 50% chance of having either gender.

DaveE