Ratio of Boys and Girls in a Country where people want only boys | Puzzles for Interview - Interview Preparation PDF Download

Introduction

In a country where all families want a boy, they keep having babies until they have a baby boy. This raises the question of what the expected ratio of boys and girls would be in this country. This article will explain how to calculate the expected ratio.

Assumptions

Before explaining the solution, we need to make some assumptions. We assume that the probability of having a boy or girl is the same, and the probability of the next child being a boy does not depend on history.

Solution

To solve the problem, we need to count the expected number of girls before a baby boy is born. Let NG be the expected number of girls before a boy is born, and let p be the probability that a child is a girl, and (1-p) be the probability that a child is a boy.
NG can be written as the sum of an infinite series:
NG = 0*(1-p) + 1*p*(1-p) + 2*p*p*(1-p) + 3*p*p*p*(1-p) + 4*p*p*p*p*(1-p) + ...
Putting p = 1/2 and (1-p) = 1/2 in the formula above, we get:
NG = 0*(1/2) + 1*(1/2)2 + 2*(1/2)3 + 3*(1/2)4 + 4*(1/2)5 + ...
We can simplify this equation by multiplying both sides by 1/2:
1/2*NG = 0*(1/2)2 + 1*(1/2)3 + 2*(1/2)4 + 3*(1/2)5 + 4*(1/2)6 + ...
Now we subtract the second equation from the first:
NG - NG/2 = 1*(1/2)2 + 1*(1/2)3 + 1*(1/2)4 + 1*(1/2)5 + 1*(1/2)6 + ...
Using the sum formula of an infinite geometrical progression with a ratio less than 1, we get:
NG/2 = (1/4)/(1-1/2) = 1/2
NG = 1

Expected Ratio

Since the expected number of girls is 1, and there is always a baby boy, the expected ratio of boys and girls is 50:50.

Conclusion

In a country where all families want a boy, the expected ratio of boys and girls is still 50:50. This is because the probability of having a boy or girl is the same, and the probability of the next child being a boy does not depend on history.