Misogynist country

Question:

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?

.

.

C

A

P

T

A

I

N

I

N

T

E

R

V

I

E

W

.

.

Solution:

Following is the required calculation:

Expected Number of boys for 1 family = `1*(Probability of 1 boy) + 1*(Probability of 1 girl and a boy) + 1*(Probability of 2 girls and a boy) + …`

For C couples = `1*(C*1/2) + 1*(C*1/2*1/2) + 1*(C*1/2*1/2*1/2) + …`

Expected Number of boys = `C/2 + C/4 + C/8 + C/16 + …`

Expected Number of boys = `C`

Expected Number of girls for 1 family = `0*(Probability of 0 girls) + 1*(Probability of 1 girl and a boy) + 2*(Probability of 2 girls and a boy) + …`

For C couples  = `0*(C*1/2) + 1*(C*1/2*1/2) + 2*(C*1/2*1/2*1/2) + …`

Expected Number of girls = `0 + C/4 + 2*C/8 + 3*C/16 + …`

Expected Number of girls = `C`

Therefore, the proportion is C/C = `1:1`