Example of Conditional Probability

There are two children in a family.

[1] What is the chance of having at least one girl in this family?

Sample space { (B, B); (B, G); (G, B); (G, G)}

Obviously: the answer is 3/4.

[2] What is the chance of having two girls given that one of the children is girl?

You may argue that since one child is G, there is only one left to look at. The chance of that child's being a G is 1/2, hence the probability of having both girls is 1/2.

Is this correct? Let us look at our sample space.

Sample space {(B, G); (G, B); (G, G)}

Obviously: the answer is 1/3.

The mistake in earlier reasoning is this. We don't know which one is a G, the first child or the second.

Looking back at the second question: What is the chance of having two girls given that one of the children is girl? We have two events: A = both girls; B = one of them girl

P(AB) = P(A intersect B) / P(B) = (1/4)/(3/4) = 1/3
where P(A intersect B) = P(A) in this case.