## Genetics Paradox

Suppose N mothers live in a city. Half of them have one child and half of them have two children. That means that an average mother has 1.5 children.

Suppose we pick the sexual orientation of every child by rolling dice. Let’s assume that a child has a 10% probability of being homosexual.

The number of mothers with one child who is homosexual is 0.05N. The number of mothers with two children both of them homosexual is 0.005N. The number of mothers with two children with only the first child homosexual is 0.045N, which is the same as the number of mothers of two children with only the second child homosexual. The total number of mothers who have two children with at least one of them homosexual is 0.095N.

Let’s calculate the average fertility of a mother with at least one homosexual child. It is (1*0.05N + 2*0.095N)/(0.05N + 0.095N) = 0.24/0.145 = 1.66. The resulting number — 1.66 — is much bigger than 1.5, the average number of children for a mother.

This means there is a correlation between homosexuality and the fertility of mothers. This suggests that there is a gay gene which at the same time is responsible for female fertility.

But the model is completely random — there can’t be any correlation.

Where is the mistake?

Obviously, you can substitute homosexuality with having blue eyes or math ability or whatever, but I invented this paradox while I was working on my “Fraternal Birth Order Threatens Research into the Genetics of Homosexuality” post. Besides, there is some research on correlation between homosexuality and fertility.

I look forward to your solution to this puzzle.

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### 6 Comments

1. #### Shaun Maguire:

This metric of fertility is biased because a mother with 2 children is more likely to have at least one homosexual child. For a mother with two children, the probability that at least one of them is homosexual is 1 – .9*.9 = .19. For a mother with one child, the probability that at least one of them is homosexual is .1.

2. #### Misha:

It looks to me like the number of mothers with 2 children with only the first gay should be .045N. Here is my computation: the total number of mothers with 2 children is N/2 (because the average fertility is 1.5, so one half of the mothers have 1 child and the other half have 2), 10%*90% of them have the first child gay and the second child straight. 10%*90%*N/2=.045N.

3. #### Tanya Khovanova:

Thanks you, Misha, for finding an arithmetic mistake in my initial version. I changed it.

4. #### Misha:

Yeah, it’s like the bigger the number — the more likely it is to have at least one of the digits equal to 7. As an extreme case of this phenomenon, almost all the real numbers between 0 and 1 contain the UNICODE of all the world’s literature in their binary expansion.

5. #### Alex:

You are right in that one “can substitute homosexuality with having blue eyes or math ability or whatever.” In particular, one can choose the feature of not being homosexual, to the same effect.

For any possible feature and its probability (small or big), you may consider three random variables:

a) has the feature,
b) all have the feature,
c) at least one does not have the feature.

Then it is always the case that

Average b < Avreage a = 1.5 < Average c.

(Just simple algebra.)

And, I believe, Shaun gave a reasonable explanation for the phenomenon.

6. #### esmeyny:

You should add the calculation for the fertility of a mother with a t least one heterosexual child as explanation.