Recently, a friend shared an interesting probability problem, the solution of which led me to a horrific conclusion. The realization that struck me isn’t new, we all have read about it in newspapers & social studies text books. But somehow figuring it out like this when I wasn’t expecting to, helped me really see it. So here we go:
Problem statement: Imagine a hypothetical society where there’s a preference for male child (not that hypothetical). In this society, a family continues to have children until they have a boy child and as soon as a boy child is born, people stop having any more children. What will be the ratio of boys to girls in the population of such a society?
Assumptions: Probability of a single child being a boy or a girl is like a fair coin toss, 50-50. Only one child is born at a time.
Solution: So in this skewed society, a family can have the following number of children:
1 Boy, 1 Girl + 1 Boy, 2 Girls + 1 Boy, 3 Girls + 1 Boy, … and so on.
What we get is each family having exactly 1 boy and either 0, 1, 2, 3, … number of girls.
Ratio of boys to girls = Total number of boys in the population / Total number of girls in the population
Given X number of families to start with, we know that each will have exactly 1 boy child. So,
Total number of boys in the population = X
So all we need is the total number of girls in the population to find the ratio, which can be calculated as follows:
Total number of girls in the population = Number of families having 0 girls * 0 + Number of families having 1 girl * 1 + Number of families having 2 girls * 2 + Number of families having 3 girls * 3 + … and so on.
To calculate number of families with n number of girls, we need total number of families (which is X) multiplied by the probability of a family having n number of girls. When a family has n number of girls, they have n+1 children, last one being a boy. Probability of any child being a boy or a girl is 1/2, so this probability is 1/2 * 1/2 * … n+1 times.
So, total number of girls in the population = X/2 * 0 + X/4 * 1 + X/8 * 2 + X/16 * 3 + X/32 * 4 + … = X (1/4 + 2/8 + 3/16 + 4/32 + …) [Equation 1]
Let S = 1/4 + 2/8 + 3/16 + 4/32 + …
Then, S/2 = 1/8 + 2/16 + 3/32 + … [Divide both sides by 2]
S – S/2 = 1/4 + (2/8 – 1/8) + (3/16 – 2/16) + (4/32 – 3/32) + … [Subtract the two equations]
We get S/2 = 1/4 + 1/8 + 1/16 + 1/32 + …
S = 1/2 + 1/4 + 1/8 + 1/16 + … = 1 [Multiply both sides by 2 and we get a famous example of geometric series.]
Plugging this back into Equation 1 above, we get:
Total number of girls in the population = X
Aha! Since number of boys in the population is equal to the of number of girls, the ratio of boys is to girls is 1:1. This means that the population is still going to have 50% boys and 50% girls. In fact, since the probability of any one child being boy or a girl is 50%, no matter how many kids people choose to have for whatever reason, it can be proven that the population will always have roughly equal number of boys and girls. Neat!
So here’s the horrific conclusion: female infanticide and sex selective abortion of female fetuses is real. It’s so damn real. Anywhere there’s skewed gender ratio, you can bet that they are intentionally stopping girls from coming into existence.
It means that even with all their preference for a male child, the gender ratio will still be close to 50-50 if they just let the girls live. Just let them live.
P.S. The problem of female infanticide and sex selective abortion is often reported in the media under the headline “Missing Women”. The issue exists world over, but is more prevalent in China, India and Pakistan. Read this brilliant piece for it’s consequence here. Read more about the India specific problem here.
* I regret having to exclude non-binary genders from this post. The problems assumes binary gender and I understand it’s non inclusive.