## Aug 13, 2009

### Amazing Boy Girl Paradox

In a country, where 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. Who will be more in the country(boys or girls)? Intuitively, I and many of my friends thought it to be boys. Which seems fine as the society favours boys. But calculation shows that they would be equal. How? Suppose there are N couples. Note that each couple would have exactly one boy. So, no. of boys born is equal to N. Counting the no. of girls. N/2 parents would have no girl child. N/4 would have exactly 1 girl child, N/8 would have exactly two girl children, N/16 would have exactly 3 girl children and so on.... So, Expected no. of girls from N couples would be S: S = 1*N/4 + 2*N/8 + 3*N/16 + 4*N/32 + .......... 2S = N/2 + 2*N/4 + 3*N/8 + 4*N/16 + 5*N/32 + ............. So, S = N/2 + N/4 + N/8 + N/16 + ..... S = N So, Expected no. of boys = Expected no. of girls. Equal sex ratio. :)