Ratio of boys and girls.

In a country in which people only want boys, every family continues to have children until they have a boy. if they have a girl, they have another child. if they have a boy, they stop. what is the proportion of boys to girls in the country?

Solution:

50:50.

probability of a boy/girl is always 2 irrespective of the number of conditions.

for the above problem we can prove it with simple probability theory.

Lets assume P(b) be the probability of having a boy. P(g) be the probability of having a girl.

Total expectation of a boy is equal to (probability of having a boy at first instance + probability of having a girl then a boy + probability of having 2 girls followd by a boy + etc... )
E(b) = 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ......
it comes down to 1 ( infinite series sum = a/1-r = 1/2/1-1/2 = 1).

E(g) = 0 + 1/2 + 1/4 + 1/8....
it comes down to 1.

It means there are equal number of girls and boys in the village.