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?

Update(24/01/10): Very simple problem. Don't know why it took me time to solve this. Thanx to Maoo for the solution. Posted by me in comments!!

1. The probability that a mother with 2 children will have a homosexual child is more than the probability of a mother with one child. Hence, the average fertility of women with homosexual children had to higher than average i.e 1.5

2. http://blog.tanyakhovanova.com/?p=42

dost problem jahan se uthaya, solution bhi wahin par thi :P

3. Problem liya that to laga tha simple hai.. baad mein karunga.. phir source bhool gaya.. and so on.. in any case problem tha to tatti hi.. ditch.. chodo is baat ko :P

4. Given mother has at least one homosexual child, Probability she has 2 children > Probability she has 1 child. While when the condition was not there both were equal.